Research ArticleShannon Wavelet Precision Integration Method forPathologic Onion Image Segmentation Based on HomotopyPerturbation Technology

Haihua Wang12 and Shu-Li Mei12

1 China Agricultural University East Campus Postbox 53 17 Qinghua Donglu Road Haidian District Beijing 100083 China2 College of Information and Electrical Engineering China Agricultural University Beijing 100083 China

Correspondence should be addressed to Shu-Li Mei meishuli163com

Received 12 October 2013 Accepted 18 January 2014 Published 19 March 2014

Academic Editor Metin O Kaya

Copyright copy 2014 H Wang and S-L Mei This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

Image segmentation variational method is good at processing the images with blurry and complicated contours which is useful inquality identification of pathologic picture of onion An adaptive Shannon wavelet precise integration method (WPIM) on digitalimage segmentation was proposed based on the image processing variational model to improve the processing speed and eliminatethe artifacts of the images First taking full advantage of the interpolation property of the Shannon wavelet function a multiscaleShannon wavelet interpolation scheme was constructed based on the homotopy perturbation method (HPM) The image pixelsof the Burkholderia cepacia (ex-Burkholder) infected onions were taken as the collocation points of the WPIM Then with thisscheme the image segmentation model (C-V model) can be discretized into a system of nonlinear ODEs and solved by the half-analytical scheme combining the HPM and the precision integration method At last the numerical precision and efficiency ofWPIM were discussed and compared with other common segmentation methods such as OSTU method and Sobel operator Theresults show that the contour curve of the segmentation object obtained by the new method has many excellent properties such asclosed and clear topological structure and the artifacts can be eliminated

1 Introduction

Rot is the primary factor of the onion storage losses Thedamages from the pests soak and over-nitrogen often lead tothe onion infected with pathogen Cross-infection of onionsaccelerates rotting in the package Therefore grading andclassification of the onions are necessary in the postharvestprocessing to decrease the losses of onions

The manual identification is time-consuming and unreli-able Image measurement technology [1] is the new methodfor grading the quality of agricultural products Image seg-mentation is an important part of the image measurementAccurate segmentation techniques can be used for rottingdetection and grading of onions The classical segementationmethods such as Sobel Canny quadtree and OTSU algo-rithm take the gradient of the image as the feature descriptordirectly in image segmentation But the object boundary and

target pixels obtained by these methods are often unclosedwhich ismakes it difficult to analyze geometric characteristicsof the target for connection and econometric analysis ofsegmentation results In the recent years the wavelet preciseintegration method [2ndash10] has been developed to solve thenonlinear PDEs for image processing which can improve theefficiency and precision of the image processing effectivelyWPIM is an image segmentation variational method basedon the C-V model with which the segmentation resultswith closed object contour can be obtained In additionthe continuity of wavelet function is helpful to eliminatethe artifact introduced by the difference method Thereis no doubt that these merits are necessary in the imagemeasurement and product quality grading

In the recent decades many wavelets which have com-pact support smoothness and other properties have beenconstructed Cattani studied the properties of the Shannon

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 601841 10 pageshttpdxdoiorg1011552014601841

2 Mathematical Problems in Engineering

wavelet function which possesses many advantages such asorthogonality continuity and differentiability [11] It alsohas the advantage over the Hermite DAF in that it isan interpolating function producing matrix equations thathave the potential to be relatively sparse In addition thesecond order approximation of a 119862

2-function based onShannon wavelet functions is given [12] The approximationis compared with the wavelet reconstruction formula andthe error of approximation is explicitly computed [13 14]Furthermore Shannon wavelet has been used to solve thefractional calculus problems in the recent years [15ndash17]A perceived disadvantage of the Shannon scaling functionis that it tends to zero quite slowly as |119909| rarr infin Adirect consequence of this is that when calculating thederivatives a large number of the nodal values will contributesignificantly However comparing with other wavelets theShannon scaling function is one that possessesmore excellentnumerical properties such as the interpolation orthogonalityand smoothness Therefore Shannon wavelet is employed toconstruct the multiscale interpolator in our research

The purpose of this research is to construct a multiscaletwo-dimension wavelet interpolation operator based on thehomotopy perturbation method [18ndash20] with which theimage segmentation PDE (C-V model) can be discretizedinto a system of nonlinear ODEs Combining the preciseintegrationmethodwith homotopy perturbationmethod [2122] for solving nonlinear problems a fast wavelet numericalalgorithm for P-M and C-V model in image processing canbe obtained With this new image segmentation method thepathologic onion classification and grading can be processedprecisely and efficiency

2 Chan-Vese Model

In order to solve Mumford-Shah model with Euler-Lagrangemethod a simplified model was deduced by Chan and Vesein which Euclid length was employed instead of Hausdorfflength This simplified model can also be called Chan-Vesemodel which can be expressed as follows

119864CV

(1198881 1198882 119862) = 120582

1intΩ1

(1198680minus 1198881)2

119889119909 119889119910

+ 1205822intΩ2

(1198680minus 1198882)2

119889119909 119889119910 + ] |119862|

(1)

119888119894= mean

Ω119894

(1199060) =

intΩ119894

1199060(119909 119910) 119889119909 119889119910

Area (Ω119894)

119894 = 1 2 (2)

where 1205821and 120582

2are positive constants and 119888

1and 1198882are the

average gray level values inside (Ω1) and outside (Ω

2) of the

object contour respectively 1198680denotes the image to process

|119862| is the length of the object contour and ] is the weightparameter According to the level set method the contour

curve of the objects should be embedded into the level setfunction as follows

119862 = (119909 119910) | (119909 119910) isin Ω 120601 (119909 119910) = 0

Ω1= (119909 119910) | (119909 119910) isin Ω 120601 (119909 119910) gt 0

Ω2= (119909 119910) | (119909 119910) isin Ω 120601 (119909 119910) lt 0

(3)

Then the level set-based C-V model can be rewritten asfollows

119864 (1198881 1198882 120601) = 120582

1intΩ

10038161003816100381610038161198680 minus 11988811003816100381610038161003816

2

119867(120601) 119889119909 119889119910

+ 1205822intΩ

10038161003816100381610038161198680 minus 11988821003816100381610038161003816

2

(1 minus 119867 (120601)) 119889119909 119889119910

+ ]intΩ

1003816100381610038161003816119867 (120601)1003816100381610038161003816 119889119909 119889119910

119867 (120601) = 1 120601 ge 0

0 120601 lt 0

120575120576=

120576

120587 (1205762 + 1206012)

(4)

Using the variational method the PDEs with respect to thevariable 120601 can be obtained as follows

120597120601

120597119905= 120575120576(120601) [] div(

nabla120601

1003816100381610038161003816nabla1206011003816100381610038161003816

) minus 1205821

10038161003816100381610038161198680 minus 11988811003816100381610038161003816

2

+ 1205822

10038161003816100381610038161198680 minus 11988821003816100381610038161003816

2

]

(5)

Obviously div(nabla120601|nabla120601|) is the curvature of the level setfunction 120601 and 120575

120576(120601) is used to constrain the growth of the

level set functionThe solution of (5) is the level set function 120601(119909 119910 119905) at

time 119905 The zero level set is the object contour curve whichcan be obtained by solving the equation 120601(119909 119910 119905) = 0We focus on multiscale wavelet image segmentation discreteformat and the corresponding numerical solution method

3 HPM-Based Wavelet Interpolation OperatorConstruction Schemes

Let the definition domain of the image be (119909min 119909max) times(119910min 119910max) the discretization points can be defined as(119909119895

1198961

119910119895

1198962

) where 119895 is a scale parameter and 1198961and 119896

2are

position parameters So

119909119895

1198961

= 119909min + 1198961119909max minus 119909min

2119895

119910119895

1198961

= 119910min + 1198962119910max minus 119910min

2119895

119895 1198961 1198962isin Z

(6)

In addition 119908119895(119898119899)11989611198962

(119909 119910) denotes the multiscale waveletfunction and the corresponding 119898th and 119899th derivativeswith respect to 119909 and 119910 respectively The level set function

Mathematical Problems in Engineering 3

120601(119909 119910 119905) and the corresponding derivative function can bediscretized as follows

120601119869(119898119899)

(119909 119910 119905)

=

1

sum

11989601=0

1

sum

11989602=0

120601 (1199090

11989601

1199100

11989602

)1199080(119898119899)

1198960111989602

(119909 119910)

+

119869minus1

sum

119895=0

2119895minus1

sum

11989611=0

2119895minus1

sum

11989612=0

[1205721

1198951198961111989612

(119905) 119908119895+1(119898119899)

211989611+12119896

12

(119909 119910)

+ 1205722

1198951198961111989612

(119905) 119908119895+1(119898119899)

211989611211989612+1(119909 119910)

+ 1205723

1198951198961111989612

(119905) 119908119895+1(119898119899)

211989611+12119896

12+1(119909 119910)]

(7)

where 119895 and 119869 are constants which denote the wavelet scalenumber and the maximum of the scale number respectively1205721

1198951198961111989612

12057221198951198961111989612

and 12057231198951198961111989612

are the wavelet coefficients atthe points (119909119895

1198961

119910119895

1198962

) According to the interpolation wavelettransform theory the wavelet coefficients can be written as

1205721

11989511989611198962

= 120601 (119909119895+12119896

1+1 119910119895+12119896

2

) minus 119868119895120601 (119909119895+12119896

1+1 119910119895+12119896

2

)

1205722

11989511989611198962

= 120601 (119909119895+12119896

1

119910119895+12119896

2+1) minus 119868119895120601 (119909119895+12119896

1

119910119895+12119896

2+1)

1205723

11989511989611198962

=120601 (119909119895+12119896

1+1 119910119895+12119896

2+1)minus119868119895120601 (119909119895+12119896

1+1 119910119895+12119896

2+1)

(8)

where 119868119895denotes the multilevel interpolation operator

In order to obtain the multilevel interpolation opera-tor it is necessary to express the wavelet coefficients1205721

11989511989611198962

1205722

11989511989611198962

1205723

11989511989611198962

as a weighted sum of 119906 in all of thecollocation points in the 119869-levelTherefore we should give thedefinition of the restriction operator as follows

119877119897119897119895119895

1198961119896211989811198982

= 1 119909

119897

1198961

= 119909119895

1198981

119910119897

1198962

= 119910119895

1198982

0 otherwise(9)

Using the restriction operator 119906(119909119895+121198961+1 119910119895+1

21198962

) 119906(119909119895+1

21198961

119910119895+1

21198962+1)

and 119906(119909119895+1

21198961+1 119910119895+1

21198962+1) can be rewritten as

120601 (119909119895+1

21198961+1 119910119895+1

21198962

) =

2119869

sum

1198991=0

2119869

sum

1198992=0

119877119895+1119895+1119869119869

21198961+12119896

211989911198992

120601 (119909119869

1198991

119910119869

1198992

)

120601 (119909119895+1

21198961

119910119895+1

21198962+1) =

2119869

sum

1198991=0

2119869

sum

1198992=0

119877119895+1119895+1119869119869

2119896121198962+111989911198992

120601 (119909119869

1198991

119910119869

1198992

)

120601 (119909119895+1

21198961+1 119910119895+1

21198962+1) =

2119869

sum

1198991=0

2119869

sum

1198992=0

119877119895+1119895+1119869119869

21198961+12119896

2+111989911198992

120601 (119909119869

1198991

119910119869

1198992

)

(10)

Introducing the extension operators 1198621 1198622 and 1198623 andsubstituting (10) into (8) the wavelet coefficients can berewritten as

1205721

11989511989611198962

=

2119869

sum

1198991=0

2119869

sum

1198992=0

119877119895+1119895+1119869119869

21198961+12119896

211989911198992

120601 (119909119869

1198991

119910119869

1198992

)

minus [

[

2119869

sum

1198991=0

2119869

sum

1198992=0

21198950

sum

11989601=0

21198950

sum

11989602=0

11987711989501198950119869119869

119896011198960211989911198992

120601 (119909119869

1198991

119910119869

1198992

)1199081198950

1198960111989602

(119909119895+1

21198961+1 119910119895+1

21198962

)

+

119895minus1

sum

1198951=1198950

2119869

sum

1198991=0

2119869

sum

1198992=0

21198951

sum

11989611=0

21198951

sum

11989612=0

(119862111989511198951119869119869

119896111198961211989911198992

1199081198951+1

211989611+12119896

12

(119909119895+1

21198961+1 119910119895+1

21198962

) 120601 (119909119869

1198991

119910119869

1198992

)

+ 119862211989511198951119869119869

119896111198961211989911198992

1199081198951+1

211989611211989612+1(119909119895+1

21198961+1 119910119895+1

21198962

) 119906 (119909119869

1198991

119910119869

1198992

)

+ 119862311989511198951119869119869

119896111198961211989911198992

1199081198951+1

211989611+12119896

12+1

(119909119895+1

21198961+1 119910119895+1

21198962

) 120601 (119909119869

1198991

119910119869

1198992

))]

]

=

2119869

sum

1198991=0

2119869

sum

1198992=0

1198621119895119895119869119869

1198961119896211989911198992

120601 (119909119869

1198991

119910119869

1198992

)

(11)

4 Mathematical Problems in Engineering

1205722

11989511989611198962

and 120572311989511989611198962

are similar to 120572111989511989611198962

From the aboveequation the extension operator can be obtained as

1198621119895119895119869119869

1198961119896211989911198992

= 119877119895+1119895+1119869119869

21198961+12119896

211989911198992

minus [

[

21198950

sum

11989601=0

21198950

sum

11989602=0

11987711989501198950119869119869

119896011198960211989911198992

120601 (119909119869

1198991

119910119869

1198992

)1199081198950

1198960111989602

(119909119895+1

21198961+1 119910119895+1

21198962

)

+

119895minus1

sum

1198951=1198950

2119869

sum

1198992=0

21198951

sum

11989611=0

(119862111989511198951119869119869

119896111198961211989911198992

1199081198951+1

211989611+12119896

12

(119909119895+1

21198961+1 119910119895+1

21198962

) 120601 (119909119869

1198991

119910119869

1198992

)

+ 119862211989511198951119869119869

119896111198961211989911198992

1199081198951+1

211989611211989612+1(119909119895+1

21198961+1 119910119895+1

21198962

) 120601 (119909119869

1198991

119910119869

1198992

)

+ 119862311989511198951119869119869

119896111198961211989911198992

1199081198951+1

211989611+12119896

12+1(119909119895+1

21198961+1 119910119895+1

21198962

) 120601 (119909119869

1198991

119910119869

1198992

))]

]

(12)

1198622 and1198623 can be obtained with the samemethodThereforethe calculation time complexity of the wavelet transformcoefficients 1205721

1198951198961111989612

12057221198951198961111989612

and 12057231198951198961111989612

is 119874((13)42119869minus1)Substituting 1205721

1198951198961111989612

12057221198951198961111989612

and 1205723

1198951198961111989612

1198621 1198622 and1198623 into (2) the multilevel wavelet interpolation operator canbe obtained as

11986811989911198992

(119909 119910)

=

21198950

sum

11989601=0

21198950

sum

11989602=0

11987711989501198950119869119869

119896011198960211989911198992

1199081198950

1198960111989602

(119909 119910)

+

119869minus1

sum

119895=1198950

2119895

sum

1198961=0

2119895

sum

1198962=0

(1198621119895119895119869119869

1198961119896211989911198992

119908119895+1

21198961+12119896

2

(119909 119910) 120601 (119909119869

1198991

119910119869

1198992

)

+ 1198622119895119895119869119869

1198961119896211989911198992

119908119895+1

2119896121198962+1(119909 119910) 120601 (119909

119869

1198991

119910119869

1198992

)

+1198623119895119895119869119869

1198961119896211989911198992

119908119895+1

21198961+12119896

2+1(119909 119910) 120601 (119909

119869

1198991

119910119869

1198992

))

(13)

Then (7) can be rewritten as

120601119869(119898119899)

(119909 119910 119905) =

2119869

sum

1198991

2119869

sum

1198992

11986811989911198992

(119909 119910) 120601 (119909119869

1198991

119910119869

1198992

) (14)

Substituting (14) into (5) themultilevel wavelet discretizationscheme of PERONA-MALIK model can be obtained

The purpose of constructing the multilevel wavelet collo-cation method is to decrease the amount of the collocationpoints and then improve the efficiency of the algorithm Butthe efficiency will be eliminated if the computation com-plexity of the multilevel wavelet interpolation operator is toohigh It is easy to understand that the interpolation waveletcoefficient is the error between the interpolation result andthe exact result at the same collocation point And so thewavelet coefficientmust be the function of the parameter 119905 In

other words the wavelet coefficient should vary with the timeparameter 119905 Then the interpolation operator can be viewedas a nonlinear problem HPM is efficient and effective tool tosolve nonlinear problem Aiming to improve the efficiency ofthe multilevel wavelet interpolation operators HPM wouldbe employed to construct a novel interpolation operator inthis section

For convenience 120601 and its derivative in (5) should berewritten as

120597120601

120597119905= 119865(119905 119909 119910 120601

120597120601

120597119909120597120601

1205971199101205972120601

12059711990921205972120601

1205971199091205971199101205972120601

1205971199102) (119905 gt 0)

120601 (119909 119910 0) = 1206010(119909 119910)

119889120601119869(119909 119910 119905)

119889119905= 119865 [119905 119909 119910 120601

119869(119909 119910 119905) 120601

119869(10)(119909 119910 119905)

120601119869(01)

(119909 119910 119905) 120601119869(20)

(119909 119910 119905)

120601119869(11)

(119909 119910 119905) 120601119869(02)

(119909 119910 119905)]

(15)

respectively where

120601119869(119909 119910 119905)

=

1

sum

11989601=0

1

sum

11989602=0

120601 (1199090

11989601

1199100

11989602

)1199080

1198960111989602

(119909 119910)

+

119869minus1

sum

119895=0

2119895minus1

sum

11989611=0

2119895minus1

sum

11989612=0

[1205721

1198951198961111989612

119908119895+1

211989611+12119896

12

(119909 119910)

+ 1205722

1198951198961111989612

119908119895+1

211989611211989612+1(119909 119910)

+ 1205723

1198951198961111989612

119908119895+1

211989611+12119896

12+1(119909 119910)]

(16)

Mathematical Problems in Engineering 5

The value of 120601119869(119909 119910 119905119899) at 119905119899is denoted as 120601

119899 and

119865 [119905119899 119909 119910 120601

119869(119909 119910 119905

119899) 120601119869(10)

(119909 119910 119905119899) 120601119869(01)

(119909 119910 119905119899)

120601119869(20)

(119909 119910 119905119899) 120601119869(11)

(119909 119910 119905119899) 120601119869(02)

(119909 119910 119905119899)]

(17)

is denoted as 119865119899 And then a linear homotopy function can be

constructed as

120601119869(119909 119910 119905) = (1 minus 120576) 119865

119899+ 120576119865119899+1

(18)

It is easy to identify the homotopy parameter as

120576 (119905) =119905 minus 119905119899

119905119899+1

minus 119905119899

119905 isin [119905119899 119905119899+1

] there4 120576 isin [0 1] (19)

According to the perturbation theory the solution of (18) canbe expressed as the power series expansion of 120576 as follows

120601119869= 120601119869

0+ 120576120601119869

1+ 1205762120601119869

2+ sdot sdot sdot (20)

Substituting (20) into (18) and rearranging based on powersof 120576-terms we have

12057601206011198690= 119865119899

12057611206011198691= 119865119899+1

minus119865119899

(21)

According to HPM we obtain the wavelet coefficients1205721

11989511989611198962

(119905119899+1

) 120572211989511989611198962

(119905119899+1

) 120572311989511989611198962

(119905119899+1

) at 119905119899as follows

1205721

11989511989611198962

= 120601 (119909119895+1

21198961+1 119910119895+1

21198962

) minus 119868119895120601 (119909119895+1

21198961+1 119910119895+1

21198962

)

= 120601 (119909119895+1

21198961+1 119910119895+1

21198962

)

minus [

[

1

sum

11989601=0

1

sum

11989602=0

120601 (1199090

11989601

1199100

11989602

)1199080

1198960111989602

(119909119895+1

21198961+1 119910119895+1

21198962

)

+

119895minus1

sum

1198951=0

21198951

sum

11989611=0

21198951

sum

11989612=0

(1205721

11989511198961111989612

1199081198951+1

211989611+12119896

12

(119909119895+1

21198961+1 119910119895+1

21198962

)

+ 1205722

11989511198961111989612

1199081198951+1

211989611211989612+1(119909119895+1

21198961+1 119910119895+1

21198962

)

+ 1205723

11989511198961111989612

1199081198951+1

211989611+12119896

12+1(119909119895+1

21198961+1 119910119895+1

21198962

))]

]

1205722

11989511989611198962

= 120601 (119909119895+1

21198961

119910119895+1

21198962+1) minus 119868119895120601 (119909119895+1

21198961

119910119895+1

21198962+1)

= 120601 (119909119895+1

21198961

119910119895+1

21198962+1)

minus [

[

1

sum

11989601=0

1

sum

11989602=0

120601 (1199090

11989601

1199100

11989602

)1199080

1198960111989602

(119909119895+1

21198961

119910119895+1

21198962+1)

+

119895minus1

sum

1198951=0

21198951

sum

11989611=0

21198951

sum

11989612=0

(1205721

11989511198961111989612

1199081198951+1

211989611+12119896

12

(119909119895+1

21198961

119910119895+1

21198962+1)

+ 1205722

11989511198961111989612

1199081198951+1

211989611211989612+1(119909119895+1

21198961

119910119895+1

21198962+1)

+ 1205723

11989511198961111989612

1199081198951+1

211989611+12119896

12+1(119909119895+1

21198961

119910119895+1

21198962+1))]

]

6 Mathematical Problems in Engineering

1205723

11989511989611198962

= 120601 (119909119895+1

21198961+1 119910119895+1

21198962+1) minus 119868119895120601 (119909119895+1

21198961+1 119910119895+1

21198962+1)

= 120601 (119909119895+1

21198961+1 119910119895+1

21198962+1)

minus [

[

1

sum

11989601=0

1

sum

11989602=0

120601 (1199090

11989601

1199100

11989602

)1199080

1198960111989602

(119909119895+1

21198961+1 119910119895+1

21198962+1)

+

119895minus1

sum

1198951=0

21198951

sum

11989611=0

21198951

sum

11989612=0

(1205721

11989511198961111989612

1199081198951+1

211989611+12119896

12

(119909119895+1

21198961+1 119910119895+1

21198962+1)

+ 1205722

11989511198961111989612

1199081198951+1

211989611211989612+1(119909119895+1

21198961+1 119910119895+1

21198962+1)

+ 1205723

11989511198961111989612

1199081198951+1

211989611+12119896

12+1(119909119895+1

21198961+1 119910119895+1

21198962+1))]

]

(22)

Obviously the calculation time complexity of the wavelettransform coefficients 1205721

11989511989611198962

120572211989511989611198962

and 1205723

11989511989611198962

is 119874(4119869)which is decreased greatly than that in (8) which is119874((13)4

2119869minus1)

Substituting the wavelet transform efficient (22) into (16)we obtain120601119869(119909 119910 119905

119899+1)

= 120601119869(119909 119910 119905

119899)

+Δ119905

2[119865 (119905119899 119909 119910 120601

119869(119909 119910 119905

119899) 120601119869(10)

(119909 119910 119905119899)

120601119869(01)

(119909 119910 119905119899) 120601119869(20)

(119909 119910 119905119899)

120601119869(11)

(119909 119910 119905119899) 120601119869(02)

(119909 119910 119905119899))

+ 119865 (119905119899+1

119909 119910 120601119869

0(119909 119910 119905

119899+1)

120601119869(10)

0(119909 119910 119905

119899+1) 120601119869(01)

0(119909 119910 119905

119899+1)

120601119869(20)

0(119909 119910 119905

119899+1) 120601119869(11)

0(119909 119910 119905

119899+1)

120601119869(02)

0(119909 119910 119905

119899+1))]

(23)

and the derivative function120601119869(119898119899)

(119909 119910)

=

1

sum

11989601=0

1

sum

11989602=0

120601 (1199090

11989601

1199100

11989602

)1199080(119898119899)

1198960111989602

(119909 119910)

+

119869minus1

sum

119895=0

2119895minus1

sum

11989611=0

2119895minus1

sum

11989612=0

[1205721

1198951198961111989612

119908119895+1(119898119899)

211989611+12119896

12

(119909 119910)

+ 1205722

1198951198961111989612

119908119895+1(119898119899)

211989611211989612+1(119909 119910)

+ 1205723

1198951198961111989612

119908119895+1(119898119899)

211989611+12119896

12+1(119909 119910)]

(24)

Obviously the computation complexity is decreased greatlycomparing with (14)

31TheMultiscale InterpolationWavelet Approximation of theC-V Model There are many ways to solve partial differentialequations and the most typical method is the differencemethod This method uses the flat function to describeimage approximately the surface function But it is easyto cause artifacts phenomenon affecting the accuracy ofimage segmentation Wavelet function has both smooth andcompactly supported characteristics Besides performance ofmultiscale analysis can be used to construct the multiscaleadaptive interpolation operator for solving nonlinear partialdifferential equationsThe wavelet approximation of the levelset function and its derivative with respect to 119909 and 119910respectively can be expressed as follows

120601119869(119898119899)

(119909 119910 119905)

=

1

sum

11989601=0

1

sum

11989602=0

120601 (1199090

11989601

1199100

11989602

)1199080(119898119899)

1198960111989602

(119909 119910)

+

119869minus1

sum

119895=0

2119895minus1

sum

11989611=0

2119895minus1

sum

11989612=0

[1205721

1198951198961111989612

(119905) 119908119895+1(119898119899)

211989611+12119896

12

(119909 119910)

+ 1205722

1198951198961111989612

(119905) 119908119895+1(119898119899)

211989611211989612+1(119909 119910)

+ 1205723

1198951198961111989612

(119905) 119908119895+1(119898119899)

211989611+12119896

12+1(119909 119910)]

(25)

where 119908119895(119898119899)11989611198962

(119909 119910) is the wavelet function and its 119898- and119899-order derivative with respect to 119909 and 119910 respectively1205721

1198951198961111989612

(119905) 12057221198951198961111989612

(119905) and 12057231198951198961111989612

(119905) are wavelet transform

Mathematical Problems in Engineering 7

coefficients We convert (2) to wavelet multiscale discreteformat of the level set function by stead of (25)

119889120601119869(119909 119910 119905)

119889119905

= 120575120576(120601119869(119909 119910 119905))

times [] div(nabla120601119869(119909 119910 119905)

1003816100381610038161003816nabla120601119869 (119909 119910 119905)

1003816100381610038161003816

)minus1205821

10038161003816100381610038161198680 minus 11988811003816100381610038161003816

2

+1205822

10038161003816100381610038161198680 minus 11988821003816100381610038161003816

2

]

(26)

It is the most direct way for dynamic adaptation toonly retain the distribution points of corresponding waveletcoefficients that satisfy the condition

min (100381610038161003816100381610038161205721

1198951198961111989612

(119905)10038161003816100381610038161003816100381610038161003816100381610038161205722

1198951198961111989612

(119905)10038161003816100381610038161003816100381610038161003816100381610038161205723

1198951198961111989612

(119905)10038161003816100381610038161003816) ge 120576 (27)

Time domain numerical integration of partial differentialequations is an iterative process therefore some pointswhich are possible important next step need to be kept toenable the algorithm to track singularities of solutions Soadjacent points of distribution points also should be keptTheadjacent region can be delineated as follows

1003816100381610038161003816119904 minus 1198951003816100381610038161003816 le 119872

10038161003816100381610038161003816119909119904

119896minus 119909119895

119894

10038161003816100381610038161003816le 120576119909

10038161003816100381610038161003816119909119904

119896minus 119909119895

119894

10038161003816100381610038161003816le 120576119910

(28)

where 119904 119895 are numbers of different scale wavelet and 119896 119894119872 isin

119885 120576119909 120576119910are constant

32 Nonlinear Discrete Ordinary Differential EquationsBecause ] is a small parameter in ordinary differentialequation (26) value of ] div(nabla120601119869(119909 119910 119905)|nabla120601119869(119909 119910 119905)|) is lowEquation (26) can be converted into

119889120601119869(119909 119910 119905)

119889119905

= 120575120576(120601119869(119909 119910 119905))

times [] div(nabla120601119869(119909 119910 119905

0)

1003816100381610038161003816nabla120601119869 (119909 119910 119905

0)1003816100381610038161003816

) minus 1205821

10038161003816100381610038161198680 minus 11988811003816100381610038161003816

2

+ 1205822

10038161003816100381610038161198680 minus 11988821003816100381610038161003816

2

]

(29)

The solution of (29) is

120601119869(119909 119910 119905) = (

120576119898

2120587)

13

+ (119898

21205871205765)

minus13

(30)

where

119898 = ] div(nabla120601119869(119909 119910 119905

0)

1003816100381610038161003816nabla120601119869 (119909 119910 119905

0)1003816100381610038161003816

)

minus 1205821

10038161003816100381610038161198680 minus 11988811003816100381610038161003816

2

+ 1205822

10038161003816100381610038161198680 minus 11988821003816100381610038161003816

2

(31)

We can get the solution of ordinary differential equations(26) by iterative solution (30)

4 Experiment and Discussion ofOnion Infected Region Segmentation

Figure 1(a) is a 256 times 302 image of an onion infected bysour skin virus We noticed that the onion has a water-soaked appearance Compared with the background thegrayscale difference between the water-soaked appearanceand the healthy part is smaller So it is beneficial to compareperformance of the different algorithms Figure 1(b) is anideal segmentation results Segmentation target is infectedregions of onion however infected part is often not uniformIn the image the difference of gradient is less than 1 at upperleft part due to slight infection so the algorithm is difficult toprecise segmentation based on global threshold Thereforethe best one of the different segmentation algorithms candistinguish themajority of the infected regionwithout seriousoversegmentationThen it is easy to identify andmeasure theinfected portions by using a priori knowledge

41 Comparison among Different SegmentationMethods Thecommon image segmentation methods including water-shed algorithm Sobel operator and Canny edge detectionalgorithm Otsu algorithm and an effective and commonquad-tree decomposition algorithm were selected for com-parison Shannon wavelet was employed to construct thewavelet interpolation operator The representation of Shan-non wavelet is based upon approximating the Dirac deltafunction as a band-limited function and is given by

119908 (119909) =sin (120587119909)120587119909

(32)

Consider a one-dimensional function 119891(119909) 119909 isin [119886 119887] Adiscrete point sequence of the variable 119909 is defined as

119909119899= 119886 +

119887 minus 119886

2119895sdot 119899 119895 isin Z 119899 = 0 1 2 2

119895 (33)

and the corresponding discrete point sequence of the scalingfunction 120601(119909) can be defined as

119908119895119899(119909) = 119908

119895(119909 minus 119909

119899) =

sin (2119895120587 (119887 minus 119886)) (119909 minus 119909119899)

(2119895120587 (119887 minus 119886)) (119909 minus 119909119899)

(34)

The first and second order derivatives of 120601119895(119909 minus 119909

119899) at the

discrete point 119909119896are

1206011015840

119895(119909119896minus 119909119899) =

0 119896 = 119899

2119895 cos [120587 (119896 minus 119899)](119896 minus 119899) (119887 minus 119886)

119896 = 119899

12060110158401015840

119895(119909119896minus 119909119899) =

minus1205872

3((119887 minus 119886)2119895)2 119896 = 119899

minus2 cos [120587 (119896 minus 119899)]

((119887 minus 119886) 2119895)2

(119896 minus 119899)2 119896 = 119899

(35)

The corresponding 2-dimension weight function can be rep-resented as the tensor product form of the above equations

8 Mathematical Problems in Engineering

(a) Original image (b) Segmentation target

Figure 1 Burkholderia cepacia (ex Burkholder) infected onion and the target segmentation

(a) Grayscale image (b) Gradient image (c) OSTU

(d) Watershed 1 (e) Watershed 2 (f) Qtdecomp

(g) Sobel (h) Canny (i) Wavelet precise integration

Figure 2 Comparison of various segmentation methods

Mathematical Problems in Engineering 9

300

250

200

150

100

50

00 50 100 150 200 250 300

Figure 3 Adaptive wavelet collocation points on level set

The experimental procedure is described as follows

(1) convert the infested onion image to grayscale(Figure 2(a)) and solve for the gradient map(Figure 2(b))

(2) use the grayscale image to test the Sobel operatorCanny operator Qtdecomp algorithms and waveletprecise integration method Use the gradient mat totest the OSTU method and watershed algorithm

(3) applying the watershed method to segment the imagewhich has been processed by OSTU method in orderto avoid the oversegmentation from the watershedalgorithm

The segmentation results are shown in Figure 2 Thewatershed algorithm segmentation result shows serious over-segmentation (Figure 2(d)) and cannot recognize the infectedpart Although OSTUmethod separated part infected regionof the onion the partition boundary is discontinuity andis difficult to measure infection specific gravity To avoidoversegmentation OSTUwas overlapped with the watershedsegmentationThe result is shown in Figure 2(e) in which thepartition boundary is clear but is unable to distinguish thevirus infected part The Sobel operator recognition on thepart of the infection is also not good (Figure 2(g)) Cannyoperator and Qtdecomp algorithm identified the area ofinfection but the boundary points of segmentation regionare disorder and cannot be measuredThe precise integrationmethod presented in this paper can identify the infected areaclearly So it is helpful to onion evaluation and classification

In fact it is impossible to project image segmentationby a single algorithm The important reason that partialdifferential equations are effective for image segmentationis that the method integrated many image segmentationprinciples to the model of partial differential equations Inthis paper the C-V image segmentation model is a globalconvex optimization variational model which is establishedon image piecewise smooth (119888

1and 1198882are the average gray

values inside (Ω1) and outside (Ω

2) of the object contour

respectively) To ensure the accuracy of image segmentation

the curvature of the image the border gradient and levelset function evolution were taken into account in imagesegmentation It means that the global convex optimizationmodel of image segmentation has been built based onintegration of a variety of image segmentation theories andhas obvious advantages In addition the method of iterativesolution of the self-adaptive method can also be integratedinto the segmentation process to ensure the accuracy ofsegmentation method further However the speed of thealgorithm will be affected Therefore it is important to findefficient and accurate numerical method

42 Efficiency Comparison of Multiscale Adaptive WaveletNumerical Method and Difference Method The C-V modelwas used for 256 times 302 images segmentation and divideddifference method was used to disperse partial differentialequations So discrete 7312 (256 times 302) ordinary differentialequations are huge solvingworkload But the adaptivewaveletprecise integration method can reduce the scale to 9576equations It can improve solution efficiency greatly due toless workload and low memory demand Of course the useof adaptive wavelet precise integration method for solvingthe number of distribution points will dynamically changeas the solution process In addition as shown in Figure 3distribution points are relatively dense within the ellipsering and another location was sparse The evident grayscaledifference between the infected and the healthy parts leadto this special points distribution Furthermore distributionpoints also exhibit regular matrix form which result fromblock solving method of wavelet transform to improve theefficiency The matrix-like distribution is from the boundaryeffect among the different blocks The interval wavelet caneffectively reduce the range effect but it will also increase thecomputation work of the wavelet transform

In this paper difference method was tested in MATLABThewavelet interpolation operatorwas implementedwithVCprogramming and other parts with MATLAB programmingOn the same computer difference method takes 03 secondsthe adaptive wavelet precise integration method takes 018seconds The results also show that the wavelet transform

10 Mathematical Problems in Engineering

of the iterative process reduces the overall computationalefficiency of the algorithm

5 Conclusions

Shannon wavelet precise integration method is a new imagesegmentation method based on the C-V model which wasused to construct adaptive wavelet interpolation operator dueto multiscale characteristics of wavelet transform combinedwith the time precise integration technology The methodmakes full use of the multiscale characteristics and thehigh precise performance of precise integration methodCompared to the gradient method and wavelet transformmethod of image segmentation object boundary obtained byWPIM segmentation method is clear and closed comparedto the watershed method the WPIM method avoids over-and undersegmentation problems and is very suitable formeasurement of image segmentation such as onion qualityassessment

The adaptive interpolation operator in the Shannonwavelet precision integration method can reduce the amountof the collocation points and improve the calculation effi-ciency As the interpolation operator contains a wavelettransform process the corresponding algorithm needs to doa wavelet transform between each two iteration time stepsSo the cost of the wavelet transformation is an importantpart of the calculation amount of the algorithm Compactlysupported orthogonal wavelet function can be expected tosolve the problem

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors appreciate the funding support from NationalNatural Science Foundation of China (Award nos 41171184and 41171337) The software tools were provided by the Foodand Fiber Sensing Lab of University of Georgia and theComputer Center of China Agricultural University

References

[1] Y Chen Y Xia Y Bian and Z-P Zhong ldquoImage measurementof precision aluminum alloy forgingsrdquo Journal of PlasticityEngineering vol 17 no 6 pp 77ndash81 2010

[2] S-L Mei Q-S Lu S-W Zhang and L Jin ldquoAdaptive intervalwavelet precise integrationmethod for partial differential equa-tionsrdquo Applied Mathematics and Mechanics vol 26 no 3 pp364ndash371 2005

[3] H-H Yan ldquoAdaptive wavelet precise integration method fornonlinear black-scholes model based on variational iterationmethodrdquo Abstract and Applied Analysis vol 2013 Article ID735919 6 pages 2013

[4] S-L Pang ldquoWavelet numerical method for nonlinear randomsystemrdquo Transactions of the Chinese Society of AgriculturalMachinery vol 38 no 3 pp 168ndash170 2007

[5] Y Wang ldquoWavelet precise time-integration method for heatconduction equationrdquo Journal of Chongqing Institute of Technol-ogy vol 21 no 8 pp 130ndash132 2007

[6] L X Zhang Y Yang and S L Mei ldquoWavelet precise integrationmethod on image denoisingrdquoTransactions of the Chinese Societyof Agricultural Machinery vol 37 no 7 pp 109ndash112 2006

[7] W N Xu S L Mei P X Wang and Y Yang ldquoAdaptivewavelet precise integration method on remote sensing imagedenoisingrdquo Transactions of the Chinese Society of AgriculturalMachinery vol 42 no 4 pp 148ndash152 2011

[8] R-Y Xing ldquoWavelet-based homotopy analysismethod for non-linear matrix system and its application in burgers equationrdquoMathematical Problems in Engineering vol 2013 Article ID982810 7 pages 2013

[9] S-L Mei ldquoConstruction of target controllable image segmen-tation model based on homotopy perturbation technologyrdquoAbstract and Applied Analysis vol 2013 Article ID 131207 8pages 2013

[10] L Liu ldquoConstruction of interval shannon wavelet and itsapplication in solving nonlinear black-scholes equationrdquoMath-ematical Problems in Engineering vol 2014 Article ID 541023 8pages 2014

[11] C Cattani ldquoShannon wavelets theoryrdquo Mathematical Problemsin Engineering vol 2008 Article ID 164808 24 pages 2008

[12] C Cattani ldquoSecond order Shannon wavelet approximationof C2-functionsrdquo UPB Scientific Bulletin Series A AppliedMathematics and Physics vol 73 no 3 pp 73ndash84 2011

[13] C Cattani and L M S Ruiz ldquoDiscrete differential operators inmultidimensional haar wavelet spacesrdquo International Journal ofMathematics and Mathematical Sciences vol 2004 no 44 pp2347ndash2355 2004

[14] C Cattani A Ciancio and B Lods ldquoOn a mathematical modelof immune competitionrdquo Applied Mathematics Letters vol 19no 7 pp 678ndash683 2006

[15] C Cattani ldquoConnection coefficients of Shannon waveletsrdquoMathematical Modelling and Analysis vol 11 no 2 pp 117ndash1322006

[16] S-L Mei and D-H Zhu ldquoInterval shannon wavelet collocationmethod for fractional fokker-planck equationrdquo Advances inMathematical Physics vol 2013 Article ID 821820 12 pages2013

[17] L-W Liu ldquoInterval wavelet numerical method on fokker-planck equations for nonlinear random systemrdquo Advances inMathematical Physics vol 2013 Article ID 651357 7 pages 2013

[18] J-H He ldquoHomotopy perturbation method for bifurcation ofnonlinear problemsrdquo International Journal of Nonlinear Sciencesand Numerical Simulation vol 6 no 2 pp 207ndash208 2005

[19] J-H He ldquoSome asymptotic methods for strongly nonlinearequationsrdquo International Journal of Modern Physics B vol 20no 10 pp 1141ndash1199 2006

[20] J-H He ldquoVariational iteration method-Some recent resultsand new interpretationsrdquo Journal of Computational and AppliedMathematics vol 207 no 1 pp 3ndash17 2007

[21] S-L Mei C-J Du and S-W Zhang ldquoAsymptotic numericalmethod for multi-degree-of-freedom nonlinear dynamic sys-temsrdquo Chaos Solitons and Fractals vol 35 no 3 pp 536ndash5422008

[22] S-L Mei and S-W Zhang ldquoCoupling technique of variationaliteration and homotopy perturbation methods for nonlinearmatrix differential equationsrdquoComputers andMathematics withApplications vol 54 no 7-8 pp 1092ndash1100 2007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

wavelet function which possesses many advantages such asorthogonality continuity and differentiability [11] It alsohas the advantage over the Hermite DAF in that it isan interpolating function producing matrix equations thathave the potential to be relatively sparse In addition thesecond order approximation of a 119862

2-function based onShannon wavelet functions is given [12] The approximationis compared with the wavelet reconstruction formula andthe error of approximation is explicitly computed [13 14]Furthermore Shannon wavelet has been used to solve thefractional calculus problems in the recent years [15ndash17]A perceived disadvantage of the Shannon scaling functionis that it tends to zero quite slowly as |119909| rarr infin Adirect consequence of this is that when calculating thederivatives a large number of the nodal values will contributesignificantly However comparing with other wavelets theShannon scaling function is one that possessesmore excellentnumerical properties such as the interpolation orthogonalityand smoothness Therefore Shannon wavelet is employed toconstruct the multiscale interpolator in our research

The purpose of this research is to construct a multiscaletwo-dimension wavelet interpolation operator based on thehomotopy perturbation method [18ndash20] with which theimage segmentation PDE (C-V model) can be discretizedinto a system of nonlinear ODEs Combining the preciseintegrationmethodwith homotopy perturbationmethod [2122] for solving nonlinear problems a fast wavelet numericalalgorithm for P-M and C-V model in image processing canbe obtained With this new image segmentation method thepathologic onion classification and grading can be processedprecisely and efficiency

2 Chan-Vese Model

In order to solve Mumford-Shah model with Euler-Lagrangemethod a simplified model was deduced by Chan and Vesein which Euclid length was employed instead of Hausdorfflength This simplified model can also be called Chan-Vesemodel which can be expressed as follows

119864CV

(1198881 1198882 119862) = 120582

1intΩ1

(1198680minus 1198881)2

119889119909 119889119910

+ 1205822intΩ2

(1198680minus 1198882)2

119889119909 119889119910 + ] |119862|

(1)

119888119894= mean

Ω119894

(1199060) =

intΩ119894

1199060(119909 119910) 119889119909 119889119910

Area (Ω119894)

119894 = 1 2 (2)

where 1205821and 120582

2are positive constants and 119888

1and 1198882are the

average gray level values inside (Ω1) and outside (Ω

2) of the

object contour respectively 1198680denotes the image to process

|119862| is the length of the object contour and ] is the weightparameter According to the level set method the contour

curve of the objects should be embedded into the level setfunction as follows

119862 = (119909 119910) | (119909 119910) isin Ω 120601 (119909 119910) = 0

Ω1= (119909 119910) | (119909 119910) isin Ω 120601 (119909 119910) gt 0

Ω2= (119909 119910) | (119909 119910) isin Ω 120601 (119909 119910) lt 0

(3)

Then the level set-based C-V model can be rewritten asfollows

119864 (1198881 1198882 120601) = 120582

1intΩ

10038161003816100381610038161198680 minus 11988811003816100381610038161003816

2

119867(120601) 119889119909 119889119910

+ 1205822intΩ

10038161003816100381610038161198680 minus 11988821003816100381610038161003816

2

(1 minus 119867 (120601)) 119889119909 119889119910

+ ]intΩ

1003816100381610038161003816119867 (120601)1003816100381610038161003816 119889119909 119889119910

119867 (120601) = 1 120601 ge 0

0 120601 lt 0

120575120576=

120576

120587 (1205762 + 1206012)

(4)

Using the variational method the PDEs with respect to thevariable 120601 can be obtained as follows

120597120601

120597119905= 120575120576(120601) [] div(

nabla120601

1003816100381610038161003816nabla1206011003816100381610038161003816

) minus 1205821

10038161003816100381610038161198680 minus 11988811003816100381610038161003816

2

+ 1205822

10038161003816100381610038161198680 minus 11988821003816100381610038161003816

2

]

(5)

Obviously div(nabla120601|nabla120601|) is the curvature of the level setfunction 120601 and 120575

120576(120601) is used to constrain the growth of the

level set functionThe solution of (5) is the level set function 120601(119909 119910 119905) at

time 119905 The zero level set is the object contour curve whichcan be obtained by solving the equation 120601(119909 119910 119905) = 0We focus on multiscale wavelet image segmentation discreteformat and the corresponding numerical solution method

3 HPM-Based Wavelet Interpolation OperatorConstruction Schemes

Let the definition domain of the image be (119909min 119909max) times(119910min 119910max) the discretization points can be defined as(119909119895

1198961

119910119895

1198962

) where 119895 is a scale parameter and 1198961and 119896

2are

position parameters So

119909119895

1198961

= 119909min + 1198961119909max minus 119909min

2119895

119910119895

1198961

= 119910min + 1198962119910max minus 119910min

2119895

119895 1198961 1198962isin Z

(6)

In addition 119908119895(119898119899)11989611198962

(119909 119910) denotes the multiscale waveletfunction and the corresponding 119898th and 119899th derivativeswith respect to 119909 and 119910 respectively The level set function

Mathematical Problems in Engineering 3

120601(119909 119910 119905) and the corresponding derivative function can bediscretized as follows

120601119869(119898119899)

(119909 119910 119905)

=

1

sum

11989601=0

1

sum

11989602=0

120601 (1199090

11989601

1199100

11989602

)1199080(119898119899)

1198960111989602

(119909 119910)

+

119869minus1

sum

119895=0

2119895minus1

sum

11989611=0

2119895minus1

sum

11989612=0

[1205721

1198951198961111989612

(119905) 119908119895+1(119898119899)

211989611+12119896

12

(119909 119910)

+ 1205722

1198951198961111989612

(119905) 119908119895+1(119898119899)

211989611211989612+1(119909 119910)

+ 1205723

1198951198961111989612

(119905) 119908119895+1(119898119899)

211989611+12119896

12+1(119909 119910)]

(7)

where 119895 and 119869 are constants which denote the wavelet scalenumber and the maximum of the scale number respectively1205721

1198951198961111989612

12057221198951198961111989612

and 12057231198951198961111989612

are the wavelet coefficients atthe points (119909119895

1198961

119910119895

1198962

) According to the interpolation wavelettransform theory the wavelet coefficients can be written as

1205721

11989511989611198962

= 120601 (119909119895+12119896

1+1 119910119895+12119896

2

) minus 119868119895120601 (119909119895+12119896

1+1 119910119895+12119896

2

)

1205722

11989511989611198962

= 120601 (119909119895+12119896

1

119910119895+12119896

2+1) minus 119868119895120601 (119909119895+12119896

1

119910119895+12119896

2+1)

1205723

11989511989611198962

=120601 (119909119895+12119896

1+1 119910119895+12119896

2+1)minus119868119895120601 (119909119895+12119896

1+1 119910119895+12119896

2+1)

(8)

where 119868119895denotes the multilevel interpolation operator

In order to obtain the multilevel interpolation opera-tor it is necessary to express the wavelet coefficients1205721

11989511989611198962

1205722

11989511989611198962

1205723

11989511989611198962

as a weighted sum of 119906 in all of thecollocation points in the 119869-levelTherefore we should give thedefinition of the restriction operator as follows

119877119897119897119895119895

1198961119896211989811198982

= 1 119909

119897

1198961

= 119909119895

1198981

119910119897

1198962

= 119910119895

1198982

0 otherwise(9)

Using the restriction operator 119906(119909119895+121198961+1 119910119895+1

21198962

) 119906(119909119895+1

21198961

119910119895+1

21198962+1)

and 119906(119909119895+1

21198961+1 119910119895+1

21198962+1) can be rewritten as

120601 (119909119895+1

21198961+1 119910119895+1

21198962

) =

2119869

sum

1198991=0

2119869

sum

1198992=0

119877119895+1119895+1119869119869

21198961+12119896

211989911198992

120601 (119909119869

1198991

119910119869

1198992

)

120601 (119909119895+1

21198961

119910119895+1

21198962+1) =

2119869

sum

1198991=0

2119869

sum

1198992=0

119877119895+1119895+1119869119869

2119896121198962+111989911198992

120601 (119909119869

1198991

119910119869

1198992

)

120601 (119909119895+1

21198961+1 119910119895+1

21198962+1) =

2119869

sum

1198991=0

2119869

sum

1198992=0

119877119895+1119895+1119869119869

21198961+12119896

2+111989911198992

120601 (119909119869

1198991

119910119869

1198992

)

(10)

Introducing the extension operators 1198621 1198622 and 1198623 andsubstituting (10) into (8) the wavelet coefficients can berewritten as

1205721

11989511989611198962

=

2119869

sum

1198991=0

2119869

sum

1198992=0

119877119895+1119895+1119869119869

21198961+12119896

211989911198992

120601 (119909119869

1198991

119910119869

1198992

)

minus [

[

2119869

sum

1198991=0

2119869

sum

1198992=0

21198950

sum

11989601=0

21198950

sum

11989602=0

11987711989501198950119869119869

119896011198960211989911198992

120601 (119909119869

1198991

119910119869

1198992

)1199081198950

1198960111989602

(119909119895+1

21198961+1 119910119895+1

21198962

)

+

119895minus1

sum

1198951=1198950

2119869

sum

1198991=0

2119869

sum

1198992=0

21198951

sum

11989611=0

21198951

sum

11989612=0

(119862111989511198951119869119869

119896111198961211989911198992

1199081198951+1

211989611+12119896

12

(119909119895+1

21198961+1 119910119895+1

21198962

) 120601 (119909119869

1198991

119910119869

1198992

)

+ 119862211989511198951119869119869

119896111198961211989911198992

1199081198951+1

211989611211989612+1(119909119895+1

21198961+1 119910119895+1

21198962

) 119906 (119909119869

1198991

119910119869

1198992

)

+ 119862311989511198951119869119869

119896111198961211989911198992

1199081198951+1

211989611+12119896

12+1

(119909119895+1

21198961+1 119910119895+1

21198962

) 120601 (119909119869

1198991

119910119869

1198992

))]

]

=

2119869

sum

1198991=0

2119869

sum

1198992=0

1198621119895119895119869119869

1198961119896211989911198992

120601 (119909119869

1198991

119910119869

1198992

)

(11)

4 Mathematical Problems in Engineering

1205722

11989511989611198962

and 120572311989511989611198962

are similar to 120572111989511989611198962

From the aboveequation the extension operator can be obtained as

1198621119895119895119869119869

1198961119896211989911198992

= 119877119895+1119895+1119869119869

21198961+12119896

211989911198992

minus [

[

21198950

sum

11989601=0

21198950

sum

11989602=0

11987711989501198950119869119869

119896011198960211989911198992

120601 (119909119869

1198991

119910119869

1198992

)1199081198950

1198960111989602

(119909119895+1

21198961+1 119910119895+1

21198962

)

+

119895minus1

sum

1198951=1198950

2119869

sum

1198992=0

21198951

sum

11989611=0

(119862111989511198951119869119869

119896111198961211989911198992

1199081198951+1

211989611+12119896

12

(119909119895+1

21198961+1 119910119895+1

21198962

) 120601 (119909119869

1198991

119910119869

1198992

)

+ 119862211989511198951119869119869

119896111198961211989911198992

1199081198951+1

211989611211989612+1(119909119895+1

21198961+1 119910119895+1

21198962

) 120601 (119909119869

1198991

119910119869

1198992

)

+ 119862311989511198951119869119869

119896111198961211989911198992

1199081198951+1

211989611+12119896

12+1(119909119895+1

21198961+1 119910119895+1

21198962

) 120601 (119909119869

1198991

119910119869

1198992

))]

]

(12)

1198622 and1198623 can be obtained with the samemethodThereforethe calculation time complexity of the wavelet transformcoefficients 1205721

1198951198961111989612

12057221198951198961111989612

and 12057231198951198961111989612

is 119874((13)42119869minus1)Substituting 1205721

1198951198961111989612

12057221198951198961111989612

and 1205723

1198951198961111989612

1198621 1198622 and1198623 into (2) the multilevel wavelet interpolation operator canbe obtained as

11986811989911198992

(119909 119910)

=

21198950

sum

11989601=0

21198950

sum

11989602=0

11987711989501198950119869119869

119896011198960211989911198992

1199081198950

1198960111989602

(119909 119910)

+

119869minus1

sum

119895=1198950

2119895

sum

1198961=0

2119895

sum

1198962=0

(1198621119895119895119869119869

1198961119896211989911198992

119908119895+1

21198961+12119896

2

(119909 119910) 120601 (119909119869

1198991

119910119869

1198992

)

+ 1198622119895119895119869119869

1198961119896211989911198992

119908119895+1

2119896121198962+1(119909 119910) 120601 (119909

119869

1198991

119910119869

1198992

)

+1198623119895119895119869119869

1198961119896211989911198992

119908119895+1

21198961+12119896

2+1(119909 119910) 120601 (119909

119869

1198991

119910119869

1198992

))

(13)

Then (7) can be rewritten as

120601119869(119898119899)

(119909 119910 119905) =

2119869

sum

1198991

2119869

sum

1198992

11986811989911198992

(119909 119910) 120601 (119909119869

1198991

119910119869

1198992

) (14)

Substituting (14) into (5) themultilevel wavelet discretizationscheme of PERONA-MALIK model can be obtained

The purpose of constructing the multilevel wavelet collo-cation method is to decrease the amount of the collocationpoints and then improve the efficiency of the algorithm Butthe efficiency will be eliminated if the computation com-plexity of the multilevel wavelet interpolation operator is toohigh It is easy to understand that the interpolation waveletcoefficient is the error between the interpolation result andthe exact result at the same collocation point And so thewavelet coefficientmust be the function of the parameter 119905 In

other words the wavelet coefficient should vary with the timeparameter 119905 Then the interpolation operator can be viewedas a nonlinear problem HPM is efficient and effective tool tosolve nonlinear problem Aiming to improve the efficiency ofthe multilevel wavelet interpolation operators HPM wouldbe employed to construct a novel interpolation operator inthis section

For convenience 120601 and its derivative in (5) should berewritten as

120597120601

120597119905= 119865(119905 119909 119910 120601

120597120601

120597119909120597120601

1205971199101205972120601

12059711990921205972120601

1205971199091205971199101205972120601

1205971199102) (119905 gt 0)

120601 (119909 119910 0) = 1206010(119909 119910)

119889120601119869(119909 119910 119905)

119889119905= 119865 [119905 119909 119910 120601

119869(119909 119910 119905) 120601

119869(10)(119909 119910 119905)

120601119869(01)

(119909 119910 119905) 120601119869(20)

(119909 119910 119905)

120601119869(11)

(119909 119910 119905) 120601119869(02)

(119909 119910 119905)]

(15)

respectively where

120601119869(119909 119910 119905)

=

1

sum

11989601=0

1

sum

11989602=0

120601 (1199090

11989601

1199100

11989602

)1199080

1198960111989602

(119909 119910)

+

119869minus1

sum

119895=0

2119895minus1

sum

11989611=0

2119895minus1

sum

11989612=0

[1205721

1198951198961111989612

119908119895+1

211989611+12119896

12

(119909 119910)

+ 1205722

1198951198961111989612

119908119895+1

211989611211989612+1(119909 119910)

+ 1205723

1198951198961111989612

119908119895+1

211989611+12119896

12+1(119909 119910)]

(16)

Mathematical Problems in Engineering 5

The value of 120601119869(119909 119910 119905119899) at 119905119899is denoted as 120601

119899 and

119865 [119905119899 119909 119910 120601

119869(119909 119910 119905

119899) 120601119869(10)

(119909 119910 119905119899) 120601119869(01)

(119909 119910 119905119899)

120601119869(20)

(119909 119910 119905119899) 120601119869(11)

(119909 119910 119905119899) 120601119869(02)

(119909 119910 119905119899)]

(17)

is denoted as 119865119899 And then a linear homotopy function can be

constructed as

120601119869(119909 119910 119905) = (1 minus 120576) 119865

119899+ 120576119865119899+1

(18)

It is easy to identify the homotopy parameter as

120576 (119905) =119905 minus 119905119899

119905119899+1

minus 119905119899

119905 isin [119905119899 119905119899+1

] there4 120576 isin [0 1] (19)

According to the perturbation theory the solution of (18) canbe expressed as the power series expansion of 120576 as follows

120601119869= 120601119869

0+ 120576120601119869

1+ 1205762120601119869

2+ sdot sdot sdot (20)

Substituting (20) into (18) and rearranging based on powersof 120576-terms we have

12057601206011198690= 119865119899

12057611206011198691= 119865119899+1

minus119865119899

(21)

According to HPM we obtain the wavelet coefficients1205721

11989511989611198962

(119905119899+1

) 120572211989511989611198962

(119905119899+1

) 120572311989511989611198962

(119905119899+1

) at 119905119899as follows

1205721

11989511989611198962

= 120601 (119909119895+1

21198961+1 119910119895+1

21198962

) minus 119868119895120601 (119909119895+1

21198961+1 119910119895+1

21198962

)

= 120601 (119909119895+1

21198961+1 119910119895+1

21198962

)

minus [

[

1

sum

11989601=0

1

sum

11989602=0

120601 (1199090

11989601

1199100

11989602

)1199080

1198960111989602

(119909119895+1

21198961+1 119910119895+1

21198962

)

+

119895minus1

sum

1198951=0

21198951

sum

11989611=0

21198951

sum

11989612=0

(1205721

11989511198961111989612

1199081198951+1

211989611+12119896

12

(119909119895+1

21198961+1 119910119895+1

21198962

)

+ 1205722

11989511198961111989612

1199081198951+1

211989611211989612+1(119909119895+1

21198961+1 119910119895+1

21198962

)

+ 1205723

11989511198961111989612

1199081198951+1

211989611+12119896

12+1(119909119895+1

21198961+1 119910119895+1

21198962

))]

]

1205722

11989511989611198962

= 120601 (119909119895+1

21198961

119910119895+1

21198962+1) minus 119868119895120601 (119909119895+1

21198961

119910119895+1

21198962+1)

= 120601 (119909119895+1

21198961

119910119895+1

21198962+1)

minus [

[

1

sum

11989601=0

1

sum

11989602=0

120601 (1199090

11989601

1199100

11989602

)1199080

1198960111989602

(119909119895+1

21198961

119910119895+1

21198962+1)

+

119895minus1

sum

1198951=0

21198951

sum

11989611=0

21198951

sum

11989612=0

(1205721

11989511198961111989612

1199081198951+1

211989611+12119896

12

(119909119895+1

21198961

119910119895+1

21198962+1)

+ 1205722

11989511198961111989612

1199081198951+1

211989611211989612+1(119909119895+1

21198961

119910119895+1

21198962+1)

+ 1205723

11989511198961111989612

1199081198951+1

211989611+12119896

12+1(119909119895+1

21198961

119910119895+1

21198962+1))]

]

6 Mathematical Problems in Engineering

1205723

11989511989611198962

= 120601 (119909119895+1

21198961+1 119910119895+1

21198962+1) minus 119868119895120601 (119909119895+1

21198961+1 119910119895+1

21198962+1)

= 120601 (119909119895+1

21198961+1 119910119895+1

21198962+1)

minus [

[

1

sum

11989601=0

1

sum

11989602=0

120601 (1199090

11989601

1199100

11989602

)1199080

1198960111989602

(119909119895+1

21198961+1 119910119895+1

21198962+1)

+

119895minus1

sum

1198951=0

21198951

sum

11989611=0

21198951

sum

11989612=0

(1205721

11989511198961111989612

1199081198951+1

211989611+12119896

12

(119909119895+1

21198961+1 119910119895+1

21198962+1)

+ 1205722

11989511198961111989612

1199081198951+1

211989611211989612+1(119909119895+1

21198961+1 119910119895+1

21198962+1)

+ 1205723

11989511198961111989612

1199081198951+1

211989611+12119896

12+1(119909119895+1

21198961+1 119910119895+1

21198962+1))]

]

(22)

Obviously the calculation time complexity of the wavelettransform coefficients 1205721

11989511989611198962

120572211989511989611198962

and 1205723

11989511989611198962

is 119874(4119869)which is decreased greatly than that in (8) which is119874((13)4

2119869minus1)

Substituting the wavelet transform efficient (22) into (16)we obtain120601119869(119909 119910 119905

119899+1)

= 120601119869(119909 119910 119905

119899)

+Δ119905

2[119865 (119905119899 119909 119910 120601

119869(119909 119910 119905

119899) 120601119869(10)

(119909 119910 119905119899)

120601119869(01)

(119909 119910 119905119899) 120601119869(20)

(119909 119910 119905119899)

120601119869(11)

(119909 119910 119905119899) 120601119869(02)

(119909 119910 119905119899))

+ 119865 (119905119899+1

119909 119910 120601119869

0(119909 119910 119905

119899+1)

120601119869(10)

0(119909 119910 119905

119899+1) 120601119869(01)

0(119909 119910 119905

119899+1)

120601119869(20)

0(119909 119910 119905

119899+1) 120601119869(11)

0(119909 119910 119905

119899+1)

120601119869(02)

0(119909 119910 119905

119899+1))]

(23)

and the derivative function120601119869(119898119899)

(119909 119910)

=

1

sum

11989601=0

1

sum

11989602=0

120601 (1199090

11989601

1199100

11989602

)1199080(119898119899)

1198960111989602

(119909 119910)

+

119869minus1

sum

119895=0

2119895minus1

sum

11989611=0

2119895minus1

sum

11989612=0

[1205721

1198951198961111989612

119908119895+1(119898119899)

211989611+12119896

12

(119909 119910)

+ 1205722

1198951198961111989612

119908119895+1(119898119899)

211989611211989612+1(119909 119910)

+ 1205723

1198951198961111989612

119908119895+1(119898119899)

211989611+12119896

12+1(119909 119910)]

(24)

Obviously the computation complexity is decreased greatlycomparing with (14)

31TheMultiscale InterpolationWavelet Approximation of theC-V Model There are many ways to solve partial differentialequations and the most typical method is the differencemethod This method uses the flat function to describeimage approximately the surface function But it is easyto cause artifacts phenomenon affecting the accuracy ofimage segmentation Wavelet function has both smooth andcompactly supported characteristics Besides performance ofmultiscale analysis can be used to construct the multiscaleadaptive interpolation operator for solving nonlinear partialdifferential equationsThe wavelet approximation of the levelset function and its derivative with respect to 119909 and 119910respectively can be expressed as follows

120601119869(119898119899)

(119909 119910 119905)

=

1

sum

11989601=0

1

sum

11989602=0

120601 (1199090

11989601

1199100

11989602

)1199080(119898119899)

1198960111989602

(119909 119910)

+

119869minus1

sum

119895=0

2119895minus1

sum

11989611=0

2119895minus1

sum

11989612=0

[1205721

1198951198961111989612

(119905) 119908119895+1(119898119899)

211989611+12119896

12

(119909 119910)

+ 1205722

1198951198961111989612

(119905) 119908119895+1(119898119899)

211989611211989612+1(119909 119910)

+ 1205723

1198951198961111989612

(119905) 119908119895+1(119898119899)

211989611+12119896

12+1(119909 119910)]

(25)

where 119908119895(119898119899)11989611198962

(119909 119910) is the wavelet function and its 119898- and119899-order derivative with respect to 119909 and 119910 respectively1205721

1198951198961111989612

(119905) 12057221198951198961111989612

(119905) and 12057231198951198961111989612

(119905) are wavelet transform

Mathematical Problems in Engineering 7

coefficients We convert (2) to wavelet multiscale discreteformat of the level set function by stead of (25)

119889120601119869(119909 119910 119905)

119889119905

= 120575120576(120601119869(119909 119910 119905))

times [] div(nabla120601119869(119909 119910 119905)

1003816100381610038161003816nabla120601119869 (119909 119910 119905)

1003816100381610038161003816

)minus1205821

10038161003816100381610038161198680 minus 11988811003816100381610038161003816

2

+1205822

10038161003816100381610038161198680 minus 11988821003816100381610038161003816

2

]

(26)

It is the most direct way for dynamic adaptation toonly retain the distribution points of corresponding waveletcoefficients that satisfy the condition

min (100381610038161003816100381610038161205721

1198951198961111989612

(119905)10038161003816100381610038161003816100381610038161003816100381610038161205722

1198951198961111989612

(119905)10038161003816100381610038161003816100381610038161003816100381610038161205723

1198951198961111989612

(119905)10038161003816100381610038161003816) ge 120576 (27)

Time domain numerical integration of partial differentialequations is an iterative process therefore some pointswhich are possible important next step need to be kept toenable the algorithm to track singularities of solutions Soadjacent points of distribution points also should be keptTheadjacent region can be delineated as follows

1003816100381610038161003816119904 minus 1198951003816100381610038161003816 le 119872

10038161003816100381610038161003816119909119904

119896minus 119909119895

119894

10038161003816100381610038161003816le 120576119909

10038161003816100381610038161003816119909119904

119896minus 119909119895

119894

10038161003816100381610038161003816le 120576119910

(28)

where 119904 119895 are numbers of different scale wavelet and 119896 119894119872 isin

119885 120576119909 120576119910are constant

32 Nonlinear Discrete Ordinary Differential EquationsBecause ] is a small parameter in ordinary differentialequation (26) value of ] div(nabla120601119869(119909 119910 119905)|nabla120601119869(119909 119910 119905)|) is lowEquation (26) can be converted into

119889120601119869(119909 119910 119905)

119889119905

= 120575120576(120601119869(119909 119910 119905))

times [] div(nabla120601119869(119909 119910 119905

0)

1003816100381610038161003816nabla120601119869 (119909 119910 119905

0)1003816100381610038161003816

) minus 1205821

10038161003816100381610038161198680 minus 11988811003816100381610038161003816

2

+ 1205822

10038161003816100381610038161198680 minus 11988821003816100381610038161003816

2

]

(29)

The solution of (29) is

120601119869(119909 119910 119905) = (

120576119898

2120587)

13

+ (119898

21205871205765)

minus13

(30)

where

119898 = ] div(nabla120601119869(119909 119910 119905

0)

1003816100381610038161003816nabla120601119869 (119909 119910 119905

0)1003816100381610038161003816

)

minus 1205821

10038161003816100381610038161198680 minus 11988811003816100381610038161003816

2

+ 1205822

10038161003816100381610038161198680 minus 11988821003816100381610038161003816

2

(31)

We can get the solution of ordinary differential equations(26) by iterative solution (30)

4 Experiment and Discussion ofOnion Infected Region Segmentation

Figure 1(a) is a 256 times 302 image of an onion infected bysour skin virus We noticed that the onion has a water-soaked appearance Compared with the background thegrayscale difference between the water-soaked appearanceand the healthy part is smaller So it is beneficial to compareperformance of the different algorithms Figure 1(b) is anideal segmentation results Segmentation target is infectedregions of onion however infected part is often not uniformIn the image the difference of gradient is less than 1 at upperleft part due to slight infection so the algorithm is difficult toprecise segmentation based on global threshold Thereforethe best one of the different segmentation algorithms candistinguish themajority of the infected regionwithout seriousoversegmentationThen it is easy to identify andmeasure theinfected portions by using a priori knowledge

41 Comparison among Different SegmentationMethods Thecommon image segmentation methods including water-shed algorithm Sobel operator and Canny edge detectionalgorithm Otsu algorithm and an effective and commonquad-tree decomposition algorithm were selected for com-parison Shannon wavelet was employed to construct thewavelet interpolation operator The representation of Shan-non wavelet is based upon approximating the Dirac deltafunction as a band-limited function and is given by

119908 (119909) =sin (120587119909)120587119909

(32)

Consider a one-dimensional function 119891(119909) 119909 isin [119886 119887] Adiscrete point sequence of the variable 119909 is defined as

119909119899= 119886 +

119887 minus 119886

2119895sdot 119899 119895 isin Z 119899 = 0 1 2 2

119895 (33)

and the corresponding discrete point sequence of the scalingfunction 120601(119909) can be defined as

119908119895119899(119909) = 119908

119895(119909 minus 119909

119899) =

sin (2119895120587 (119887 minus 119886)) (119909 minus 119909119899)

(2119895120587 (119887 minus 119886)) (119909 minus 119909119899)

(34)

The first and second order derivatives of 120601119895(119909 minus 119909

119899) at the

discrete point 119909119896are

1206011015840

119895(119909119896minus 119909119899) =

0 119896 = 119899

2119895 cos [120587 (119896 minus 119899)](119896 minus 119899) (119887 minus 119886)

119896 = 119899

12060110158401015840

119895(119909119896minus 119909119899) =

minus1205872

3((119887 minus 119886)2119895)2 119896 = 119899

minus2 cos [120587 (119896 minus 119899)]

((119887 minus 119886) 2119895)2

(119896 minus 119899)2 119896 = 119899

(35)

The corresponding 2-dimension weight function can be rep-resented as the tensor product form of the above equations

8 Mathematical Problems in Engineering

(a) Original image (b) Segmentation target

Figure 1 Burkholderia cepacia (ex Burkholder) infected onion and the target segmentation

(a) Grayscale image (b) Gradient image (c) OSTU

(d) Watershed 1 (e) Watershed 2 (f) Qtdecomp

(g) Sobel (h) Canny (i) Wavelet precise integration

Figure 2 Comparison of various segmentation methods

Mathematical Problems in Engineering 9

300

250

200

150

100

50

00 50 100 150 200 250 300

Figure 3 Adaptive wavelet collocation points on level set

The experimental procedure is described as follows

(1) convert the infested onion image to grayscale(Figure 2(a)) and solve for the gradient map(Figure 2(b))

(2) use the grayscale image to test the Sobel operatorCanny operator Qtdecomp algorithms and waveletprecise integration method Use the gradient mat totest the OSTU method and watershed algorithm

(3) applying the watershed method to segment the imagewhich has been processed by OSTU method in orderto avoid the oversegmentation from the watershedalgorithm

The segmentation results are shown in Figure 2 Thewatershed algorithm segmentation result shows serious over-segmentation (Figure 2(d)) and cannot recognize the infectedpart Although OSTUmethod separated part infected regionof the onion the partition boundary is discontinuity andis difficult to measure infection specific gravity To avoidoversegmentation OSTUwas overlapped with the watershedsegmentationThe result is shown in Figure 2(e) in which thepartition boundary is clear but is unable to distinguish thevirus infected part The Sobel operator recognition on thepart of the infection is also not good (Figure 2(g)) Cannyoperator and Qtdecomp algorithm identified the area ofinfection but the boundary points of segmentation regionare disorder and cannot be measuredThe precise integrationmethod presented in this paper can identify the infected areaclearly So it is helpful to onion evaluation and classification

In fact it is impossible to project image segmentationby a single algorithm The important reason that partialdifferential equations are effective for image segmentationis that the method integrated many image segmentationprinciples to the model of partial differential equations Inthis paper the C-V image segmentation model is a globalconvex optimization variational model which is establishedon image piecewise smooth (119888

1and 1198882are the average gray

values inside (Ω1) and outside (Ω

2) of the object contour

respectively) To ensure the accuracy of image segmentation

the curvature of the image the border gradient and levelset function evolution were taken into account in imagesegmentation It means that the global convex optimizationmodel of image segmentation has been built based onintegration of a variety of image segmentation theories andhas obvious advantages In addition the method of iterativesolution of the self-adaptive method can also be integratedinto the segmentation process to ensure the accuracy ofsegmentation method further However the speed of thealgorithm will be affected Therefore it is important to findefficient and accurate numerical method

42 Efficiency Comparison of Multiscale Adaptive WaveletNumerical Method and Difference Method The C-V modelwas used for 256 times 302 images segmentation and divideddifference method was used to disperse partial differentialequations So discrete 7312 (256 times 302) ordinary differentialequations are huge solvingworkload But the adaptivewaveletprecise integration method can reduce the scale to 9576equations It can improve solution efficiency greatly due toless workload and low memory demand Of course the useof adaptive wavelet precise integration method for solvingthe number of distribution points will dynamically changeas the solution process In addition as shown in Figure 3distribution points are relatively dense within the ellipsering and another location was sparse The evident grayscaledifference between the infected and the healthy parts leadto this special points distribution Furthermore distributionpoints also exhibit regular matrix form which result fromblock solving method of wavelet transform to improve theefficiency The matrix-like distribution is from the boundaryeffect among the different blocks The interval wavelet caneffectively reduce the range effect but it will also increase thecomputation work of the wavelet transform

In this paper difference method was tested in MATLABThewavelet interpolation operatorwas implementedwithVCprogramming and other parts with MATLAB programmingOn the same computer difference method takes 03 secondsthe adaptive wavelet precise integration method takes 018seconds The results also show that the wavelet transform

10 Mathematical Problems in Engineering

of the iterative process reduces the overall computationalefficiency of the algorithm

5 Conclusions

Shannon wavelet precise integration method is a new imagesegmentation method based on the C-V model which wasused to construct adaptive wavelet interpolation operator dueto multiscale characteristics of wavelet transform combinedwith the time precise integration technology The methodmakes full use of the multiscale characteristics and thehigh precise performance of precise integration methodCompared to the gradient method and wavelet transformmethod of image segmentation object boundary obtained byWPIM segmentation method is clear and closed comparedto the watershed method the WPIM method avoids over-and undersegmentation problems and is very suitable formeasurement of image segmentation such as onion qualityassessment

The adaptive interpolation operator in the Shannonwavelet precision integration method can reduce the amountof the collocation points and improve the calculation effi-ciency As the interpolation operator contains a wavelettransform process the corresponding algorithm needs to doa wavelet transform between each two iteration time stepsSo the cost of the wavelet transformation is an importantpart of the calculation amount of the algorithm Compactlysupported orthogonal wavelet function can be expected tosolve the problem

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors appreciate the funding support from NationalNatural Science Foundation of China (Award nos 41171184and 41171337) The software tools were provided by the Foodand Fiber Sensing Lab of University of Georgia and theComputer Center of China Agricultural University

References

[1] Y Chen Y Xia Y Bian and Z-P Zhong ldquoImage measurementof precision aluminum alloy forgingsrdquo Journal of PlasticityEngineering vol 17 no 6 pp 77ndash81 2010

[2] S-L Mei Q-S Lu S-W Zhang and L Jin ldquoAdaptive intervalwavelet precise integrationmethod for partial differential equa-tionsrdquo Applied Mathematics and Mechanics vol 26 no 3 pp364ndash371 2005

[3] H-H Yan ldquoAdaptive wavelet precise integration method fornonlinear black-scholes model based on variational iterationmethodrdquo Abstract and Applied Analysis vol 2013 Article ID735919 6 pages 2013

[4] S-L Pang ldquoWavelet numerical method for nonlinear randomsystemrdquo Transactions of the Chinese Society of AgriculturalMachinery vol 38 no 3 pp 168ndash170 2007

[5] Y Wang ldquoWavelet precise time-integration method for heatconduction equationrdquo Journal of Chongqing Institute of Technol-ogy vol 21 no 8 pp 130ndash132 2007

[6] L X Zhang Y Yang and S L Mei ldquoWavelet precise integrationmethod on image denoisingrdquoTransactions of the Chinese Societyof Agricultural Machinery vol 37 no 7 pp 109ndash112 2006

[7] W N Xu S L Mei P X Wang and Y Yang ldquoAdaptivewavelet precise integration method on remote sensing imagedenoisingrdquo Transactions of the Chinese Society of AgriculturalMachinery vol 42 no 4 pp 148ndash152 2011

[8] R-Y Xing ldquoWavelet-based homotopy analysismethod for non-linear matrix system and its application in burgers equationrdquoMathematical Problems in Engineering vol 2013 Article ID982810 7 pages 2013

[9] S-L Mei ldquoConstruction of target controllable image segmen-tation model based on homotopy perturbation technologyrdquoAbstract and Applied Analysis vol 2013 Article ID 131207 8pages 2013

[10] L Liu ldquoConstruction of interval shannon wavelet and itsapplication in solving nonlinear black-scholes equationrdquoMath-ematical Problems in Engineering vol 2014 Article ID 541023 8pages 2014

[11] C Cattani ldquoShannon wavelets theoryrdquo Mathematical Problemsin Engineering vol 2008 Article ID 164808 24 pages 2008

[12] C Cattani ldquoSecond order Shannon wavelet approximationof C2-functionsrdquo UPB Scientific Bulletin Series A AppliedMathematics and Physics vol 73 no 3 pp 73ndash84 2011

[13] C Cattani and L M S Ruiz ldquoDiscrete differential operators inmultidimensional haar wavelet spacesrdquo International Journal ofMathematics and Mathematical Sciences vol 2004 no 44 pp2347ndash2355 2004

[14] C Cattani A Ciancio and B Lods ldquoOn a mathematical modelof immune competitionrdquo Applied Mathematics Letters vol 19no 7 pp 678ndash683 2006

[15] C Cattani ldquoConnection coefficients of Shannon waveletsrdquoMathematical Modelling and Analysis vol 11 no 2 pp 117ndash1322006

[16] S-L Mei and D-H Zhu ldquoInterval shannon wavelet collocationmethod for fractional fokker-planck equationrdquo Advances inMathematical Physics vol 2013 Article ID 821820 12 pages2013

[17] L-W Liu ldquoInterval wavelet numerical method on fokker-planck equations for nonlinear random systemrdquo Advances inMathematical Physics vol 2013 Article ID 651357 7 pages 2013

[18] J-H He ldquoHomotopy perturbation method for bifurcation ofnonlinear problemsrdquo International Journal of Nonlinear Sciencesand Numerical Simulation vol 6 no 2 pp 207ndash208 2005

[19] J-H He ldquoSome asymptotic methods for strongly nonlinearequationsrdquo International Journal of Modern Physics B vol 20no 10 pp 1141ndash1199 2006

[20] J-H He ldquoVariational iteration method-Some recent resultsand new interpretationsrdquo Journal of Computational and AppliedMathematics vol 207 no 1 pp 3ndash17 2007

[21] S-L Mei C-J Du and S-W Zhang ldquoAsymptotic numericalmethod for multi-degree-of-freedom nonlinear dynamic sys-temsrdquo Chaos Solitons and Fractals vol 35 no 3 pp 536ndash5422008

[22] S-L Mei and S-W Zhang ldquoCoupling technique of variationaliteration and homotopy perturbation methods for nonlinearmatrix differential equationsrdquoComputers andMathematics withApplications vol 54 no 7-8 pp 1092ndash1100 2007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

120601(119909 119910 119905) and the corresponding derivative function can bediscretized as follows

120601119869(119898119899)

(119909 119910 119905)

=

1

sum

11989601=0

1

sum

11989602=0

120601 (1199090

11989601

1199100

11989602

)1199080(119898119899)

1198960111989602

(119909 119910)

+

119869minus1

sum

119895=0

2119895minus1

sum

11989611=0

2119895minus1

sum

11989612=0

[1205721

1198951198961111989612

(119905) 119908119895+1(119898119899)

211989611+12119896

12

(119909 119910)

+ 1205722

1198951198961111989612

(119905) 119908119895+1(119898119899)

211989611211989612+1(119909 119910)

+ 1205723

1198951198961111989612

(119905) 119908119895+1(119898119899)

211989611+12119896

12+1(119909 119910)]

(7)

where 119895 and 119869 are constants which denote the wavelet scalenumber and the maximum of the scale number respectively1205721

1198951198961111989612

12057221198951198961111989612

and 12057231198951198961111989612

are the wavelet coefficients atthe points (119909119895

1198961

119910119895

1198962

) According to the interpolation wavelettransform theory the wavelet coefficients can be written as

1205721

11989511989611198962

= 120601 (119909119895+12119896

1+1 119910119895+12119896

2

) minus 119868119895120601 (119909119895+12119896

1+1 119910119895+12119896

2

)

1205722

11989511989611198962

= 120601 (119909119895+12119896

1

119910119895+12119896

2+1) minus 119868119895120601 (119909119895+12119896

1

119910119895+12119896

2+1)

1205723

11989511989611198962

=120601 (119909119895+12119896

1+1 119910119895+12119896

2+1)minus119868119895120601 (119909119895+12119896

1+1 119910119895+12119896

2+1)

(8)

where 119868119895denotes the multilevel interpolation operator

In order to obtain the multilevel interpolation opera-tor it is necessary to express the wavelet coefficients1205721

11989511989611198962

1205722

11989511989611198962

1205723

11989511989611198962

as a weighted sum of 119906 in all of thecollocation points in the 119869-levelTherefore we should give thedefinition of the restriction operator as follows

119877119897119897119895119895

1198961119896211989811198982

= 1 119909

119897

1198961

= 119909119895

1198981

119910119897

1198962

= 119910119895

1198982

0 otherwise(9)

Using the restriction operator 119906(119909119895+121198961+1 119910119895+1

21198962

) 119906(119909119895+1

21198961

119910119895+1

21198962+1)

and 119906(119909119895+1

21198961+1 119910119895+1

21198962+1) can be rewritten as

120601 (119909119895+1

21198961+1 119910119895+1

21198962

) =

2119869

sum

1198991=0

2119869

sum

1198992=0

119877119895+1119895+1119869119869

21198961+12119896

211989911198992

120601 (119909119869

1198991

119910119869

1198992

)

120601 (119909119895+1

21198961

119910119895+1

21198962+1) =

2119869

sum

1198991=0

2119869

sum

1198992=0

119877119895+1119895+1119869119869

2119896121198962+111989911198992

120601 (119909119869

1198991

119910119869

1198992

)

120601 (119909119895+1

21198961+1 119910119895+1

21198962+1) =

2119869

sum

1198991=0

2119869

sum

1198992=0

119877119895+1119895+1119869119869

21198961+12119896

2+111989911198992

120601 (119909119869

1198991

119910119869

1198992

)

(10)

Introducing the extension operators 1198621 1198622 and 1198623 andsubstituting (10) into (8) the wavelet coefficients can berewritten as

1205721

11989511989611198962

=

2119869

sum

1198991=0

2119869

sum

1198992=0

119877119895+1119895+1119869119869

21198961+12119896

211989911198992

120601 (119909119869

1198991

119910119869

1198992

)

minus [

[

2119869

sum

1198991=0

2119869

sum

1198992=0

21198950

sum

11989601=0

21198950

sum

11989602=0

11987711989501198950119869119869

119896011198960211989911198992

120601 (119909119869

1198991

119910119869

1198992

)1199081198950

1198960111989602

(119909119895+1

21198961+1 119910119895+1

21198962

)

+

119895minus1

sum

1198951=1198950

2119869

sum

1198991=0

2119869

sum

1198992=0

21198951

sum

11989611=0

21198951

sum

11989612=0

(119862111989511198951119869119869

119896111198961211989911198992

1199081198951+1

211989611+12119896

12

(119909119895+1

21198961+1 119910119895+1

21198962

) 120601 (119909119869

1198991

119910119869

1198992

)

+ 119862211989511198951119869119869

119896111198961211989911198992

1199081198951+1

211989611211989612+1(119909119895+1

21198961+1 119910119895+1

21198962

) 119906 (119909119869

1198991

119910119869

1198992

)

+ 119862311989511198951119869119869

119896111198961211989911198992

1199081198951+1

211989611+12119896

12+1

(119909119895+1

21198961+1 119910119895+1

21198962

) 120601 (119909119869

1198991

119910119869

1198992

))]

]

=

2119869

sum

1198991=0

2119869

sum

1198992=0

1198621119895119895119869119869

1198961119896211989911198992

120601 (119909119869

1198991

119910119869

1198992

)

(11)

4 Mathematical Problems in Engineering

1205722

11989511989611198962

and 120572311989511989611198962

are similar to 120572111989511989611198962

From the aboveequation the extension operator can be obtained as

1198621119895119895119869119869

1198961119896211989911198992

= 119877119895+1119895+1119869119869

21198961+12119896

211989911198992

minus [

[

21198950

sum

11989601=0

21198950

sum

11989602=0

11987711989501198950119869119869

119896011198960211989911198992

120601 (119909119869

1198991

119910119869

1198992

)1199081198950

1198960111989602

(119909119895+1

21198961+1 119910119895+1

21198962

)

+

119895minus1

sum

1198951=1198950

2119869

sum

1198992=0

21198951

sum

11989611=0

(119862111989511198951119869119869

119896111198961211989911198992

1199081198951+1

211989611+12119896

12

(119909119895+1

21198961+1 119910119895+1

21198962

) 120601 (119909119869

1198991

119910119869

1198992

)

+ 119862211989511198951119869119869

119896111198961211989911198992

1199081198951+1

211989611211989612+1(119909119895+1

21198961+1 119910119895+1

21198962

) 120601 (119909119869

1198991

119910119869

1198992

)

+ 119862311989511198951119869119869

119896111198961211989911198992

1199081198951+1

211989611+12119896

12+1(119909119895+1

21198961+1 119910119895+1

21198962

) 120601 (119909119869

1198991

119910119869

1198992

))]

]

(12)

1198622 and1198623 can be obtained with the samemethodThereforethe calculation time complexity of the wavelet transformcoefficients 1205721

1198951198961111989612

12057221198951198961111989612

and 12057231198951198961111989612

is 119874((13)42119869minus1)Substituting 1205721

1198951198961111989612

12057221198951198961111989612

and 1205723

1198951198961111989612

1198621 1198622 and1198623 into (2) the multilevel wavelet interpolation operator canbe obtained as

11986811989911198992

(119909 119910)

=

21198950

sum

11989601=0

21198950

sum

11989602=0

11987711989501198950119869119869

119896011198960211989911198992

1199081198950

1198960111989602

(119909 119910)

+

119869minus1

sum

119895=1198950

2119895

sum

1198961=0

2119895

sum

1198962=0

(1198621119895119895119869119869

1198961119896211989911198992

119908119895+1

21198961+12119896

2

(119909 119910) 120601 (119909119869

1198991

119910119869

1198992

)

+ 1198622119895119895119869119869

1198961119896211989911198992

119908119895+1

2119896121198962+1(119909 119910) 120601 (119909

119869

1198991

119910119869

1198992

)

+1198623119895119895119869119869

1198961119896211989911198992

119908119895+1

21198961+12119896

2+1(119909 119910) 120601 (119909

119869

1198991

119910119869

1198992

))

(13)

Then (7) can be rewritten as

120601119869(119898119899)

(119909 119910 119905) =

2119869

sum

1198991

2119869

sum

1198992

11986811989911198992

(119909 119910) 120601 (119909119869

1198991

119910119869

1198992

) (14)

Substituting (14) into (5) themultilevel wavelet discretizationscheme of PERONA-MALIK model can be obtained

The purpose of constructing the multilevel wavelet collo-cation method is to decrease the amount of the collocationpoints and then improve the efficiency of the algorithm Butthe efficiency will be eliminated if the computation com-plexity of the multilevel wavelet interpolation operator is toohigh It is easy to understand that the interpolation waveletcoefficient is the error between the interpolation result andthe exact result at the same collocation point And so thewavelet coefficientmust be the function of the parameter 119905 In

other words the wavelet coefficient should vary with the timeparameter 119905 Then the interpolation operator can be viewedas a nonlinear problem HPM is efficient and effective tool tosolve nonlinear problem Aiming to improve the efficiency ofthe multilevel wavelet interpolation operators HPM wouldbe employed to construct a novel interpolation operator inthis section

For convenience 120601 and its derivative in (5) should berewritten as

120597120601

120597119905= 119865(119905 119909 119910 120601

120597120601

120597119909120597120601

1205971199101205972120601

12059711990921205972120601

1205971199091205971199101205972120601

1205971199102) (119905 gt 0)

120601 (119909 119910 0) = 1206010(119909 119910)

119889120601119869(119909 119910 119905)

119889119905= 119865 [119905 119909 119910 120601

119869(119909 119910 119905) 120601

119869(10)(119909 119910 119905)

120601119869(01)

(119909 119910 119905) 120601119869(20)

(119909 119910 119905)

120601119869(11)

(119909 119910 119905) 120601119869(02)

(119909 119910 119905)]

(15)

respectively where

120601119869(119909 119910 119905)

=

1

sum

11989601=0

1

sum

11989602=0

120601 (1199090

11989601

1199100

11989602

)1199080

1198960111989602

(119909 119910)

+

119869minus1

sum

119895=0

2119895minus1

sum

11989611=0

2119895minus1

sum

11989612=0

[1205721

1198951198961111989612

119908119895+1

211989611+12119896

12

(119909 119910)

+ 1205722

1198951198961111989612

119908119895+1

211989611211989612+1(119909 119910)

+ 1205723

1198951198961111989612

119908119895+1

211989611+12119896

12+1(119909 119910)]

(16)

Mathematical Problems in Engineering 5

The value of 120601119869(119909 119910 119905119899) at 119905119899is denoted as 120601

119899 and

119865 [119905119899 119909 119910 120601

119869(119909 119910 119905

119899) 120601119869(10)

(119909 119910 119905119899) 120601119869(01)

(119909 119910 119905119899)

120601119869(20)

(119909 119910 119905119899) 120601119869(11)

(119909 119910 119905119899) 120601119869(02)

(119909 119910 119905119899)]

(17)

is denoted as 119865119899 And then a linear homotopy function can be

constructed as

120601119869(119909 119910 119905) = (1 minus 120576) 119865

119899+ 120576119865119899+1

(18)

It is easy to identify the homotopy parameter as

120576 (119905) =119905 minus 119905119899

119905119899+1

minus 119905119899

119905 isin [119905119899 119905119899+1

] there4 120576 isin [0 1] (19)

According to the perturbation theory the solution of (18) canbe expressed as the power series expansion of 120576 as follows

120601119869= 120601119869

0+ 120576120601119869

1+ 1205762120601119869

2+ sdot sdot sdot (20)

Substituting (20) into (18) and rearranging based on powersof 120576-terms we have

12057601206011198690= 119865119899

12057611206011198691= 119865119899+1

minus119865119899

(21)

According to HPM we obtain the wavelet coefficients1205721

11989511989611198962

(119905119899+1

) 120572211989511989611198962

(119905119899+1

) 120572311989511989611198962

(119905119899+1

) at 119905119899as follows

1205721

11989511989611198962

= 120601 (119909119895+1

21198961+1 119910119895+1

21198962

) minus 119868119895120601 (119909119895+1

21198961+1 119910119895+1

21198962

)

= 120601 (119909119895+1

21198961+1 119910119895+1

21198962

)

minus [

[

1

sum

11989601=0

1

sum

11989602=0

120601 (1199090

11989601

1199100

11989602

)1199080

1198960111989602

(119909119895+1

21198961+1 119910119895+1

21198962

)

+

119895minus1

sum

1198951=0

21198951

sum

11989611=0

21198951

sum

11989612=0

(1205721

11989511198961111989612

1199081198951+1

211989611+12119896

12

(119909119895+1

21198961+1 119910119895+1

21198962

)

+ 1205722

11989511198961111989612

1199081198951+1

211989611211989612+1(119909119895+1

21198961+1 119910119895+1

21198962

)

+ 1205723

11989511198961111989612

1199081198951+1

211989611+12119896

12+1(119909119895+1

21198961+1 119910119895+1

21198962

))]

]

1205722

11989511989611198962

= 120601 (119909119895+1

21198961

119910119895+1

21198962+1) minus 119868119895120601 (119909119895+1

21198961

119910119895+1

21198962+1)

= 120601 (119909119895+1

21198961

119910119895+1

21198962+1)

minus [

[

1

sum

11989601=0

1

sum

11989602=0

120601 (1199090

11989601

1199100

11989602

)1199080

1198960111989602

(119909119895+1

21198961

119910119895+1

21198962+1)

+

119895minus1

sum

1198951=0

21198951

sum

11989611=0

21198951

sum

11989612=0

(1205721

11989511198961111989612

1199081198951+1

211989611+12119896

12

(119909119895+1

21198961

119910119895+1

21198962+1)

+ 1205722

11989511198961111989612

1199081198951+1

211989611211989612+1(119909119895+1

21198961

119910119895+1

21198962+1)

+ 1205723

11989511198961111989612

1199081198951+1

211989611+12119896

12+1(119909119895+1

21198961

119910119895+1

21198962+1))]

]

6 Mathematical Problems in Engineering

1205723

11989511989611198962

= 120601 (119909119895+1

21198961+1 119910119895+1

21198962+1) minus 119868119895120601 (119909119895+1

21198961+1 119910119895+1

21198962+1)

= 120601 (119909119895+1

21198961+1 119910119895+1

21198962+1)

minus [

[

1

sum

11989601=0

1

sum

11989602=0

120601 (1199090

11989601

1199100

11989602

)1199080

1198960111989602

(119909119895+1

21198961+1 119910119895+1

21198962+1)

+

119895minus1

sum

1198951=0

21198951

sum

11989611=0

21198951

sum

11989612=0

(1205721

11989511198961111989612

1199081198951+1

211989611+12119896

12

(119909119895+1

21198961+1 119910119895+1

21198962+1)

+ 1205722

11989511198961111989612

1199081198951+1

211989611211989612+1(119909119895+1

21198961+1 119910119895+1

21198962+1)

+ 1205723

11989511198961111989612

1199081198951+1

211989611+12119896

12+1(119909119895+1

21198961+1 119910119895+1

21198962+1))]

]

(22)

Obviously the calculation time complexity of the wavelettransform coefficients 1205721

11989511989611198962

120572211989511989611198962

and 1205723

11989511989611198962

is 119874(4119869)which is decreased greatly than that in (8) which is119874((13)4

2119869minus1)

Substituting the wavelet transform efficient (22) into (16)we obtain120601119869(119909 119910 119905

119899+1)

= 120601119869(119909 119910 119905

119899)

+Δ119905

2[119865 (119905119899 119909 119910 120601

119869(119909 119910 119905

119899) 120601119869(10)

(119909 119910 119905119899)

120601119869(01)

(119909 119910 119905119899) 120601119869(20)

(119909 119910 119905119899)

120601119869(11)

(119909 119910 119905119899) 120601119869(02)

(119909 119910 119905119899))

+ 119865 (119905119899+1

119909 119910 120601119869

0(119909 119910 119905

119899+1)

120601119869(10)

0(119909 119910 119905

119899+1) 120601119869(01)

0(119909 119910 119905

119899+1)

120601119869(20)

0(119909 119910 119905

119899+1) 120601119869(11)

0(119909 119910 119905

119899+1)

120601119869(02)

0(119909 119910 119905

119899+1))]

(23)

and the derivative function120601119869(119898119899)

(119909 119910)

=

1

sum

11989601=0

1

sum

11989602=0

120601 (1199090

11989601

1199100

11989602

)1199080(119898119899)

1198960111989602

(119909 119910)

+

119869minus1

sum

119895=0

2119895minus1

sum

11989611=0

2119895minus1

sum

11989612=0

[1205721

1198951198961111989612

119908119895+1(119898119899)

211989611+12119896

12

(119909 119910)

+ 1205722

1198951198961111989612

119908119895+1(119898119899)

211989611211989612+1(119909 119910)

+ 1205723

1198951198961111989612

119908119895+1(119898119899)

211989611+12119896

12+1(119909 119910)]

(24)

Obviously the computation complexity is decreased greatlycomparing with (14)

31TheMultiscale InterpolationWavelet Approximation of theC-V Model There are many ways to solve partial differentialequations and the most typical method is the differencemethod This method uses the flat function to describeimage approximately the surface function But it is easyto cause artifacts phenomenon affecting the accuracy ofimage segmentation Wavelet function has both smooth andcompactly supported characteristics Besides performance ofmultiscale analysis can be used to construct the multiscaleadaptive interpolation operator for solving nonlinear partialdifferential equationsThe wavelet approximation of the levelset function and its derivative with respect to 119909 and 119910respectively can be expressed as follows

120601119869(119898119899)

(119909 119910 119905)

=

1

sum

11989601=0

1

sum

11989602=0

120601 (1199090

11989601

1199100

11989602

)1199080(119898119899)

1198960111989602

(119909 119910)

+

119869minus1

sum

119895=0

2119895minus1

sum

11989611=0

2119895minus1

sum

11989612=0

[1205721

1198951198961111989612

(119905) 119908119895+1(119898119899)

211989611+12119896

12

(119909 119910)

+ 1205722

1198951198961111989612

(119905) 119908119895+1(119898119899)

211989611211989612+1(119909 119910)

+ 1205723

1198951198961111989612

(119905) 119908119895+1(119898119899)

211989611+12119896

12+1(119909 119910)]

(25)

where 119908119895(119898119899)11989611198962

(119909 119910) is the wavelet function and its 119898- and119899-order derivative with respect to 119909 and 119910 respectively1205721

1198951198961111989612

(119905) 12057221198951198961111989612

(119905) and 12057231198951198961111989612

(119905) are wavelet transform

Mathematical Problems in Engineering 7

coefficients We convert (2) to wavelet multiscale discreteformat of the level set function by stead of (25)

119889120601119869(119909 119910 119905)

119889119905

= 120575120576(120601119869(119909 119910 119905))

times [] div(nabla120601119869(119909 119910 119905)

1003816100381610038161003816nabla120601119869 (119909 119910 119905)

1003816100381610038161003816

)minus1205821

10038161003816100381610038161198680 minus 11988811003816100381610038161003816

2

+1205822

10038161003816100381610038161198680 minus 11988821003816100381610038161003816

2

]

(26)

It is the most direct way for dynamic adaptation toonly retain the distribution points of corresponding waveletcoefficients that satisfy the condition

min (100381610038161003816100381610038161205721

1198951198961111989612

(119905)10038161003816100381610038161003816100381610038161003816100381610038161205722

1198951198961111989612

(119905)10038161003816100381610038161003816100381610038161003816100381610038161205723

1198951198961111989612

(119905)10038161003816100381610038161003816) ge 120576 (27)

Time domain numerical integration of partial differentialequations is an iterative process therefore some pointswhich are possible important next step need to be kept toenable the algorithm to track singularities of solutions Soadjacent points of distribution points also should be keptTheadjacent region can be delineated as follows

1003816100381610038161003816119904 minus 1198951003816100381610038161003816 le 119872

10038161003816100381610038161003816119909119904

119896minus 119909119895

119894

10038161003816100381610038161003816le 120576119909

10038161003816100381610038161003816119909119904

119896minus 119909119895

119894

10038161003816100381610038161003816le 120576119910

(28)

where 119904 119895 are numbers of different scale wavelet and 119896 119894119872 isin

119885 120576119909 120576119910are constant

32 Nonlinear Discrete Ordinary Differential EquationsBecause ] is a small parameter in ordinary differentialequation (26) value of ] div(nabla120601119869(119909 119910 119905)|nabla120601119869(119909 119910 119905)|) is lowEquation (26) can be converted into

119889120601119869(119909 119910 119905)

119889119905

= 120575120576(120601119869(119909 119910 119905))

times [] div(nabla120601119869(119909 119910 119905

0)

1003816100381610038161003816nabla120601119869 (119909 119910 119905

0)1003816100381610038161003816

) minus 1205821

10038161003816100381610038161198680 minus 11988811003816100381610038161003816

2

+ 1205822

10038161003816100381610038161198680 minus 11988821003816100381610038161003816

2

]

(29)

The solution of (29) is

120601119869(119909 119910 119905) = (

120576119898

2120587)

13

+ (119898

21205871205765)

minus13

(30)

where

119898 = ] div(nabla120601119869(119909 119910 119905

0)

1003816100381610038161003816nabla120601119869 (119909 119910 119905

0)1003816100381610038161003816

)

minus 1205821

10038161003816100381610038161198680 minus 11988811003816100381610038161003816

2

+ 1205822

10038161003816100381610038161198680 minus 11988821003816100381610038161003816

2

(31)

We can get the solution of ordinary differential equations(26) by iterative solution (30)

4 Experiment and Discussion ofOnion Infected Region Segmentation

Figure 1(a) is a 256 times 302 image of an onion infected bysour skin virus We noticed that the onion has a water-soaked appearance Compared with the background thegrayscale difference between the water-soaked appearanceand the healthy part is smaller So it is beneficial to compareperformance of the different algorithms Figure 1(b) is anideal segmentation results Segmentation target is infectedregions of onion however infected part is often not uniformIn the image the difference of gradient is less than 1 at upperleft part due to slight infection so the algorithm is difficult toprecise segmentation based on global threshold Thereforethe best one of the different segmentation algorithms candistinguish themajority of the infected regionwithout seriousoversegmentationThen it is easy to identify andmeasure theinfected portions by using a priori knowledge

41 Comparison among Different SegmentationMethods Thecommon image segmentation methods including water-shed algorithm Sobel operator and Canny edge detectionalgorithm Otsu algorithm and an effective and commonquad-tree decomposition algorithm were selected for com-parison Shannon wavelet was employed to construct thewavelet interpolation operator The representation of Shan-non wavelet is based upon approximating the Dirac deltafunction as a band-limited function and is given by

119908 (119909) =sin (120587119909)120587119909

(32)

Consider a one-dimensional function 119891(119909) 119909 isin [119886 119887] Adiscrete point sequence of the variable 119909 is defined as

119909119899= 119886 +

119887 minus 119886

2119895sdot 119899 119895 isin Z 119899 = 0 1 2 2

119895 (33)

and the corresponding discrete point sequence of the scalingfunction 120601(119909) can be defined as

119908119895119899(119909) = 119908

119895(119909 minus 119909

119899) =

sin (2119895120587 (119887 minus 119886)) (119909 minus 119909119899)

(2119895120587 (119887 minus 119886)) (119909 minus 119909119899)

(34)

The first and second order derivatives of 120601119895(119909 minus 119909

119899) at the

discrete point 119909119896are

1206011015840

119895(119909119896minus 119909119899) =

0 119896 = 119899

2119895 cos [120587 (119896 minus 119899)](119896 minus 119899) (119887 minus 119886)

119896 = 119899

12060110158401015840

119895(119909119896minus 119909119899) =

minus1205872

3((119887 minus 119886)2119895)2 119896 = 119899

minus2 cos [120587 (119896 minus 119899)]

((119887 minus 119886) 2119895)2

(119896 minus 119899)2 119896 = 119899

(35)

The corresponding 2-dimension weight function can be rep-resented as the tensor product form of the above equations

8 Mathematical Problems in Engineering

(a) Original image (b) Segmentation target

Figure 1 Burkholderia cepacia (ex Burkholder) infected onion and the target segmentation

(a) Grayscale image (b) Gradient image (c) OSTU

(d) Watershed 1 (e) Watershed 2 (f) Qtdecomp

(g) Sobel (h) Canny (i) Wavelet precise integration

Figure 2 Comparison of various segmentation methods

Mathematical Problems in Engineering 9

300

250

200

150

100

50

00 50 100 150 200 250 300

Figure 3 Adaptive wavelet collocation points on level set

The experimental procedure is described as follows

(1) convert the infested onion image to grayscale(Figure 2(a)) and solve for the gradient map(Figure 2(b))

(2) use the grayscale image to test the Sobel operatorCanny operator Qtdecomp algorithms and waveletprecise integration method Use the gradient mat totest the OSTU method and watershed algorithm

(3) applying the watershed method to segment the imagewhich has been processed by OSTU method in orderto avoid the oversegmentation from the watershedalgorithm

The segmentation results are shown in Figure 2 Thewatershed algorithm segmentation result shows serious over-segmentation (Figure 2(d)) and cannot recognize the infectedpart Although OSTUmethod separated part infected regionof the onion the partition boundary is discontinuity andis difficult to measure infection specific gravity To avoidoversegmentation OSTUwas overlapped with the watershedsegmentationThe result is shown in Figure 2(e) in which thepartition boundary is clear but is unable to distinguish thevirus infected part The Sobel operator recognition on thepart of the infection is also not good (Figure 2(g)) Cannyoperator and Qtdecomp algorithm identified the area ofinfection but the boundary points of segmentation regionare disorder and cannot be measuredThe precise integrationmethod presented in this paper can identify the infected areaclearly So it is helpful to onion evaluation and classification

In fact it is impossible to project image segmentationby a single algorithm The important reason that partialdifferential equations are effective for image segmentationis that the method integrated many image segmentationprinciples to the model of partial differential equations Inthis paper the C-V image segmentation model is a globalconvex optimization variational model which is establishedon image piecewise smooth (119888

1and 1198882are the average gray

values inside (Ω1) and outside (Ω

2) of the object contour

respectively) To ensure the accuracy of image segmentation

the curvature of the image the border gradient and levelset function evolution were taken into account in imagesegmentation It means that the global convex optimizationmodel of image segmentation has been built based onintegration of a variety of image segmentation theories andhas obvious advantages In addition the method of iterativesolution of the self-adaptive method can also be integratedinto the segmentation process to ensure the accuracy ofsegmentation method further However the speed of thealgorithm will be affected Therefore it is important to findefficient and accurate numerical method

42 Efficiency Comparison of Multiscale Adaptive WaveletNumerical Method and Difference Method The C-V modelwas used for 256 times 302 images segmentation and divideddifference method was used to disperse partial differentialequations So discrete 7312 (256 times 302) ordinary differentialequations are huge solvingworkload But the adaptivewaveletprecise integration method can reduce the scale to 9576equations It can improve solution efficiency greatly due toless workload and low memory demand Of course the useof adaptive wavelet precise integration method for solvingthe number of distribution points will dynamically changeas the solution process In addition as shown in Figure 3distribution points are relatively dense within the ellipsering and another location was sparse The evident grayscaledifference between the infected and the healthy parts leadto this special points distribution Furthermore distributionpoints also exhibit regular matrix form which result fromblock solving method of wavelet transform to improve theefficiency The matrix-like distribution is from the boundaryeffect among the different blocks The interval wavelet caneffectively reduce the range effect but it will also increase thecomputation work of the wavelet transform

In this paper difference method was tested in MATLABThewavelet interpolation operatorwas implementedwithVCprogramming and other parts with MATLAB programmingOn the same computer difference method takes 03 secondsthe adaptive wavelet precise integration method takes 018seconds The results also show that the wavelet transform

10 Mathematical Problems in Engineering

of the iterative process reduces the overall computationalefficiency of the algorithm

5 Conclusions

Shannon wavelet precise integration method is a new imagesegmentation method based on the C-V model which wasused to construct adaptive wavelet interpolation operator dueto multiscale characteristics of wavelet transform combinedwith the time precise integration technology The methodmakes full use of the multiscale characteristics and thehigh precise performance of precise integration methodCompared to the gradient method and wavelet transformmethod of image segmentation object boundary obtained byWPIM segmentation method is clear and closed comparedto the watershed method the WPIM method avoids over-and undersegmentation problems and is very suitable formeasurement of image segmentation such as onion qualityassessment

The adaptive interpolation operator in the Shannonwavelet precision integration method can reduce the amountof the collocation points and improve the calculation effi-ciency As the interpolation operator contains a wavelettransform process the corresponding algorithm needs to doa wavelet transform between each two iteration time stepsSo the cost of the wavelet transformation is an importantpart of the calculation amount of the algorithm Compactlysupported orthogonal wavelet function can be expected tosolve the problem

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors appreciate the funding support from NationalNatural Science Foundation of China (Award nos 41171184and 41171337) The software tools were provided by the Foodand Fiber Sensing Lab of University of Georgia and theComputer Center of China Agricultural University

References

[1] Y Chen Y Xia Y Bian and Z-P Zhong ldquoImage measurementof precision aluminum alloy forgingsrdquo Journal of PlasticityEngineering vol 17 no 6 pp 77ndash81 2010

[2] S-L Mei Q-S Lu S-W Zhang and L Jin ldquoAdaptive intervalwavelet precise integrationmethod for partial differential equa-tionsrdquo Applied Mathematics and Mechanics vol 26 no 3 pp364ndash371 2005

[3] H-H Yan ldquoAdaptive wavelet precise integration method fornonlinear black-scholes model based on variational iterationmethodrdquo Abstract and Applied Analysis vol 2013 Article ID735919 6 pages 2013

[4] S-L Pang ldquoWavelet numerical method for nonlinear randomsystemrdquo Transactions of the Chinese Society of AgriculturalMachinery vol 38 no 3 pp 168ndash170 2007

[5] Y Wang ldquoWavelet precise time-integration method for heatconduction equationrdquo Journal of Chongqing Institute of Technol-ogy vol 21 no 8 pp 130ndash132 2007

[6] L X Zhang Y Yang and S L Mei ldquoWavelet precise integrationmethod on image denoisingrdquoTransactions of the Chinese Societyof Agricultural Machinery vol 37 no 7 pp 109ndash112 2006

[7] W N Xu S L Mei P X Wang and Y Yang ldquoAdaptivewavelet precise integration method on remote sensing imagedenoisingrdquo Transactions of the Chinese Society of AgriculturalMachinery vol 42 no 4 pp 148ndash152 2011

[8] R-Y Xing ldquoWavelet-based homotopy analysismethod for non-linear matrix system and its application in burgers equationrdquoMathematical Problems in Engineering vol 2013 Article ID982810 7 pages 2013

[9] S-L Mei ldquoConstruction of target controllable image segmen-tation model based on homotopy perturbation technologyrdquoAbstract and Applied Analysis vol 2013 Article ID 131207 8pages 2013

[10] L Liu ldquoConstruction of interval shannon wavelet and itsapplication in solving nonlinear black-scholes equationrdquoMath-ematical Problems in Engineering vol 2014 Article ID 541023 8pages 2014

[11] C Cattani ldquoShannon wavelets theoryrdquo Mathematical Problemsin Engineering vol 2008 Article ID 164808 24 pages 2008

[12] C Cattani ldquoSecond order Shannon wavelet approximationof C2-functionsrdquo UPB Scientific Bulletin Series A AppliedMathematics and Physics vol 73 no 3 pp 73ndash84 2011

[13] C Cattani and L M S Ruiz ldquoDiscrete differential operators inmultidimensional haar wavelet spacesrdquo International Journal ofMathematics and Mathematical Sciences vol 2004 no 44 pp2347ndash2355 2004

[14] C Cattani A Ciancio and B Lods ldquoOn a mathematical modelof immune competitionrdquo Applied Mathematics Letters vol 19no 7 pp 678ndash683 2006

[15] C Cattani ldquoConnection coefficients of Shannon waveletsrdquoMathematical Modelling and Analysis vol 11 no 2 pp 117ndash1322006

[16] S-L Mei and D-H Zhu ldquoInterval shannon wavelet collocationmethod for fractional fokker-planck equationrdquo Advances inMathematical Physics vol 2013 Article ID 821820 12 pages2013

[17] L-W Liu ldquoInterval wavelet numerical method on fokker-planck equations for nonlinear random systemrdquo Advances inMathematical Physics vol 2013 Article ID 651357 7 pages 2013

[18] J-H He ldquoHomotopy perturbation method for bifurcation ofnonlinear problemsrdquo International Journal of Nonlinear Sciencesand Numerical Simulation vol 6 no 2 pp 207ndash208 2005

[19] J-H He ldquoSome asymptotic methods for strongly nonlinearequationsrdquo International Journal of Modern Physics B vol 20no 10 pp 1141ndash1199 2006

[20] J-H He ldquoVariational iteration method-Some recent resultsand new interpretationsrdquo Journal of Computational and AppliedMathematics vol 207 no 1 pp 3ndash17 2007

[21] S-L Mei C-J Du and S-W Zhang ldquoAsymptotic numericalmethod for multi-degree-of-freedom nonlinear dynamic sys-temsrdquo Chaos Solitons and Fractals vol 35 no 3 pp 536ndash5422008

[22] S-L Mei and S-W Zhang ldquoCoupling technique of variationaliteration and homotopy perturbation methods for nonlinearmatrix differential equationsrdquoComputers andMathematics withApplications vol 54 no 7-8 pp 1092ndash1100 2007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

1205722

11989511989611198962

and 120572311989511989611198962

are similar to 120572111989511989611198962

From the aboveequation the extension operator can be obtained as

1198621119895119895119869119869

1198961119896211989911198992

= 119877119895+1119895+1119869119869

21198961+12119896

211989911198992

minus [

[

21198950

sum

11989601=0

21198950

sum

11989602=0

11987711989501198950119869119869

119896011198960211989911198992

120601 (119909119869

1198991

119910119869

1198992

)1199081198950

1198960111989602

(119909119895+1

21198961+1 119910119895+1

21198962

)

+

119895minus1

sum

1198951=1198950

2119869

sum

1198992=0

21198951

sum

11989611=0

(119862111989511198951119869119869

119896111198961211989911198992

1199081198951+1

211989611+12119896

12

(119909119895+1

21198961+1 119910119895+1

21198962

) 120601 (119909119869

1198991

119910119869

1198992

)

+ 119862211989511198951119869119869

119896111198961211989911198992

1199081198951+1

211989611211989612+1(119909119895+1

21198961+1 119910119895+1

21198962

) 120601 (119909119869

1198991

119910119869

1198992

)

+ 119862311989511198951119869119869

119896111198961211989911198992

1199081198951+1

211989611+12119896

12+1(119909119895+1

21198961+1 119910119895+1

21198962

) 120601 (119909119869

1198991

119910119869

1198992

))]

]

(12)

1198622 and1198623 can be obtained with the samemethodThereforethe calculation time complexity of the wavelet transformcoefficients 1205721

1198951198961111989612

12057221198951198961111989612

and 12057231198951198961111989612

is 119874((13)42119869minus1)Substituting 1205721

1198951198961111989612

12057221198951198961111989612

and 1205723

1198951198961111989612

1198621 1198622 and1198623 into (2) the multilevel wavelet interpolation operator canbe obtained as

11986811989911198992

(119909 119910)

=

21198950

sum

11989601=0

21198950

sum

11989602=0

11987711989501198950119869119869

119896011198960211989911198992

1199081198950

1198960111989602

(119909 119910)

+

119869minus1

sum

119895=1198950

2119895

sum

1198961=0

2119895

sum

1198962=0

(1198621119895119895119869119869

1198961119896211989911198992

119908119895+1

21198961+12119896

2

(119909 119910) 120601 (119909119869

1198991

119910119869

1198992

)

+ 1198622119895119895119869119869

1198961119896211989911198992

119908119895+1

2119896121198962+1(119909 119910) 120601 (119909

119869

1198991

119910119869

1198992

)

+1198623119895119895119869119869

1198961119896211989911198992

119908119895+1

21198961+12119896

2+1(119909 119910) 120601 (119909

119869

1198991

119910119869

1198992

))

(13)

Then (7) can be rewritten as

120601119869(119898119899)

(119909 119910 119905) =

2119869

sum

1198991

2119869

sum

1198992

11986811989911198992

(119909 119910) 120601 (119909119869

1198991

119910119869

1198992

) (14)

Substituting (14) into (5) themultilevel wavelet discretizationscheme of PERONA-MALIK model can be obtained

The purpose of constructing the multilevel wavelet collo-cation method is to decrease the amount of the collocationpoints and then improve the efficiency of the algorithm Butthe efficiency will be eliminated if the computation com-plexity of the multilevel wavelet interpolation operator is toohigh It is easy to understand that the interpolation waveletcoefficient is the error between the interpolation result andthe exact result at the same collocation point And so thewavelet coefficientmust be the function of the parameter 119905 In

other words the wavelet coefficient should vary with the timeparameter 119905 Then the interpolation operator can be viewedas a nonlinear problem HPM is efficient and effective tool tosolve nonlinear problem Aiming to improve the efficiency ofthe multilevel wavelet interpolation operators HPM wouldbe employed to construct a novel interpolation operator inthis section

For convenience 120601 and its derivative in (5) should berewritten as

120597120601

120597119905= 119865(119905 119909 119910 120601

120597120601

120597119909120597120601

1205971199101205972120601

12059711990921205972120601

1205971199091205971199101205972120601

1205971199102) (119905 gt 0)

120601 (119909 119910 0) = 1206010(119909 119910)

119889120601119869(119909 119910 119905)

119889119905= 119865 [119905 119909 119910 120601

119869(119909 119910 119905) 120601

119869(10)(119909 119910 119905)

120601119869(01)

(119909 119910 119905) 120601119869(20)

(119909 119910 119905)

120601119869(11)

(119909 119910 119905) 120601119869(02)

(119909 119910 119905)]

(15)

respectively where

120601119869(119909 119910 119905)

=

1

sum

11989601=0

1

sum

11989602=0

120601 (1199090

11989601

1199100

11989602

)1199080

1198960111989602

(119909 119910)

+

119869minus1

sum

119895=0

2119895minus1

sum

11989611=0

2119895minus1

sum

11989612=0

[1205721

1198951198961111989612

119908119895+1

211989611+12119896

12

(119909 119910)

+ 1205722

1198951198961111989612

119908119895+1

211989611211989612+1(119909 119910)

+ 1205723

1198951198961111989612

119908119895+1

211989611+12119896

12+1(119909 119910)]

(16)

Mathematical Problems in Engineering 5

The value of 120601119869(119909 119910 119905119899) at 119905119899is denoted as 120601

119899 and

119865 [119905119899 119909 119910 120601

119869(119909 119910 119905

119899) 120601119869(10)

(119909 119910 119905119899) 120601119869(01)

(119909 119910 119905119899)

120601119869(20)

(119909 119910 119905119899) 120601119869(11)

(119909 119910 119905119899) 120601119869(02)

(119909 119910 119905119899)]

(17)

is denoted as 119865119899 And then a linear homotopy function can be

constructed as

120601119869(119909 119910 119905) = (1 minus 120576) 119865

119899+ 120576119865119899+1

(18)

It is easy to identify the homotopy parameter as

120576 (119905) =119905 minus 119905119899

119905119899+1

minus 119905119899

119905 isin [119905119899 119905119899+1

] there4 120576 isin [0 1] (19)

According to the perturbation theory the solution of (18) canbe expressed as the power series expansion of 120576 as follows

120601119869= 120601119869

0+ 120576120601119869

1+ 1205762120601119869

2+ sdot sdot sdot (20)

Substituting (20) into (18) and rearranging based on powersof 120576-terms we have

12057601206011198690= 119865119899

12057611206011198691= 119865119899+1

minus119865119899

(21)

According to HPM we obtain the wavelet coefficients1205721

11989511989611198962

(119905119899+1

) 120572211989511989611198962

(119905119899+1

) 120572311989511989611198962

(119905119899+1

) at 119905119899as follows

1205721

11989511989611198962

= 120601 (119909119895+1

21198961+1 119910119895+1

21198962

) minus 119868119895120601 (119909119895+1

21198961+1 119910119895+1

21198962

)

= 120601 (119909119895+1

21198961+1 119910119895+1

21198962

)

minus [

[

1

sum

11989601=0

1

sum

11989602=0

120601 (1199090

11989601

1199100

11989602

)1199080

1198960111989602

(119909119895+1

21198961+1 119910119895+1

21198962

)

+

119895minus1

sum

1198951=0

21198951

sum

11989611=0

21198951

sum

11989612=0

(1205721

11989511198961111989612

1199081198951+1

211989611+12119896

12

(119909119895+1

21198961+1 119910119895+1

21198962

)

+ 1205722

11989511198961111989612

1199081198951+1

211989611211989612+1(119909119895+1

21198961+1 119910119895+1

21198962

)

+ 1205723

11989511198961111989612

1199081198951+1

211989611+12119896

12+1(119909119895+1

21198961+1 119910119895+1

21198962

))]

]

1205722

11989511989611198962

= 120601 (119909119895+1

21198961

119910119895+1

21198962+1) minus 119868119895120601 (119909119895+1

21198961

119910119895+1

21198962+1)

= 120601 (119909119895+1

21198961

119910119895+1

21198962+1)

minus [

[

1

sum

11989601=0

1

sum

11989602=0

120601 (1199090

11989601

1199100

11989602

)1199080

1198960111989602

(119909119895+1

21198961

119910119895+1

21198962+1)

+

119895minus1

sum

1198951=0

21198951

sum

11989611=0

21198951

sum

11989612=0

(1205721

11989511198961111989612

1199081198951+1

211989611+12119896

12

(119909119895+1

21198961

119910119895+1

21198962+1)

+ 1205722

11989511198961111989612

1199081198951+1

211989611211989612+1(119909119895+1

21198961

119910119895+1

21198962+1)

+ 1205723

11989511198961111989612

1199081198951+1

211989611+12119896

12+1(119909119895+1

21198961

119910119895+1

21198962+1))]

]

6 Mathematical Problems in Engineering

1205723

11989511989611198962

= 120601 (119909119895+1

21198961+1 119910119895+1

21198962+1) minus 119868119895120601 (119909119895+1

21198961+1 119910119895+1

21198962+1)

= 120601 (119909119895+1

21198961+1 119910119895+1

21198962+1)

minus [

[

1

sum

11989601=0

1

sum

11989602=0

120601 (1199090

11989601

1199100

11989602

)1199080

1198960111989602

(119909119895+1

21198961+1 119910119895+1

21198962+1)

+

119895minus1

sum

1198951=0

21198951

sum

11989611=0

21198951

sum

11989612=0

(1205721

11989511198961111989612

1199081198951+1

211989611+12119896

12

(119909119895+1

21198961+1 119910119895+1

21198962+1)

+ 1205722

11989511198961111989612

1199081198951+1

211989611211989612+1(119909119895+1

21198961+1 119910119895+1

21198962+1)

+ 1205723

11989511198961111989612

1199081198951+1

211989611+12119896

12+1(119909119895+1

21198961+1 119910119895+1

21198962+1))]

]

(22)

Obviously the calculation time complexity of the wavelettransform coefficients 1205721

11989511989611198962

120572211989511989611198962

and 1205723

11989511989611198962

is 119874(4119869)which is decreased greatly than that in (8) which is119874((13)4

2119869minus1)

Substituting the wavelet transform efficient (22) into (16)we obtain120601119869(119909 119910 119905

119899+1)

= 120601119869(119909 119910 119905

119899)

+Δ119905

2[119865 (119905119899 119909 119910 120601

119869(119909 119910 119905

119899) 120601119869(10)

(119909 119910 119905119899)

120601119869(01)

(119909 119910 119905119899) 120601119869(20)

(119909 119910 119905119899)

120601119869(11)

(119909 119910 119905119899) 120601119869(02)

(119909 119910 119905119899))

+ 119865 (119905119899+1

119909 119910 120601119869

0(119909 119910 119905

119899+1)

120601119869(10)

0(119909 119910 119905

119899+1) 120601119869(01)

0(119909 119910 119905

119899+1)

120601119869(20)

0(119909 119910 119905

119899+1) 120601119869(11)

0(119909 119910 119905

119899+1)

120601119869(02)

0(119909 119910 119905

119899+1))]

(23)

and the derivative function120601119869(119898119899)

(119909 119910)

=

1

sum

11989601=0

1

sum

11989602=0

120601 (1199090

11989601

1199100

11989602

)1199080(119898119899)

1198960111989602

(119909 119910)

+

119869minus1

sum

119895=0

2119895minus1

sum

11989611=0

2119895minus1

sum

11989612=0

[1205721

1198951198961111989612

119908119895+1(119898119899)

211989611+12119896

12

(119909 119910)

+ 1205722

1198951198961111989612

119908119895+1(119898119899)

211989611211989612+1(119909 119910)

+ 1205723

1198951198961111989612

119908119895+1(119898119899)

211989611+12119896

12+1(119909 119910)]

(24)

Obviously the computation complexity is decreased greatlycomparing with (14)

31TheMultiscale InterpolationWavelet Approximation of theC-V Model There are many ways to solve partial differentialequations and the most typical method is the differencemethod This method uses the flat function to describeimage approximately the surface function But it is easyto cause artifacts phenomenon affecting the accuracy ofimage segmentation Wavelet function has both smooth andcompactly supported characteristics Besides performance ofmultiscale analysis can be used to construct the multiscaleadaptive interpolation operator for solving nonlinear partialdifferential equationsThe wavelet approximation of the levelset function and its derivative with respect to 119909 and 119910respectively can be expressed as follows

120601119869(119898119899)

(119909 119910 119905)

=

1

sum

11989601=0

1

sum

11989602=0

120601 (1199090

11989601

1199100

11989602

)1199080(119898119899)

1198960111989602

(119909 119910)

+

119869minus1

sum

119895=0

2119895minus1

sum

11989611=0

2119895minus1

sum

11989612=0

[1205721

1198951198961111989612

(119905) 119908119895+1(119898119899)

211989611+12119896

12

(119909 119910)

+ 1205722

1198951198961111989612

(119905) 119908119895+1(119898119899)

211989611211989612+1(119909 119910)

+ 1205723

1198951198961111989612

(119905) 119908119895+1(119898119899)

211989611+12119896

12+1(119909 119910)]

(25)

where 119908119895(119898119899)11989611198962

(119909 119910) is the wavelet function and its 119898- and119899-order derivative with respect to 119909 and 119910 respectively1205721

1198951198961111989612

(119905) 12057221198951198961111989612

(119905) and 12057231198951198961111989612

(119905) are wavelet transform

Mathematical Problems in Engineering 7

coefficients We convert (2) to wavelet multiscale discreteformat of the level set function by stead of (25)

119889120601119869(119909 119910 119905)

119889119905

= 120575120576(120601119869(119909 119910 119905))

times [] div(nabla120601119869(119909 119910 119905)

1003816100381610038161003816nabla120601119869 (119909 119910 119905)

1003816100381610038161003816

)minus1205821

10038161003816100381610038161198680 minus 11988811003816100381610038161003816

2

+1205822

10038161003816100381610038161198680 minus 11988821003816100381610038161003816

2

]

(26)

It is the most direct way for dynamic adaptation toonly retain the distribution points of corresponding waveletcoefficients that satisfy the condition

min (100381610038161003816100381610038161205721

1198951198961111989612

(119905)10038161003816100381610038161003816100381610038161003816100381610038161205722

1198951198961111989612

(119905)10038161003816100381610038161003816100381610038161003816100381610038161205723

1198951198961111989612

(119905)10038161003816100381610038161003816) ge 120576 (27)

Time domain numerical integration of partial differentialequations is an iterative process therefore some pointswhich are possible important next step need to be kept toenable the algorithm to track singularities of solutions Soadjacent points of distribution points also should be keptTheadjacent region can be delineated as follows

1003816100381610038161003816119904 minus 1198951003816100381610038161003816 le 119872

10038161003816100381610038161003816119909119904

119896minus 119909119895

119894

10038161003816100381610038161003816le 120576119909

10038161003816100381610038161003816119909119904

119896minus 119909119895

119894

10038161003816100381610038161003816le 120576119910

(28)

where 119904 119895 are numbers of different scale wavelet and 119896 119894119872 isin

119885 120576119909 120576119910are constant

32 Nonlinear Discrete Ordinary Differential EquationsBecause ] is a small parameter in ordinary differentialequation (26) value of ] div(nabla120601119869(119909 119910 119905)|nabla120601119869(119909 119910 119905)|) is lowEquation (26) can be converted into

119889120601119869(119909 119910 119905)

119889119905

= 120575120576(120601119869(119909 119910 119905))

times [] div(nabla120601119869(119909 119910 119905

0)

1003816100381610038161003816nabla120601119869 (119909 119910 119905

0)1003816100381610038161003816

) minus 1205821

10038161003816100381610038161198680 minus 11988811003816100381610038161003816

2

+ 1205822

10038161003816100381610038161198680 minus 11988821003816100381610038161003816

2

]

(29)

The solution of (29) is

120601119869(119909 119910 119905) = (

120576119898

2120587)

13

+ (119898

21205871205765)

minus13

(30)

where

119898 = ] div(nabla120601119869(119909 119910 119905

0)

1003816100381610038161003816nabla120601119869 (119909 119910 119905

0)1003816100381610038161003816

)

minus 1205821

10038161003816100381610038161198680 minus 11988811003816100381610038161003816

2

+ 1205822

10038161003816100381610038161198680 minus 11988821003816100381610038161003816

2

(31)

We can get the solution of ordinary differential equations(26) by iterative solution (30)

4 Experiment and Discussion ofOnion Infected Region Segmentation

Figure 1(a) is a 256 times 302 image of an onion infected bysour skin virus We noticed that the onion has a water-soaked appearance Compared with the background thegrayscale difference between the water-soaked appearanceand the healthy part is smaller So it is beneficial to compareperformance of the different algorithms Figure 1(b) is anideal segmentation results Segmentation target is infectedregions of onion however infected part is often not uniformIn the image the difference of gradient is less than 1 at upperleft part due to slight infection so the algorithm is difficult toprecise segmentation based on global threshold Thereforethe best one of the different segmentation algorithms candistinguish themajority of the infected regionwithout seriousoversegmentationThen it is easy to identify andmeasure theinfected portions by using a priori knowledge

41 Comparison among Different SegmentationMethods Thecommon image segmentation methods including water-shed algorithm Sobel operator and Canny edge detectionalgorithm Otsu algorithm and an effective and commonquad-tree decomposition algorithm were selected for com-parison Shannon wavelet was employed to construct thewavelet interpolation operator The representation of Shan-non wavelet is based upon approximating the Dirac deltafunction as a band-limited function and is given by

119908 (119909) =sin (120587119909)120587119909

(32)

Consider a one-dimensional function 119891(119909) 119909 isin [119886 119887] Adiscrete point sequence of the variable 119909 is defined as

119909119899= 119886 +

119887 minus 119886

2119895sdot 119899 119895 isin Z 119899 = 0 1 2 2

119895 (33)

and the corresponding discrete point sequence of the scalingfunction 120601(119909) can be defined as

119908119895119899(119909) = 119908

119895(119909 minus 119909

119899) =

sin (2119895120587 (119887 minus 119886)) (119909 minus 119909119899)

(2119895120587 (119887 minus 119886)) (119909 minus 119909119899)

(34)

The first and second order derivatives of 120601119895(119909 minus 119909

119899) at the

discrete point 119909119896are

1206011015840

119895(119909119896minus 119909119899) =

0 119896 = 119899

2119895 cos [120587 (119896 minus 119899)](119896 minus 119899) (119887 minus 119886)

119896 = 119899

12060110158401015840

119895(119909119896minus 119909119899) =

minus1205872

3((119887 minus 119886)2119895)2 119896 = 119899

minus2 cos [120587 (119896 minus 119899)]

((119887 minus 119886) 2119895)2

(119896 minus 119899)2 119896 = 119899

(35)

The corresponding 2-dimension weight function can be rep-resented as the tensor product form of the above equations

8 Mathematical Problems in Engineering

(a) Original image (b) Segmentation target

Figure 1 Burkholderia cepacia (ex Burkholder) infected onion and the target segmentation

(a) Grayscale image (b) Gradient image (c) OSTU

(d) Watershed 1 (e) Watershed 2 (f) Qtdecomp

(g) Sobel (h) Canny (i) Wavelet precise integration

Figure 2 Comparison of various segmentation methods

Mathematical Problems in Engineering 9

300

250

200

150

100

50

00 50 100 150 200 250 300

Figure 3 Adaptive wavelet collocation points on level set

The experimental procedure is described as follows

(1) convert the infested onion image to grayscale(Figure 2(a)) and solve for the gradient map(Figure 2(b))

(2) use the grayscale image to test the Sobel operatorCanny operator Qtdecomp algorithms and waveletprecise integration method Use the gradient mat totest the OSTU method and watershed algorithm

(3) applying the watershed method to segment the imagewhich has been processed by OSTU method in orderto avoid the oversegmentation from the watershedalgorithm

The segmentation results are shown in Figure 2 Thewatershed algorithm segmentation result shows serious over-segmentation (Figure 2(d)) and cannot recognize the infectedpart Although OSTUmethod separated part infected regionof the onion the partition boundary is discontinuity andis difficult to measure infection specific gravity To avoidoversegmentation OSTUwas overlapped with the watershedsegmentationThe result is shown in Figure 2(e) in which thepartition boundary is clear but is unable to distinguish thevirus infected part The Sobel operator recognition on thepart of the infection is also not good (Figure 2(g)) Cannyoperator and Qtdecomp algorithm identified the area ofinfection but the boundary points of segmentation regionare disorder and cannot be measuredThe precise integrationmethod presented in this paper can identify the infected areaclearly So it is helpful to onion evaluation and classification

In fact it is impossible to project image segmentationby a single algorithm The important reason that partialdifferential equations are effective for image segmentationis that the method integrated many image segmentationprinciples to the model of partial differential equations Inthis paper the C-V image segmentation model is a globalconvex optimization variational model which is establishedon image piecewise smooth (119888

1and 1198882are the average gray

values inside (Ω1) and outside (Ω

2) of the object contour

respectively) To ensure the accuracy of image segmentation

the curvature of the image the border gradient and levelset function evolution were taken into account in imagesegmentation It means that the global convex optimizationmodel of image segmentation has been built based onintegration of a variety of image segmentation theories andhas obvious advantages In addition the method of iterativesolution of the self-adaptive method can also be integratedinto the segmentation process to ensure the accuracy ofsegmentation method further However the speed of thealgorithm will be affected Therefore it is important to findefficient and accurate numerical method

42 Efficiency Comparison of Multiscale Adaptive WaveletNumerical Method and Difference Method The C-V modelwas used for 256 times 302 images segmentation and divideddifference method was used to disperse partial differentialequations So discrete 7312 (256 times 302) ordinary differentialequations are huge solvingworkload But the adaptivewaveletprecise integration method can reduce the scale to 9576equations It can improve solution efficiency greatly due toless workload and low memory demand Of course the useof adaptive wavelet precise integration method for solvingthe number of distribution points will dynamically changeas the solution process In addition as shown in Figure 3distribution points are relatively dense within the ellipsering and another location was sparse The evident grayscaledifference between the infected and the healthy parts leadto this special points distribution Furthermore distributionpoints also exhibit regular matrix form which result fromblock solving method of wavelet transform to improve theefficiency The matrix-like distribution is from the boundaryeffect among the different blocks The interval wavelet caneffectively reduce the range effect but it will also increase thecomputation work of the wavelet transform

In this paper difference method was tested in MATLABThewavelet interpolation operatorwas implementedwithVCprogramming and other parts with MATLAB programmingOn the same computer difference method takes 03 secondsthe adaptive wavelet precise integration method takes 018seconds The results also show that the wavelet transform

10 Mathematical Problems in Engineering

of the iterative process reduces the overall computationalefficiency of the algorithm

5 Conclusions

Shannon wavelet precise integration method is a new imagesegmentation method based on the C-V model which wasused to construct adaptive wavelet interpolation operator dueto multiscale characteristics of wavelet transform combinedwith the time precise integration technology The methodmakes full use of the multiscale characteristics and thehigh precise performance of precise integration methodCompared to the gradient method and wavelet transformmethod of image segmentation object boundary obtained byWPIM segmentation method is clear and closed comparedto the watershed method the WPIM method avoids over-and undersegmentation problems and is very suitable formeasurement of image segmentation such as onion qualityassessment

The adaptive interpolation operator in the Shannonwavelet precision integration method can reduce the amountof the collocation points and improve the calculation effi-ciency As the interpolation operator contains a wavelettransform process the corresponding algorithm needs to doa wavelet transform between each two iteration time stepsSo the cost of the wavelet transformation is an importantpart of the calculation amount of the algorithm Compactlysupported orthogonal wavelet function can be expected tosolve the problem

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors appreciate the funding support from NationalNatural Science Foundation of China (Award nos 41171184and 41171337) The software tools were provided by the Foodand Fiber Sensing Lab of University of Georgia and theComputer Center of China Agricultural University

References

[1] Y Chen Y Xia Y Bian and Z-P Zhong ldquoImage measurementof precision aluminum alloy forgingsrdquo Journal of PlasticityEngineering vol 17 no 6 pp 77ndash81 2010

[2] S-L Mei Q-S Lu S-W Zhang and L Jin ldquoAdaptive intervalwavelet precise integrationmethod for partial differential equa-tionsrdquo Applied Mathematics and Mechanics vol 26 no 3 pp364ndash371 2005

[3] H-H Yan ldquoAdaptive wavelet precise integration method fornonlinear black-scholes model based on variational iterationmethodrdquo Abstract and Applied Analysis vol 2013 Article ID735919 6 pages 2013

[4] S-L Pang ldquoWavelet numerical method for nonlinear randomsystemrdquo Transactions of the Chinese Society of AgriculturalMachinery vol 38 no 3 pp 168ndash170 2007

[5] Y Wang ldquoWavelet precise time-integration method for heatconduction equationrdquo Journal of Chongqing Institute of Technol-ogy vol 21 no 8 pp 130ndash132 2007

[6] L X Zhang Y Yang and S L Mei ldquoWavelet precise integrationmethod on image denoisingrdquoTransactions of the Chinese Societyof Agricultural Machinery vol 37 no 7 pp 109ndash112 2006

[7] W N Xu S L Mei P X Wang and Y Yang ldquoAdaptivewavelet precise integration method on remote sensing imagedenoisingrdquo Transactions of the Chinese Society of AgriculturalMachinery vol 42 no 4 pp 148ndash152 2011

[8] R-Y Xing ldquoWavelet-based homotopy analysismethod for non-linear matrix system and its application in burgers equationrdquoMathematical Problems in Engineering vol 2013 Article ID982810 7 pages 2013

[9] S-L Mei ldquoConstruction of target controllable image segmen-tation model based on homotopy perturbation technologyrdquoAbstract and Applied Analysis vol 2013 Article ID 131207 8pages 2013

[10] L Liu ldquoConstruction of interval shannon wavelet and itsapplication in solving nonlinear black-scholes equationrdquoMath-ematical Problems in Engineering vol 2014 Article ID 541023 8pages 2014

[11] C Cattani ldquoShannon wavelets theoryrdquo Mathematical Problemsin Engineering vol 2008 Article ID 164808 24 pages 2008

[12] C Cattani ldquoSecond order Shannon wavelet approximationof C2-functionsrdquo UPB Scientific Bulletin Series A AppliedMathematics and Physics vol 73 no 3 pp 73ndash84 2011

[13] C Cattani and L M S Ruiz ldquoDiscrete differential operators inmultidimensional haar wavelet spacesrdquo International Journal ofMathematics and Mathematical Sciences vol 2004 no 44 pp2347ndash2355 2004

[14] C Cattani A Ciancio and B Lods ldquoOn a mathematical modelof immune competitionrdquo Applied Mathematics Letters vol 19no 7 pp 678ndash683 2006

[15] C Cattani ldquoConnection coefficients of Shannon waveletsrdquoMathematical Modelling and Analysis vol 11 no 2 pp 117ndash1322006

[16] S-L Mei and D-H Zhu ldquoInterval shannon wavelet collocationmethod for fractional fokker-planck equationrdquo Advances inMathematical Physics vol 2013 Article ID 821820 12 pages2013

[17] L-W Liu ldquoInterval wavelet numerical method on fokker-planck equations for nonlinear random systemrdquo Advances inMathematical Physics vol 2013 Article ID 651357 7 pages 2013

[18] J-H He ldquoHomotopy perturbation method for bifurcation ofnonlinear problemsrdquo International Journal of Nonlinear Sciencesand Numerical Simulation vol 6 no 2 pp 207ndash208 2005

[19] J-H He ldquoSome asymptotic methods for strongly nonlinearequationsrdquo International Journal of Modern Physics B vol 20no 10 pp 1141ndash1199 2006

[20] J-H He ldquoVariational iteration method-Some recent resultsand new interpretationsrdquo Journal of Computational and AppliedMathematics vol 207 no 1 pp 3ndash17 2007

[21] S-L Mei C-J Du and S-W Zhang ldquoAsymptotic numericalmethod for multi-degree-of-freedom nonlinear dynamic sys-temsrdquo Chaos Solitons and Fractals vol 35 no 3 pp 536ndash5422008

[22] S-L Mei and S-W Zhang ldquoCoupling technique of variationaliteration and homotopy perturbation methods for nonlinearmatrix differential equationsrdquoComputers andMathematics withApplications vol 54 no 7-8 pp 1092ndash1100 2007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

The value of 120601119869(119909 119910 119905119899) at 119905119899is denoted as 120601

119899 and

119865 [119905119899 119909 119910 120601

119869(119909 119910 119905

119899) 120601119869(10)

(119909 119910 119905119899) 120601119869(01)

(119909 119910 119905119899)

120601119869(20)

(119909 119910 119905119899) 120601119869(11)

(119909 119910 119905119899) 120601119869(02)

(119909 119910 119905119899)]

(17)

is denoted as 119865119899 And then a linear homotopy function can be

constructed as

120601119869(119909 119910 119905) = (1 minus 120576) 119865

119899+ 120576119865119899+1

(18)

It is easy to identify the homotopy parameter as

120576 (119905) =119905 minus 119905119899

119905119899+1

minus 119905119899

119905 isin [119905119899 119905119899+1

] there4 120576 isin [0 1] (19)

According to the perturbation theory the solution of (18) canbe expressed as the power series expansion of 120576 as follows

120601119869= 120601119869

0+ 120576120601119869

1+ 1205762120601119869

2+ sdot sdot sdot (20)

Substituting (20) into (18) and rearranging based on powersof 120576-terms we have

12057601206011198690= 119865119899

12057611206011198691= 119865119899+1

minus119865119899

(21)

According to HPM we obtain the wavelet coefficients1205721

11989511989611198962

(119905119899+1

) 120572211989511989611198962

(119905119899+1

) 120572311989511989611198962

(119905119899+1

) at 119905119899as follows

1205721

11989511989611198962

= 120601 (119909119895+1

21198961+1 119910119895+1

21198962

) minus 119868119895120601 (119909119895+1

21198961+1 119910119895+1

21198962

)

= 120601 (119909119895+1

21198961+1 119910119895+1

21198962

)

minus [

[

1

sum

11989601=0

1

sum

11989602=0

120601 (1199090

11989601

1199100

11989602

)1199080

1198960111989602

(119909119895+1

21198961+1 119910119895+1

21198962

)

+

119895minus1

sum

1198951=0

21198951

sum

11989611=0

21198951

sum

11989612=0

(1205721

11989511198961111989612

1199081198951+1

211989611+12119896

12

(119909119895+1

21198961+1 119910119895+1

21198962

)

+ 1205722

11989511198961111989612

1199081198951+1

211989611211989612+1(119909119895+1

21198961+1 119910119895+1

21198962

)

+ 1205723

11989511198961111989612

1199081198951+1

211989611+12119896

12+1(119909119895+1

21198961+1 119910119895+1

21198962

))]

]

1205722

11989511989611198962

= 120601 (119909119895+1

21198961

119910119895+1

21198962+1) minus 119868119895120601 (119909119895+1

21198961

119910119895+1

21198962+1)

= 120601 (119909119895+1

21198961

119910119895+1

21198962+1)

minus [

[

1

sum

11989601=0

1

sum

11989602=0

120601 (1199090

11989601

1199100

11989602

)1199080

1198960111989602

(119909119895+1

21198961

119910119895+1

21198962+1)

+

119895minus1

sum

1198951=0

21198951

sum

11989611=0

21198951

sum

11989612=0

(1205721

11989511198961111989612

1199081198951+1

211989611+12119896

12

(119909119895+1

21198961

119910119895+1

21198962+1)

+ 1205722

11989511198961111989612

1199081198951+1

211989611211989612+1(119909119895+1

21198961

119910119895+1

21198962+1)

+ 1205723

11989511198961111989612

1199081198951+1

211989611+12119896

12+1(119909119895+1

21198961

119910119895+1

21198962+1))]

]

6 Mathematical Problems in Engineering

1205723

11989511989611198962

= 120601 (119909119895+1

21198961+1 119910119895+1

21198962+1) minus 119868119895120601 (119909119895+1

21198961+1 119910119895+1

21198962+1)

= 120601 (119909119895+1

21198961+1 119910119895+1

21198962+1)

minus [

[

1

sum

11989601=0

1

sum

11989602=0

120601 (1199090

11989601

1199100

11989602

)1199080

1198960111989602

(119909119895+1

21198961+1 119910119895+1

21198962+1)

+

119895minus1

sum

1198951=0

21198951

sum

11989611=0

21198951

sum

11989612=0

(1205721

11989511198961111989612

1199081198951+1

211989611+12119896

12

(119909119895+1

21198961+1 119910119895+1

21198962+1)

+ 1205722

11989511198961111989612

1199081198951+1

211989611211989612+1(119909119895+1

21198961+1 119910119895+1

21198962+1)

+ 1205723

11989511198961111989612

1199081198951+1

211989611+12119896

12+1(119909119895+1

21198961+1 119910119895+1

21198962+1))]

]

(22)

Obviously the calculation time complexity of the wavelettransform coefficients 1205721

11989511989611198962

120572211989511989611198962

and 1205723

11989511989611198962

is 119874(4119869)which is decreased greatly than that in (8) which is119874((13)4

2119869minus1)

Substituting the wavelet transform efficient (22) into (16)we obtain120601119869(119909 119910 119905

119899+1)

= 120601119869(119909 119910 119905

119899)

+Δ119905

2[119865 (119905119899 119909 119910 120601

119869(119909 119910 119905

119899) 120601119869(10)

(119909 119910 119905119899)

120601119869(01)

(119909 119910 119905119899) 120601119869(20)

(119909 119910 119905119899)

120601119869(11)

(119909 119910 119905119899) 120601119869(02)

(119909 119910 119905119899))

+ 119865 (119905119899+1

119909 119910 120601119869

0(119909 119910 119905

119899+1)

120601119869(10)

0(119909 119910 119905

119899+1) 120601119869(01)

0(119909 119910 119905

119899+1)

120601119869(20)

0(119909 119910 119905

119899+1) 120601119869(11)

0(119909 119910 119905

119899+1)

120601119869(02)

0(119909 119910 119905

119899+1))]

(23)

and the derivative function120601119869(119898119899)

(119909 119910)

=

1

sum

11989601=0

1

sum

11989602=0

120601 (1199090

11989601

1199100

11989602

)1199080(119898119899)

1198960111989602

(119909 119910)

+

119869minus1

sum

119895=0

2119895minus1

sum

11989611=0

2119895minus1

sum

11989612=0

[1205721

1198951198961111989612

119908119895+1(119898119899)

211989611+12119896

12

(119909 119910)

+ 1205722

1198951198961111989612

119908119895+1(119898119899)

211989611211989612+1(119909 119910)

+ 1205723

1198951198961111989612

119908119895+1(119898119899)

211989611+12119896

12+1(119909 119910)]

(24)

Obviously the computation complexity is decreased greatlycomparing with (14)

31TheMultiscale InterpolationWavelet Approximation of theC-V Model There are many ways to solve partial differentialequations and the most typical method is the differencemethod This method uses the flat function to describeimage approximately the surface function But it is easyto cause artifacts phenomenon affecting the accuracy ofimage segmentation Wavelet function has both smooth andcompactly supported characteristics Besides performance ofmultiscale analysis can be used to construct the multiscaleadaptive interpolation operator for solving nonlinear partialdifferential equationsThe wavelet approximation of the levelset function and its derivative with respect to 119909 and 119910respectively can be expressed as follows

120601119869(119898119899)

(119909 119910 119905)

=

1

sum

11989601=0

1

sum

11989602=0

120601 (1199090

11989601

1199100

11989602

)1199080(119898119899)

1198960111989602

(119909 119910)

+

119869minus1

sum

119895=0

2119895minus1

sum

11989611=0

2119895minus1

sum

11989612=0

[1205721

1198951198961111989612

(119905) 119908119895+1(119898119899)

211989611+12119896

12

(119909 119910)

+ 1205722

1198951198961111989612

(119905) 119908119895+1(119898119899)

211989611211989612+1(119909 119910)

+ 1205723

1198951198961111989612

(119905) 119908119895+1(119898119899)

211989611+12119896

12+1(119909 119910)]

(25)

where 119908119895(119898119899)11989611198962

(119909 119910) is the wavelet function and its 119898- and119899-order derivative with respect to 119909 and 119910 respectively1205721

1198951198961111989612

(119905) 12057221198951198961111989612

(119905) and 12057231198951198961111989612

(119905) are wavelet transform

Mathematical Problems in Engineering 7

coefficients We convert (2) to wavelet multiscale discreteformat of the level set function by stead of (25)

119889120601119869(119909 119910 119905)

119889119905

= 120575120576(120601119869(119909 119910 119905))

times [] div(nabla120601119869(119909 119910 119905)

1003816100381610038161003816nabla120601119869 (119909 119910 119905)

1003816100381610038161003816

)minus1205821

10038161003816100381610038161198680 minus 11988811003816100381610038161003816

2

+1205822

10038161003816100381610038161198680 minus 11988821003816100381610038161003816

2

]

(26)

It is the most direct way for dynamic adaptation toonly retain the distribution points of corresponding waveletcoefficients that satisfy the condition

min (100381610038161003816100381610038161205721

1198951198961111989612

(119905)10038161003816100381610038161003816100381610038161003816100381610038161205722

1198951198961111989612

(119905)10038161003816100381610038161003816100381610038161003816100381610038161205723

1198951198961111989612

(119905)10038161003816100381610038161003816) ge 120576 (27)

Time domain numerical integration of partial differentialequations is an iterative process therefore some pointswhich are possible important next step need to be kept toenable the algorithm to track singularities of solutions Soadjacent points of distribution points also should be keptTheadjacent region can be delineated as follows

1003816100381610038161003816119904 minus 1198951003816100381610038161003816 le 119872

10038161003816100381610038161003816119909119904

119896minus 119909119895

119894

10038161003816100381610038161003816le 120576119909

10038161003816100381610038161003816119909119904

119896minus 119909119895

119894

10038161003816100381610038161003816le 120576119910

(28)

where 119904 119895 are numbers of different scale wavelet and 119896 119894119872 isin

119885 120576119909 120576119910are constant

32 Nonlinear Discrete Ordinary Differential EquationsBecause ] is a small parameter in ordinary differentialequation (26) value of ] div(nabla120601119869(119909 119910 119905)|nabla120601119869(119909 119910 119905)|) is lowEquation (26) can be converted into

119889120601119869(119909 119910 119905)

119889119905

= 120575120576(120601119869(119909 119910 119905))

times [] div(nabla120601119869(119909 119910 119905

0)

1003816100381610038161003816nabla120601119869 (119909 119910 119905

0)1003816100381610038161003816

) minus 1205821

10038161003816100381610038161198680 minus 11988811003816100381610038161003816

2

+ 1205822

10038161003816100381610038161198680 minus 11988821003816100381610038161003816

2

]

(29)

The solution of (29) is

120601119869(119909 119910 119905) = (

120576119898

2120587)

13

+ (119898

21205871205765)

minus13

(30)

where

119898 = ] div(nabla120601119869(119909 119910 119905

0)

1003816100381610038161003816nabla120601119869 (119909 119910 119905

0)1003816100381610038161003816

)

minus 1205821

10038161003816100381610038161198680 minus 11988811003816100381610038161003816

2

+ 1205822

10038161003816100381610038161198680 minus 11988821003816100381610038161003816

2

(31)

We can get the solution of ordinary differential equations(26) by iterative solution (30)

4 Experiment and Discussion ofOnion Infected Region Segmentation

Figure 1(a) is a 256 times 302 image of an onion infected bysour skin virus We noticed that the onion has a water-soaked appearance Compared with the background thegrayscale difference between the water-soaked appearanceand the healthy part is smaller So it is beneficial to compareperformance of the different algorithms Figure 1(b) is anideal segmentation results Segmentation target is infectedregions of onion however infected part is often not uniformIn the image the difference of gradient is less than 1 at upperleft part due to slight infection so the algorithm is difficult toprecise segmentation based on global threshold Thereforethe best one of the different segmentation algorithms candistinguish themajority of the infected regionwithout seriousoversegmentationThen it is easy to identify andmeasure theinfected portions by using a priori knowledge

41 Comparison among Different SegmentationMethods Thecommon image segmentation methods including water-shed algorithm Sobel operator and Canny edge detectionalgorithm Otsu algorithm and an effective and commonquad-tree decomposition algorithm were selected for com-parison Shannon wavelet was employed to construct thewavelet interpolation operator The representation of Shan-non wavelet is based upon approximating the Dirac deltafunction as a band-limited function and is given by

119908 (119909) =sin (120587119909)120587119909

(32)

Consider a one-dimensional function 119891(119909) 119909 isin [119886 119887] Adiscrete point sequence of the variable 119909 is defined as

119909119899= 119886 +

119887 minus 119886

2119895sdot 119899 119895 isin Z 119899 = 0 1 2 2

119895 (33)

and the corresponding discrete point sequence of the scalingfunction 120601(119909) can be defined as

119908119895119899(119909) = 119908

119895(119909 minus 119909

119899) =

sin (2119895120587 (119887 minus 119886)) (119909 minus 119909119899)

(2119895120587 (119887 minus 119886)) (119909 minus 119909119899)

(34)

The first and second order derivatives of 120601119895(119909 minus 119909

119899) at the

discrete point 119909119896are

1206011015840

119895(119909119896minus 119909119899) =

0 119896 = 119899

2119895 cos [120587 (119896 minus 119899)](119896 minus 119899) (119887 minus 119886)

119896 = 119899

12060110158401015840

119895(119909119896minus 119909119899) =

minus1205872

3((119887 minus 119886)2119895)2 119896 = 119899

minus2 cos [120587 (119896 minus 119899)]

((119887 minus 119886) 2119895)2

(119896 minus 119899)2 119896 = 119899

(35)

The corresponding 2-dimension weight function can be rep-resented as the tensor product form of the above equations

8 Mathematical Problems in Engineering

(a) Original image (b) Segmentation target

Figure 1 Burkholderia cepacia (ex Burkholder) infected onion and the target segmentation

(a) Grayscale image (b) Gradient image (c) OSTU

(d) Watershed 1 (e) Watershed 2 (f) Qtdecomp

(g) Sobel (h) Canny (i) Wavelet precise integration

Figure 2 Comparison of various segmentation methods

Mathematical Problems in Engineering 9

300

250

200

150

100

50

00 50 100 150 200 250 300

Figure 3 Adaptive wavelet collocation points on level set

The experimental procedure is described as follows

(1) convert the infested onion image to grayscale(Figure 2(a)) and solve for the gradient map(Figure 2(b))

(2) use the grayscale image to test the Sobel operatorCanny operator Qtdecomp algorithms and waveletprecise integration method Use the gradient mat totest the OSTU method and watershed algorithm

(3) applying the watershed method to segment the imagewhich has been processed by OSTU method in orderto avoid the oversegmentation from the watershedalgorithm

The segmentation results are shown in Figure 2 Thewatershed algorithm segmentation result shows serious over-segmentation (Figure 2(d)) and cannot recognize the infectedpart Although OSTUmethod separated part infected regionof the onion the partition boundary is discontinuity andis difficult to measure infection specific gravity To avoidoversegmentation OSTUwas overlapped with the watershedsegmentationThe result is shown in Figure 2(e) in which thepartition boundary is clear but is unable to distinguish thevirus infected part The Sobel operator recognition on thepart of the infection is also not good (Figure 2(g)) Cannyoperator and Qtdecomp algorithm identified the area ofinfection but the boundary points of segmentation regionare disorder and cannot be measuredThe precise integrationmethod presented in this paper can identify the infected areaclearly So it is helpful to onion evaluation and classification

In fact it is impossible to project image segmentationby a single algorithm The important reason that partialdifferential equations are effective for image segmentationis that the method integrated many image segmentationprinciples to the model of partial differential equations Inthis paper the C-V image segmentation model is a globalconvex optimization variational model which is establishedon image piecewise smooth (119888

1and 1198882are the average gray

values inside (Ω1) and outside (Ω

2) of the object contour

respectively) To ensure the accuracy of image segmentation

the curvature of the image the border gradient and levelset function evolution were taken into account in imagesegmentation It means that the global convex optimizationmodel of image segmentation has been built based onintegration of a variety of image segmentation theories andhas obvious advantages In addition the method of iterativesolution of the self-adaptive method can also be integratedinto the segmentation process to ensure the accuracy ofsegmentation method further However the speed of thealgorithm will be affected Therefore it is important to findefficient and accurate numerical method

42 Efficiency Comparison of Multiscale Adaptive WaveletNumerical Method and Difference Method The C-V modelwas used for 256 times 302 images segmentation and divideddifference method was used to disperse partial differentialequations So discrete 7312 (256 times 302) ordinary differentialequations are huge solvingworkload But the adaptivewaveletprecise integration method can reduce the scale to 9576equations It can improve solution efficiency greatly due toless workload and low memory demand Of course the useof adaptive wavelet precise integration method for solvingthe number of distribution points will dynamically changeas the solution process In addition as shown in Figure 3distribution points are relatively dense within the ellipsering and another location was sparse The evident grayscaledifference between the infected and the healthy parts leadto this special points distribution Furthermore distributionpoints also exhibit regular matrix form which result fromblock solving method of wavelet transform to improve theefficiency The matrix-like distribution is from the boundaryeffect among the different blocks The interval wavelet caneffectively reduce the range effect but it will also increase thecomputation work of the wavelet transform

In this paper difference method was tested in MATLABThewavelet interpolation operatorwas implementedwithVCprogramming and other parts with MATLAB programmingOn the same computer difference method takes 03 secondsthe adaptive wavelet precise integration method takes 018seconds The results also show that the wavelet transform

10 Mathematical Problems in Engineering

of the iterative process reduces the overall computationalefficiency of the algorithm

5 Conclusions

Shannon wavelet precise integration method is a new imagesegmentation method based on the C-V model which wasused to construct adaptive wavelet interpolation operator dueto multiscale characteristics of wavelet transform combinedwith the time precise integration technology The methodmakes full use of the multiscale characteristics and thehigh precise performance of precise integration methodCompared to the gradient method and wavelet transformmethod of image segmentation object boundary obtained byWPIM segmentation method is clear and closed comparedto the watershed method the WPIM method avoids over-and undersegmentation problems and is very suitable formeasurement of image segmentation such as onion qualityassessment

The adaptive interpolation operator in the Shannonwavelet precision integration method can reduce the amountof the collocation points and improve the calculation effi-ciency As the interpolation operator contains a wavelettransform process the corresponding algorithm needs to doa wavelet transform between each two iteration time stepsSo the cost of the wavelet transformation is an importantpart of the calculation amount of the algorithm Compactlysupported orthogonal wavelet function can be expected tosolve the problem

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors appreciate the funding support from NationalNatural Science Foundation of China (Award nos 41171184and 41171337) The software tools were provided by the Foodand Fiber Sensing Lab of University of Georgia and theComputer Center of China Agricultural University

References

[1] Y Chen Y Xia Y Bian and Z-P Zhong ldquoImage measurementof precision aluminum alloy forgingsrdquo Journal of PlasticityEngineering vol 17 no 6 pp 77ndash81 2010

[2] S-L Mei Q-S Lu S-W Zhang and L Jin ldquoAdaptive intervalwavelet precise integrationmethod for partial differential equa-tionsrdquo Applied Mathematics and Mechanics vol 26 no 3 pp364ndash371 2005

[3] H-H Yan ldquoAdaptive wavelet precise integration method fornonlinear black-scholes model based on variational iterationmethodrdquo Abstract and Applied Analysis vol 2013 Article ID735919 6 pages 2013

[4] S-L Pang ldquoWavelet numerical method for nonlinear randomsystemrdquo Transactions of the Chinese Society of AgriculturalMachinery vol 38 no 3 pp 168ndash170 2007

[5] Y Wang ldquoWavelet precise time-integration method for heatconduction equationrdquo Journal of Chongqing Institute of Technol-ogy vol 21 no 8 pp 130ndash132 2007

[6] L X Zhang Y Yang and S L Mei ldquoWavelet precise integrationmethod on image denoisingrdquoTransactions of the Chinese Societyof Agricultural Machinery vol 37 no 7 pp 109ndash112 2006

[7] W N Xu S L Mei P X Wang and Y Yang ldquoAdaptivewavelet precise integration method on remote sensing imagedenoisingrdquo Transactions of the Chinese Society of AgriculturalMachinery vol 42 no 4 pp 148ndash152 2011

[8] R-Y Xing ldquoWavelet-based homotopy analysismethod for non-linear matrix system and its application in burgers equationrdquoMathematical Problems in Engineering vol 2013 Article ID982810 7 pages 2013

[9] S-L Mei ldquoConstruction of target controllable image segmen-tation model based on homotopy perturbation technologyrdquoAbstract and Applied Analysis vol 2013 Article ID 131207 8pages 2013

[10] L Liu ldquoConstruction of interval shannon wavelet and itsapplication in solving nonlinear black-scholes equationrdquoMath-ematical Problems in Engineering vol 2014 Article ID 541023 8pages 2014

[11] C Cattani ldquoShannon wavelets theoryrdquo Mathematical Problemsin Engineering vol 2008 Article ID 164808 24 pages 2008

[12] C Cattani ldquoSecond order Shannon wavelet approximationof C2-functionsrdquo UPB Scientific Bulletin Series A AppliedMathematics and Physics vol 73 no 3 pp 73ndash84 2011

[13] C Cattani and L M S Ruiz ldquoDiscrete differential operators inmultidimensional haar wavelet spacesrdquo International Journal ofMathematics and Mathematical Sciences vol 2004 no 44 pp2347ndash2355 2004

[14] C Cattani A Ciancio and B Lods ldquoOn a mathematical modelof immune competitionrdquo Applied Mathematics Letters vol 19no 7 pp 678ndash683 2006

[15] C Cattani ldquoConnection coefficients of Shannon waveletsrdquoMathematical Modelling and Analysis vol 11 no 2 pp 117ndash1322006

[16] S-L Mei and D-H Zhu ldquoInterval shannon wavelet collocationmethod for fractional fokker-planck equationrdquo Advances inMathematical Physics vol 2013 Article ID 821820 12 pages2013

[17] L-W Liu ldquoInterval wavelet numerical method on fokker-planck equations for nonlinear random systemrdquo Advances inMathematical Physics vol 2013 Article ID 651357 7 pages 2013

[18] J-H He ldquoHomotopy perturbation method for bifurcation ofnonlinear problemsrdquo International Journal of Nonlinear Sciencesand Numerical Simulation vol 6 no 2 pp 207ndash208 2005

[19] J-H He ldquoSome asymptotic methods for strongly nonlinearequationsrdquo International Journal of Modern Physics B vol 20no 10 pp 1141ndash1199 2006

[20] J-H He ldquoVariational iteration method-Some recent resultsand new interpretationsrdquo Journal of Computational and AppliedMathematics vol 207 no 1 pp 3ndash17 2007

[21] S-L Mei C-J Du and S-W Zhang ldquoAsymptotic numericalmethod for multi-degree-of-freedom nonlinear dynamic sys-temsrdquo Chaos Solitons and Fractals vol 35 no 3 pp 536ndash5422008

[22] S-L Mei and S-W Zhang ldquoCoupling technique of variationaliteration and homotopy perturbation methods for nonlinearmatrix differential equationsrdquoComputers andMathematics withApplications vol 54 no 7-8 pp 1092ndash1100 2007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

1205723

11989511989611198962

= 120601 (119909119895+1

21198961+1 119910119895+1

21198962+1) minus 119868119895120601 (119909119895+1

21198961+1 119910119895+1

21198962+1)

= 120601 (119909119895+1

21198961+1 119910119895+1

21198962+1)

minus [

[

1

sum

11989601=0

1

sum

11989602=0

120601 (1199090

11989601

1199100

11989602

)1199080

1198960111989602

(119909119895+1

21198961+1 119910119895+1

21198962+1)

+

119895minus1

sum

1198951=0

21198951

sum

11989611=0

21198951

sum

11989612=0

(1205721

11989511198961111989612

1199081198951+1

211989611+12119896

12

(119909119895+1

21198961+1 119910119895+1

21198962+1)

+ 1205722

11989511198961111989612

1199081198951+1

211989611211989612+1(119909119895+1

21198961+1 119910119895+1

21198962+1)

+ 1205723

11989511198961111989612

1199081198951+1

211989611+12119896

12+1(119909119895+1

21198961+1 119910119895+1

21198962+1))]

]

(22)

Obviously the calculation time complexity of the wavelettransform coefficients 1205721

11989511989611198962

120572211989511989611198962

and 1205723

11989511989611198962

is 119874(4119869)which is decreased greatly than that in (8) which is119874((13)4

2119869minus1)

Substituting the wavelet transform efficient (22) into (16)we obtain120601119869(119909 119910 119905

119899+1)

= 120601119869(119909 119910 119905

119899)

+Δ119905

2[119865 (119905119899 119909 119910 120601

119869(119909 119910 119905

119899) 120601119869(10)

(119909 119910 119905119899)

120601119869(01)

(119909 119910 119905119899) 120601119869(20)

(119909 119910 119905119899)

120601119869(11)

(119909 119910 119905119899) 120601119869(02)

(119909 119910 119905119899))

+ 119865 (119905119899+1

119909 119910 120601119869

0(119909 119910 119905

119899+1)

120601119869(10)

0(119909 119910 119905

119899+1) 120601119869(01)

0(119909 119910 119905

119899+1)

120601119869(20)

0(119909 119910 119905

119899+1) 120601119869(11)

0(119909 119910 119905

119899+1)

120601119869(02)

0(119909 119910 119905

119899+1))]

(23)

and the derivative function120601119869(119898119899)

(119909 119910)

=

1

sum

11989601=0

1

sum

11989602=0

120601 (1199090

11989601

1199100

11989602

)1199080(119898119899)

1198960111989602

(119909 119910)

+

119869minus1

sum

119895=0

2119895minus1

sum

11989611=0

2119895minus1

sum

11989612=0

[1205721

1198951198961111989612

119908119895+1(119898119899)

211989611+12119896

12

(119909 119910)

+ 1205722

1198951198961111989612

119908119895+1(119898119899)

211989611211989612+1(119909 119910)

+ 1205723

1198951198961111989612

119908119895+1(119898119899)

211989611+12119896

12+1(119909 119910)]

(24)

Obviously the computation complexity is decreased greatlycomparing with (14)

31TheMultiscale InterpolationWavelet Approximation of theC-V Model There are many ways to solve partial differentialequations and the most typical method is the differencemethod This method uses the flat function to describeimage approximately the surface function But it is easyto cause artifacts phenomenon affecting the accuracy ofimage segmentation Wavelet function has both smooth andcompactly supported characteristics Besides performance ofmultiscale analysis can be used to construct the multiscaleadaptive interpolation operator for solving nonlinear partialdifferential equationsThe wavelet approximation of the levelset function and its derivative with respect to 119909 and 119910respectively can be expressed as follows

120601119869(119898119899)

(119909 119910 119905)

=

1

sum

11989601=0

1

sum

11989602=0

120601 (1199090

11989601

1199100

11989602

)1199080(119898119899)

1198960111989602

(119909 119910)

+

119869minus1

sum

119895=0

2119895minus1

sum

11989611=0

2119895minus1

sum

11989612=0

[1205721

1198951198961111989612

(119905) 119908119895+1(119898119899)

211989611+12119896

12

(119909 119910)

+ 1205722

1198951198961111989612

(119905) 119908119895+1(119898119899)

211989611211989612+1(119909 119910)

+ 1205723

1198951198961111989612

(119905) 119908119895+1(119898119899)

211989611+12119896

12+1(119909 119910)]

(25)

where 119908119895(119898119899)11989611198962

(119909 119910) is the wavelet function and its 119898- and119899-order derivative with respect to 119909 and 119910 respectively1205721

1198951198961111989612

(119905) 12057221198951198961111989612

(119905) and 12057231198951198961111989612

(119905) are wavelet transform

Mathematical Problems in Engineering 7

coefficients We convert (2) to wavelet multiscale discreteformat of the level set function by stead of (25)

119889120601119869(119909 119910 119905)

119889119905

= 120575120576(120601119869(119909 119910 119905))

times [] div(nabla120601119869(119909 119910 119905)

1003816100381610038161003816nabla120601119869 (119909 119910 119905)

1003816100381610038161003816

)minus1205821

10038161003816100381610038161198680 minus 11988811003816100381610038161003816

2

+1205822

10038161003816100381610038161198680 minus 11988821003816100381610038161003816

2

]

(26)

It is the most direct way for dynamic adaptation toonly retain the distribution points of corresponding waveletcoefficients that satisfy the condition

min (100381610038161003816100381610038161205721

1198951198961111989612

(119905)10038161003816100381610038161003816100381610038161003816100381610038161205722

1198951198961111989612

(119905)10038161003816100381610038161003816100381610038161003816100381610038161205723

1198951198961111989612

(119905)10038161003816100381610038161003816) ge 120576 (27)

Time domain numerical integration of partial differentialequations is an iterative process therefore some pointswhich are possible important next step need to be kept toenable the algorithm to track singularities of solutions Soadjacent points of distribution points also should be keptTheadjacent region can be delineated as follows

1003816100381610038161003816119904 minus 1198951003816100381610038161003816 le 119872

10038161003816100381610038161003816119909119904

119896minus 119909119895

119894

10038161003816100381610038161003816le 120576119909

10038161003816100381610038161003816119909119904

119896minus 119909119895

119894

10038161003816100381610038161003816le 120576119910

(28)

where 119904 119895 are numbers of different scale wavelet and 119896 119894119872 isin

119885 120576119909 120576119910are constant

32 Nonlinear Discrete Ordinary Differential EquationsBecause ] is a small parameter in ordinary differentialequation (26) value of ] div(nabla120601119869(119909 119910 119905)|nabla120601119869(119909 119910 119905)|) is lowEquation (26) can be converted into

119889120601119869(119909 119910 119905)

119889119905

= 120575120576(120601119869(119909 119910 119905))

times [] div(nabla120601119869(119909 119910 119905

0)

1003816100381610038161003816nabla120601119869 (119909 119910 119905

0)1003816100381610038161003816

) minus 1205821

10038161003816100381610038161198680 minus 11988811003816100381610038161003816

2

+ 1205822

10038161003816100381610038161198680 minus 11988821003816100381610038161003816

2

]

(29)

The solution of (29) is

120601119869(119909 119910 119905) = (

120576119898

2120587)

13

+ (119898

21205871205765)

minus13

(30)

where

119898 = ] div(nabla120601119869(119909 119910 119905

0)

1003816100381610038161003816nabla120601119869 (119909 119910 119905

0)1003816100381610038161003816

)

minus 1205821

10038161003816100381610038161198680 minus 11988811003816100381610038161003816

2

+ 1205822

10038161003816100381610038161198680 minus 11988821003816100381610038161003816

2

(31)

We can get the solution of ordinary differential equations(26) by iterative solution (30)

4 Experiment and Discussion ofOnion Infected Region Segmentation

Figure 1(a) is a 256 times 302 image of an onion infected bysour skin virus We noticed that the onion has a water-soaked appearance Compared with the background thegrayscale difference between the water-soaked appearanceand the healthy part is smaller So it is beneficial to compareperformance of the different algorithms Figure 1(b) is anideal segmentation results Segmentation target is infectedregions of onion however infected part is often not uniformIn the image the difference of gradient is less than 1 at upperleft part due to slight infection so the algorithm is difficult toprecise segmentation based on global threshold Thereforethe best one of the different segmentation algorithms candistinguish themajority of the infected regionwithout seriousoversegmentationThen it is easy to identify andmeasure theinfected portions by using a priori knowledge

41 Comparison among Different SegmentationMethods Thecommon image segmentation methods including water-shed algorithm Sobel operator and Canny edge detectionalgorithm Otsu algorithm and an effective and commonquad-tree decomposition algorithm were selected for com-parison Shannon wavelet was employed to construct thewavelet interpolation operator The representation of Shan-non wavelet is based upon approximating the Dirac deltafunction as a band-limited function and is given by

119908 (119909) =sin (120587119909)120587119909

(32)

Consider a one-dimensional function 119891(119909) 119909 isin [119886 119887] Adiscrete point sequence of the variable 119909 is defined as

119909119899= 119886 +

119887 minus 119886

2119895sdot 119899 119895 isin Z 119899 = 0 1 2 2

119895 (33)

and the corresponding discrete point sequence of the scalingfunction 120601(119909) can be defined as

119908119895119899(119909) = 119908

119895(119909 minus 119909

119899) =

sin (2119895120587 (119887 minus 119886)) (119909 minus 119909119899)

(2119895120587 (119887 minus 119886)) (119909 minus 119909119899)

(34)

The first and second order derivatives of 120601119895(119909 minus 119909

119899) at the

discrete point 119909119896are

1206011015840

119895(119909119896minus 119909119899) =

0 119896 = 119899

2119895 cos [120587 (119896 minus 119899)](119896 minus 119899) (119887 minus 119886)

119896 = 119899

12060110158401015840

119895(119909119896minus 119909119899) =

minus1205872

3((119887 minus 119886)2119895)2 119896 = 119899

minus2 cos [120587 (119896 minus 119899)]

((119887 minus 119886) 2119895)2

(119896 minus 119899)2 119896 = 119899

(35)

The corresponding 2-dimension weight function can be rep-resented as the tensor product form of the above equations

8 Mathematical Problems in Engineering

(a) Original image (b) Segmentation target

Figure 1 Burkholderia cepacia (ex Burkholder) infected onion and the target segmentation

(a) Grayscale image (b) Gradient image (c) OSTU

(d) Watershed 1 (e) Watershed 2 (f) Qtdecomp

(g) Sobel (h) Canny (i) Wavelet precise integration

Figure 2 Comparison of various segmentation methods

Mathematical Problems in Engineering 9

300

250

200

150

100

50

00 50 100 150 200 250 300

Figure 3 Adaptive wavelet collocation points on level set

The experimental procedure is described as follows

(1) convert the infested onion image to grayscale(Figure 2(a)) and solve for the gradient map(Figure 2(b))

(2) use the grayscale image to test the Sobel operatorCanny operator Qtdecomp algorithms and waveletprecise integration method Use the gradient mat totest the OSTU method and watershed algorithm

(3) applying the watershed method to segment the imagewhich has been processed by OSTU method in orderto avoid the oversegmentation from the watershedalgorithm

The segmentation results are shown in Figure 2 Thewatershed algorithm segmentation result shows serious over-segmentation (Figure 2(d)) and cannot recognize the infectedpart Although OSTUmethod separated part infected regionof the onion the partition boundary is discontinuity andis difficult to measure infection specific gravity To avoidoversegmentation OSTUwas overlapped with the watershedsegmentationThe result is shown in Figure 2(e) in which thepartition boundary is clear but is unable to distinguish thevirus infected part The Sobel operator recognition on thepart of the infection is also not good (Figure 2(g)) Cannyoperator and Qtdecomp algorithm identified the area ofinfection but the boundary points of segmentation regionare disorder and cannot be measuredThe precise integrationmethod presented in this paper can identify the infected areaclearly So it is helpful to onion evaluation and classification

In fact it is impossible to project image segmentationby a single algorithm The important reason that partialdifferential equations are effective for image segmentationis that the method integrated many image segmentationprinciples to the model of partial differential equations Inthis paper the C-V image segmentation model is a globalconvex optimization variational model which is establishedon image piecewise smooth (119888

1and 1198882are the average gray

values inside (Ω1) and outside (Ω

2) of the object contour

respectively) To ensure the accuracy of image segmentation

the curvature of the image the border gradient and levelset function evolution were taken into account in imagesegmentation It means that the global convex optimizationmodel of image segmentation has been built based onintegration of a variety of image segmentation theories andhas obvious advantages In addition the method of iterativesolution of the self-adaptive method can also be integratedinto the segmentation process to ensure the accuracy ofsegmentation method further However the speed of thealgorithm will be affected Therefore it is important to findefficient and accurate numerical method

42 Efficiency Comparison of Multiscale Adaptive WaveletNumerical Method and Difference Method The C-V modelwas used for 256 times 302 images segmentation and divideddifference method was used to disperse partial differentialequations So discrete 7312 (256 times 302) ordinary differentialequations are huge solvingworkload But the adaptivewaveletprecise integration method can reduce the scale to 9576equations It can improve solution efficiency greatly due toless workload and low memory demand Of course the useof adaptive wavelet precise integration method for solvingthe number of distribution points will dynamically changeas the solution process In addition as shown in Figure 3distribution points are relatively dense within the ellipsering and another location was sparse The evident grayscaledifference between the infected and the healthy parts leadto this special points distribution Furthermore distributionpoints also exhibit regular matrix form which result fromblock solving method of wavelet transform to improve theefficiency The matrix-like distribution is from the boundaryeffect among the different blocks The interval wavelet caneffectively reduce the range effect but it will also increase thecomputation work of the wavelet transform

In this paper difference method was tested in MATLABThewavelet interpolation operatorwas implementedwithVCprogramming and other parts with MATLAB programmingOn the same computer difference method takes 03 secondsthe adaptive wavelet precise integration method takes 018seconds The results also show that the wavelet transform

10 Mathematical Problems in Engineering

of the iterative process reduces the overall computationalefficiency of the algorithm

5 Conclusions

Shannon wavelet precise integration method is a new imagesegmentation method based on the C-V model which wasused to construct adaptive wavelet interpolation operator dueto multiscale characteristics of wavelet transform combinedwith the time precise integration technology The methodmakes full use of the multiscale characteristics and thehigh precise performance of precise integration methodCompared to the gradient method and wavelet transformmethod of image segmentation object boundary obtained byWPIM segmentation method is clear and closed comparedto the watershed method the WPIM method avoids over-and undersegmentation problems and is very suitable formeasurement of image segmentation such as onion qualityassessment

The adaptive interpolation operator in the Shannonwavelet precision integration method can reduce the amountof the collocation points and improve the calculation effi-ciency As the interpolation operator contains a wavelettransform process the corresponding algorithm needs to doa wavelet transform between each two iteration time stepsSo the cost of the wavelet transformation is an importantpart of the calculation amount of the algorithm Compactlysupported orthogonal wavelet function can be expected tosolve the problem

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors appreciate the funding support from NationalNatural Science Foundation of China (Award nos 41171184and 41171337) The software tools were provided by the Foodand Fiber Sensing Lab of University of Georgia and theComputer Center of China Agricultural University

References

[1] Y Chen Y Xia Y Bian and Z-P Zhong ldquoImage measurementof precision aluminum alloy forgingsrdquo Journal of PlasticityEngineering vol 17 no 6 pp 77ndash81 2010

[2] S-L Mei Q-S Lu S-W Zhang and L Jin ldquoAdaptive intervalwavelet precise integrationmethod for partial differential equa-tionsrdquo Applied Mathematics and Mechanics vol 26 no 3 pp364ndash371 2005

[3] H-H Yan ldquoAdaptive wavelet precise integration method fornonlinear black-scholes model based on variational iterationmethodrdquo Abstract and Applied Analysis vol 2013 Article ID735919 6 pages 2013

[4] S-L Pang ldquoWavelet numerical method for nonlinear randomsystemrdquo Transactions of the Chinese Society of AgriculturalMachinery vol 38 no 3 pp 168ndash170 2007

[5] Y Wang ldquoWavelet precise time-integration method for heatconduction equationrdquo Journal of Chongqing Institute of Technol-ogy vol 21 no 8 pp 130ndash132 2007

[6] L X Zhang Y Yang and S L Mei ldquoWavelet precise integrationmethod on image denoisingrdquoTransactions of the Chinese Societyof Agricultural Machinery vol 37 no 7 pp 109ndash112 2006

[7] W N Xu S L Mei P X Wang and Y Yang ldquoAdaptivewavelet precise integration method on remote sensing imagedenoisingrdquo Transactions of the Chinese Society of AgriculturalMachinery vol 42 no 4 pp 148ndash152 2011

[8] R-Y Xing ldquoWavelet-based homotopy analysismethod for non-linear matrix system and its application in burgers equationrdquoMathematical Problems in Engineering vol 2013 Article ID982810 7 pages 2013

[9] S-L Mei ldquoConstruction of target controllable image segmen-tation model based on homotopy perturbation technologyrdquoAbstract and Applied Analysis vol 2013 Article ID 131207 8pages 2013

[10] L Liu ldquoConstruction of interval shannon wavelet and itsapplication in solving nonlinear black-scholes equationrdquoMath-ematical Problems in Engineering vol 2014 Article ID 541023 8pages 2014

[11] C Cattani ldquoShannon wavelets theoryrdquo Mathematical Problemsin Engineering vol 2008 Article ID 164808 24 pages 2008

[12] C Cattani ldquoSecond order Shannon wavelet approximationof C2-functionsrdquo UPB Scientific Bulletin Series A AppliedMathematics and Physics vol 73 no 3 pp 73ndash84 2011

[13] C Cattani and L M S Ruiz ldquoDiscrete differential operators inmultidimensional haar wavelet spacesrdquo International Journal ofMathematics and Mathematical Sciences vol 2004 no 44 pp2347ndash2355 2004

[14] C Cattani A Ciancio and B Lods ldquoOn a mathematical modelof immune competitionrdquo Applied Mathematics Letters vol 19no 7 pp 678ndash683 2006

[15] C Cattani ldquoConnection coefficients of Shannon waveletsrdquoMathematical Modelling and Analysis vol 11 no 2 pp 117ndash1322006

[16] S-L Mei and D-H Zhu ldquoInterval shannon wavelet collocationmethod for fractional fokker-planck equationrdquo Advances inMathematical Physics vol 2013 Article ID 821820 12 pages2013

[17] L-W Liu ldquoInterval wavelet numerical method on fokker-planck equations for nonlinear random systemrdquo Advances inMathematical Physics vol 2013 Article ID 651357 7 pages 2013

[18] J-H He ldquoHomotopy perturbation method for bifurcation ofnonlinear problemsrdquo International Journal of Nonlinear Sciencesand Numerical Simulation vol 6 no 2 pp 207ndash208 2005

[19] J-H He ldquoSome asymptotic methods for strongly nonlinearequationsrdquo International Journal of Modern Physics B vol 20no 10 pp 1141ndash1199 2006

[20] J-H He ldquoVariational iteration method-Some recent resultsand new interpretationsrdquo Journal of Computational and AppliedMathematics vol 207 no 1 pp 3ndash17 2007

[21] S-L Mei C-J Du and S-W Zhang ldquoAsymptotic numericalmethod for multi-degree-of-freedom nonlinear dynamic sys-temsrdquo Chaos Solitons and Fractals vol 35 no 3 pp 536ndash5422008

[22] S-L Mei and S-W Zhang ldquoCoupling technique of variationaliteration and homotopy perturbation methods for nonlinearmatrix differential equationsrdquoComputers andMathematics withApplications vol 54 no 7-8 pp 1092ndash1100 2007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

coefficients We convert (2) to wavelet multiscale discreteformat of the level set function by stead of (25)

119889120601119869(119909 119910 119905)

119889119905

= 120575120576(120601119869(119909 119910 119905))

times [] div(nabla120601119869(119909 119910 119905)

1003816100381610038161003816nabla120601119869 (119909 119910 119905)

1003816100381610038161003816

)minus1205821

10038161003816100381610038161198680 minus 11988811003816100381610038161003816

2

+1205822

10038161003816100381610038161198680 minus 11988821003816100381610038161003816

2

]

(26)

It is the most direct way for dynamic adaptation toonly retain the distribution points of corresponding waveletcoefficients that satisfy the condition

min (100381610038161003816100381610038161205721

1198951198961111989612

(119905)10038161003816100381610038161003816100381610038161003816100381610038161205722

1198951198961111989612

(119905)10038161003816100381610038161003816100381610038161003816100381610038161205723

1198951198961111989612

(119905)10038161003816100381610038161003816) ge 120576 (27)

Time domain numerical integration of partial differentialequations is an iterative process therefore some pointswhich are possible important next step need to be kept toenable the algorithm to track singularities of solutions Soadjacent points of distribution points also should be keptTheadjacent region can be delineated as follows

1003816100381610038161003816119904 minus 1198951003816100381610038161003816 le 119872

10038161003816100381610038161003816119909119904

119896minus 119909119895

119894

10038161003816100381610038161003816le 120576119909

10038161003816100381610038161003816119909119904

119896minus 119909119895

119894

10038161003816100381610038161003816le 120576119910

(28)

where 119904 119895 are numbers of different scale wavelet and 119896 119894119872 isin

119885 120576119909 120576119910are constant

32 Nonlinear Discrete Ordinary Differential EquationsBecause ] is a small parameter in ordinary differentialequation (26) value of ] div(nabla120601119869(119909 119910 119905)|nabla120601119869(119909 119910 119905)|) is lowEquation (26) can be converted into

119889120601119869(119909 119910 119905)

119889119905

= 120575120576(120601119869(119909 119910 119905))

times [] div(nabla120601119869(119909 119910 119905

0)

1003816100381610038161003816nabla120601119869 (119909 119910 119905

0)1003816100381610038161003816

) minus 1205821

10038161003816100381610038161198680 minus 11988811003816100381610038161003816

2

+ 1205822

10038161003816100381610038161198680 minus 11988821003816100381610038161003816

2

]

(29)

The solution of (29) is

120601119869(119909 119910 119905) = (

120576119898

2120587)

13

+ (119898

21205871205765)

minus13

(30)

where

119898 = ] div(nabla120601119869(119909 119910 119905

0)

1003816100381610038161003816nabla120601119869 (119909 119910 119905

0)1003816100381610038161003816

)

minus 1205821

10038161003816100381610038161198680 minus 11988811003816100381610038161003816

2

+ 1205822

10038161003816100381610038161198680 minus 11988821003816100381610038161003816

2

(31)

We can get the solution of ordinary differential equations(26) by iterative solution (30)

4 Experiment and Discussion ofOnion Infected Region Segmentation

Figure 1(a) is a 256 times 302 image of an onion infected bysour skin virus We noticed that the onion has a water-soaked appearance Compared with the background thegrayscale difference between the water-soaked appearanceand the healthy part is smaller So it is beneficial to compareperformance of the different algorithms Figure 1(b) is anideal segmentation results Segmentation target is infectedregions of onion however infected part is often not uniformIn the image the difference of gradient is less than 1 at upperleft part due to slight infection so the algorithm is difficult toprecise segmentation based on global threshold Thereforethe best one of the different segmentation algorithms candistinguish themajority of the infected regionwithout seriousoversegmentationThen it is easy to identify andmeasure theinfected portions by using a priori knowledge

41 Comparison among Different SegmentationMethods Thecommon image segmentation methods including water-shed algorithm Sobel operator and Canny edge detectionalgorithm Otsu algorithm and an effective and commonquad-tree decomposition algorithm were selected for com-parison Shannon wavelet was employed to construct thewavelet interpolation operator The representation of Shan-non wavelet is based upon approximating the Dirac deltafunction as a band-limited function and is given by

119908 (119909) =sin (120587119909)120587119909

(32)

Consider a one-dimensional function 119891(119909) 119909 isin [119886 119887] Adiscrete point sequence of the variable 119909 is defined as

119909119899= 119886 +

119887 minus 119886

2119895sdot 119899 119895 isin Z 119899 = 0 1 2 2

119895 (33)

and the corresponding discrete point sequence of the scalingfunction 120601(119909) can be defined as

119908119895119899(119909) = 119908

119895(119909 minus 119909

119899) =

sin (2119895120587 (119887 minus 119886)) (119909 minus 119909119899)

(2119895120587 (119887 minus 119886)) (119909 minus 119909119899)

(34)

The first and second order derivatives of 120601119895(119909 minus 119909

119899) at the

discrete point 119909119896are

1206011015840

119895(119909119896minus 119909119899) =

0 119896 = 119899

2119895 cos [120587 (119896 minus 119899)](119896 minus 119899) (119887 minus 119886)

119896 = 119899

12060110158401015840

119895(119909119896minus 119909119899) =

minus1205872

3((119887 minus 119886)2119895)2 119896 = 119899

minus2 cos [120587 (119896 minus 119899)]

((119887 minus 119886) 2119895)2

(119896 minus 119899)2 119896 = 119899

(35)

The corresponding 2-dimension weight function can be rep-resented as the tensor product form of the above equations

8 Mathematical Problems in Engineering

(a) Original image (b) Segmentation target

Figure 1 Burkholderia cepacia (ex Burkholder) infected onion and the target segmentation

(a) Grayscale image (b) Gradient image (c) OSTU

(d) Watershed 1 (e) Watershed 2 (f) Qtdecomp

(g) Sobel (h) Canny (i) Wavelet precise integration

Figure 2 Comparison of various segmentation methods

Mathematical Problems in Engineering 9

300

250

200

150

100

50

00 50 100 150 200 250 300

Figure 3 Adaptive wavelet collocation points on level set

The experimental procedure is described as follows

(1) convert the infested onion image to grayscale(Figure 2(a)) and solve for the gradient map(Figure 2(b))

(2) use the grayscale image to test the Sobel operatorCanny operator Qtdecomp algorithms and waveletprecise integration method Use the gradient mat totest the OSTU method and watershed algorithm

(3) applying the watershed method to segment the imagewhich has been processed by OSTU method in orderto avoid the oversegmentation from the watershedalgorithm

The segmentation results are shown in Figure 2 Thewatershed algorithm segmentation result shows serious over-segmentation (Figure 2(d)) and cannot recognize the infectedpart Although OSTUmethod separated part infected regionof the onion the partition boundary is discontinuity andis difficult to measure infection specific gravity To avoidoversegmentation OSTUwas overlapped with the watershedsegmentationThe result is shown in Figure 2(e) in which thepartition boundary is clear but is unable to distinguish thevirus infected part The Sobel operator recognition on thepart of the infection is also not good (Figure 2(g)) Cannyoperator and Qtdecomp algorithm identified the area ofinfection but the boundary points of segmentation regionare disorder and cannot be measuredThe precise integrationmethod presented in this paper can identify the infected areaclearly So it is helpful to onion evaluation and classification

In fact it is impossible to project image segmentationby a single algorithm The important reason that partialdifferential equations are effective for image segmentationis that the method integrated many image segmentationprinciples to the model of partial differential equations Inthis paper the C-V image segmentation model is a globalconvex optimization variational model which is establishedon image piecewise smooth (119888

1and 1198882are the average gray

values inside (Ω1) and outside (Ω

2) of the object contour

respectively) To ensure the accuracy of image segmentation

the curvature of the image the border gradient and levelset function evolution were taken into account in imagesegmentation It means that the global convex optimizationmodel of image segmentation has been built based onintegration of a variety of image segmentation theories andhas obvious advantages In addition the method of iterativesolution of the self-adaptive method can also be integratedinto the segmentation process to ensure the accuracy ofsegmentation method further However the speed of thealgorithm will be affected Therefore it is important to findefficient and accurate numerical method

42 Efficiency Comparison of Multiscale Adaptive WaveletNumerical Method and Difference Method The C-V modelwas used for 256 times 302 images segmentation and divideddifference method was used to disperse partial differentialequations So discrete 7312 (256 times 302) ordinary differentialequations are huge solvingworkload But the adaptivewaveletprecise integration method can reduce the scale to 9576equations It can improve solution efficiency greatly due toless workload and low memory demand Of course the useof adaptive wavelet precise integration method for solvingthe number of distribution points will dynamically changeas the solution process In addition as shown in Figure 3distribution points are relatively dense within the ellipsering and another location was sparse The evident grayscaledifference between the infected and the healthy parts leadto this special points distribution Furthermore distributionpoints also exhibit regular matrix form which result fromblock solving method of wavelet transform to improve theefficiency The matrix-like distribution is from the boundaryeffect among the different blocks The interval wavelet caneffectively reduce the range effect but it will also increase thecomputation work of the wavelet transform

In this paper difference method was tested in MATLABThewavelet interpolation operatorwas implementedwithVCprogramming and other parts with MATLAB programmingOn the same computer difference method takes 03 secondsthe adaptive wavelet precise integration method takes 018seconds The results also show that the wavelet transform

10 Mathematical Problems in Engineering

of the iterative process reduces the overall computationalefficiency of the algorithm

5 Conclusions

Shannon wavelet precise integration method is a new imagesegmentation method based on the C-V model which wasused to construct adaptive wavelet interpolation operator dueto multiscale characteristics of wavelet transform combinedwith the time precise integration technology The methodmakes full use of the multiscale characteristics and thehigh precise performance of precise integration methodCompared to the gradient method and wavelet transformmethod of image segmentation object boundary obtained byWPIM segmentation method is clear and closed comparedto the watershed method the WPIM method avoids over-and undersegmentation problems and is very suitable formeasurement of image segmentation such as onion qualityassessment

The adaptive interpolation operator in the Shannonwavelet precision integration method can reduce the amountof the collocation points and improve the calculation effi-ciency As the interpolation operator contains a wavelettransform process the corresponding algorithm needs to doa wavelet transform between each two iteration time stepsSo the cost of the wavelet transformation is an importantpart of the calculation amount of the algorithm Compactlysupported orthogonal wavelet function can be expected tosolve the problem

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors appreciate the funding support from NationalNatural Science Foundation of China (Award nos 41171184and 41171337) The software tools were provided by the Foodand Fiber Sensing Lab of University of Georgia and theComputer Center of China Agricultural University

References

[1] Y Chen Y Xia Y Bian and Z-P Zhong ldquoImage measurementof precision aluminum alloy forgingsrdquo Journal of PlasticityEngineering vol 17 no 6 pp 77ndash81 2010

[2] S-L Mei Q-S Lu S-W Zhang and L Jin ldquoAdaptive intervalwavelet precise integrationmethod for partial differential equa-tionsrdquo Applied Mathematics and Mechanics vol 26 no 3 pp364ndash371 2005

[3] H-H Yan ldquoAdaptive wavelet precise integration method fornonlinear black-scholes model based on variational iterationmethodrdquo Abstract and Applied Analysis vol 2013 Article ID735919 6 pages 2013

[4] S-L Pang ldquoWavelet numerical method for nonlinear randomsystemrdquo Transactions of the Chinese Society of AgriculturalMachinery vol 38 no 3 pp 168ndash170 2007

[5] Y Wang ldquoWavelet precise time-integration method for heatconduction equationrdquo Journal of Chongqing Institute of Technol-ogy vol 21 no 8 pp 130ndash132 2007

[6] L X Zhang Y Yang and S L Mei ldquoWavelet precise integrationmethod on image denoisingrdquoTransactions of the Chinese Societyof Agricultural Machinery vol 37 no 7 pp 109ndash112 2006

[7] W N Xu S L Mei P X Wang and Y Yang ldquoAdaptivewavelet precise integration method on remote sensing imagedenoisingrdquo Transactions of the Chinese Society of AgriculturalMachinery vol 42 no 4 pp 148ndash152 2011

[8] R-Y Xing ldquoWavelet-based homotopy analysismethod for non-linear matrix system and its application in burgers equationrdquoMathematical Problems in Engineering vol 2013 Article ID982810 7 pages 2013

[9] S-L Mei ldquoConstruction of target controllable image segmen-tation model based on homotopy perturbation technologyrdquoAbstract and Applied Analysis vol 2013 Article ID 131207 8pages 2013

[10] L Liu ldquoConstruction of interval shannon wavelet and itsapplication in solving nonlinear black-scholes equationrdquoMath-ematical Problems in Engineering vol 2014 Article ID 541023 8pages 2014

[11] C Cattani ldquoShannon wavelets theoryrdquo Mathematical Problemsin Engineering vol 2008 Article ID 164808 24 pages 2008

[12] C Cattani ldquoSecond order Shannon wavelet approximationof C2-functionsrdquo UPB Scientific Bulletin Series A AppliedMathematics and Physics vol 73 no 3 pp 73ndash84 2011

[13] C Cattani and L M S Ruiz ldquoDiscrete differential operators inmultidimensional haar wavelet spacesrdquo International Journal ofMathematics and Mathematical Sciences vol 2004 no 44 pp2347ndash2355 2004

[14] C Cattani A Ciancio and B Lods ldquoOn a mathematical modelof immune competitionrdquo Applied Mathematics Letters vol 19no 7 pp 678ndash683 2006

[15] C Cattani ldquoConnection coefficients of Shannon waveletsrdquoMathematical Modelling and Analysis vol 11 no 2 pp 117ndash1322006

[16] S-L Mei and D-H Zhu ldquoInterval shannon wavelet collocationmethod for fractional fokker-planck equationrdquo Advances inMathematical Physics vol 2013 Article ID 821820 12 pages2013

[17] L-W Liu ldquoInterval wavelet numerical method on fokker-planck equations for nonlinear random systemrdquo Advances inMathematical Physics vol 2013 Article ID 651357 7 pages 2013

[18] J-H He ldquoHomotopy perturbation method for bifurcation ofnonlinear problemsrdquo International Journal of Nonlinear Sciencesand Numerical Simulation vol 6 no 2 pp 207ndash208 2005

[19] J-H He ldquoSome asymptotic methods for strongly nonlinearequationsrdquo International Journal of Modern Physics B vol 20no 10 pp 1141ndash1199 2006

[20] J-H He ldquoVariational iteration method-Some recent resultsand new interpretationsrdquo Journal of Computational and AppliedMathematics vol 207 no 1 pp 3ndash17 2007

[21] S-L Mei C-J Du and S-W Zhang ldquoAsymptotic numericalmethod for multi-degree-of-freedom nonlinear dynamic sys-temsrdquo Chaos Solitons and Fractals vol 35 no 3 pp 536ndash5422008

[22] S-L Mei and S-W Zhang ldquoCoupling technique of variationaliteration and homotopy perturbation methods for nonlinearmatrix differential equationsrdquoComputers andMathematics withApplications vol 54 no 7-8 pp 1092ndash1100 2007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

(a) Original image (b) Segmentation target

Figure 1 Burkholderia cepacia (ex Burkholder) infected onion and the target segmentation

(a) Grayscale image (b) Gradient image (c) OSTU

(d) Watershed 1 (e) Watershed 2 (f) Qtdecomp

(g) Sobel (h) Canny (i) Wavelet precise integration

Figure 2 Comparison of various segmentation methods

Mathematical Problems in Engineering 9

300

250

200

150

100

50

00 50 100 150 200 250 300

Figure 3 Adaptive wavelet collocation points on level set

The experimental procedure is described as follows

(1) convert the infested onion image to grayscale(Figure 2(a)) and solve for the gradient map(Figure 2(b))

(2) use the grayscale image to test the Sobel operatorCanny operator Qtdecomp algorithms and waveletprecise integration method Use the gradient mat totest the OSTU method and watershed algorithm

(3) applying the watershed method to segment the imagewhich has been processed by OSTU method in orderto avoid the oversegmentation from the watershedalgorithm

The segmentation results are shown in Figure 2 Thewatershed algorithm segmentation result shows serious over-segmentation (Figure 2(d)) and cannot recognize the infectedpart Although OSTUmethod separated part infected regionof the onion the partition boundary is discontinuity andis difficult to measure infection specific gravity To avoidoversegmentation OSTUwas overlapped with the watershedsegmentationThe result is shown in Figure 2(e) in which thepartition boundary is clear but is unable to distinguish thevirus infected part The Sobel operator recognition on thepart of the infection is also not good (Figure 2(g)) Cannyoperator and Qtdecomp algorithm identified the area ofinfection but the boundary points of segmentation regionare disorder and cannot be measuredThe precise integrationmethod presented in this paper can identify the infected areaclearly So it is helpful to onion evaluation and classification

In fact it is impossible to project image segmentationby a single algorithm The important reason that partialdifferential equations are effective for image segmentationis that the method integrated many image segmentationprinciples to the model of partial differential equations Inthis paper the C-V image segmentation model is a globalconvex optimization variational model which is establishedon image piecewise smooth (119888

1and 1198882are the average gray

values inside (Ω1) and outside (Ω

2) of the object contour

respectively) To ensure the accuracy of image segmentation

the curvature of the image the border gradient and levelset function evolution were taken into account in imagesegmentation It means that the global convex optimizationmodel of image segmentation has been built based onintegration of a variety of image segmentation theories andhas obvious advantages In addition the method of iterativesolution of the self-adaptive method can also be integratedinto the segmentation process to ensure the accuracy ofsegmentation method further However the speed of thealgorithm will be affected Therefore it is important to findefficient and accurate numerical method

42 Efficiency Comparison of Multiscale Adaptive WaveletNumerical Method and Difference Method The C-V modelwas used for 256 times 302 images segmentation and divideddifference method was used to disperse partial differentialequations So discrete 7312 (256 times 302) ordinary differentialequations are huge solvingworkload But the adaptivewaveletprecise integration method can reduce the scale to 9576equations It can improve solution efficiency greatly due toless workload and low memory demand Of course the useof adaptive wavelet precise integration method for solvingthe number of distribution points will dynamically changeas the solution process In addition as shown in Figure 3distribution points are relatively dense within the ellipsering and another location was sparse The evident grayscaledifference between the infected and the healthy parts leadto this special points distribution Furthermore distributionpoints also exhibit regular matrix form which result fromblock solving method of wavelet transform to improve theefficiency The matrix-like distribution is from the boundaryeffect among the different blocks The interval wavelet caneffectively reduce the range effect but it will also increase thecomputation work of the wavelet transform

In this paper difference method was tested in MATLABThewavelet interpolation operatorwas implementedwithVCprogramming and other parts with MATLAB programmingOn the same computer difference method takes 03 secondsthe adaptive wavelet precise integration method takes 018seconds The results also show that the wavelet transform

10 Mathematical Problems in Engineering

of the iterative process reduces the overall computationalefficiency of the algorithm

5 Conclusions

Shannon wavelet precise integration method is a new imagesegmentation method based on the C-V model which wasused to construct adaptive wavelet interpolation operator dueto multiscale characteristics of wavelet transform combinedwith the time precise integration technology The methodmakes full use of the multiscale characteristics and thehigh precise performance of precise integration methodCompared to the gradient method and wavelet transformmethod of image segmentation object boundary obtained byWPIM segmentation method is clear and closed comparedto the watershed method the WPIM method avoids over-and undersegmentation problems and is very suitable formeasurement of image segmentation such as onion qualityassessment

The adaptive interpolation operator in the Shannonwavelet precision integration method can reduce the amountof the collocation points and improve the calculation effi-ciency As the interpolation operator contains a wavelettransform process the corresponding algorithm needs to doa wavelet transform between each two iteration time stepsSo the cost of the wavelet transformation is an importantpart of the calculation amount of the algorithm Compactlysupported orthogonal wavelet function can be expected tosolve the problem

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors appreciate the funding support from NationalNatural Science Foundation of China (Award nos 41171184and 41171337) The software tools were provided by the Foodand Fiber Sensing Lab of University of Georgia and theComputer Center of China Agricultural University

References

[1] Y Chen Y Xia Y Bian and Z-P Zhong ldquoImage measurementof precision aluminum alloy forgingsrdquo Journal of PlasticityEngineering vol 17 no 6 pp 77ndash81 2010

[2] S-L Mei Q-S Lu S-W Zhang and L Jin ldquoAdaptive intervalwavelet precise integrationmethod for partial differential equa-tionsrdquo Applied Mathematics and Mechanics vol 26 no 3 pp364ndash371 2005

[3] H-H Yan ldquoAdaptive wavelet precise integration method fornonlinear black-scholes model based on variational iterationmethodrdquo Abstract and Applied Analysis vol 2013 Article ID735919 6 pages 2013

[4] S-L Pang ldquoWavelet numerical method for nonlinear randomsystemrdquo Transactions of the Chinese Society of AgriculturalMachinery vol 38 no 3 pp 168ndash170 2007

[5] Y Wang ldquoWavelet precise time-integration method for heatconduction equationrdquo Journal of Chongqing Institute of Technol-ogy vol 21 no 8 pp 130ndash132 2007

[6] L X Zhang Y Yang and S L Mei ldquoWavelet precise integrationmethod on image denoisingrdquoTransactions of the Chinese Societyof Agricultural Machinery vol 37 no 7 pp 109ndash112 2006

[7] W N Xu S L Mei P X Wang and Y Yang ldquoAdaptivewavelet precise integration method on remote sensing imagedenoisingrdquo Transactions of the Chinese Society of AgriculturalMachinery vol 42 no 4 pp 148ndash152 2011

[8] R-Y Xing ldquoWavelet-based homotopy analysismethod for non-linear matrix system and its application in burgers equationrdquoMathematical Problems in Engineering vol 2013 Article ID982810 7 pages 2013

[9] S-L Mei ldquoConstruction of target controllable image segmen-tation model based on homotopy perturbation technologyrdquoAbstract and Applied Analysis vol 2013 Article ID 131207 8pages 2013

[10] L Liu ldquoConstruction of interval shannon wavelet and itsapplication in solving nonlinear black-scholes equationrdquoMath-ematical Problems in Engineering vol 2014 Article ID 541023 8pages 2014

[11] C Cattani ldquoShannon wavelets theoryrdquo Mathematical Problemsin Engineering vol 2008 Article ID 164808 24 pages 2008

[12] C Cattani ldquoSecond order Shannon wavelet approximationof C2-functionsrdquo UPB Scientific Bulletin Series A AppliedMathematics and Physics vol 73 no 3 pp 73ndash84 2011

[13] C Cattani and L M S Ruiz ldquoDiscrete differential operators inmultidimensional haar wavelet spacesrdquo International Journal ofMathematics and Mathematical Sciences vol 2004 no 44 pp2347ndash2355 2004

[14] C Cattani A Ciancio and B Lods ldquoOn a mathematical modelof immune competitionrdquo Applied Mathematics Letters vol 19no 7 pp 678ndash683 2006

[15] C Cattani ldquoConnection coefficients of Shannon waveletsrdquoMathematical Modelling and Analysis vol 11 no 2 pp 117ndash1322006

[16] S-L Mei and D-H Zhu ldquoInterval shannon wavelet collocationmethod for fractional fokker-planck equationrdquo Advances inMathematical Physics vol 2013 Article ID 821820 12 pages2013

[17] L-W Liu ldquoInterval wavelet numerical method on fokker-planck equations for nonlinear random systemrdquo Advances inMathematical Physics vol 2013 Article ID 651357 7 pages 2013

[18] J-H He ldquoHomotopy perturbation method for bifurcation ofnonlinear problemsrdquo International Journal of Nonlinear Sciencesand Numerical Simulation vol 6 no 2 pp 207ndash208 2005

[19] J-H He ldquoSome asymptotic methods for strongly nonlinearequationsrdquo International Journal of Modern Physics B vol 20no 10 pp 1141ndash1199 2006

[20] J-H He ldquoVariational iteration method-Some recent resultsand new interpretationsrdquo Journal of Computational and AppliedMathematics vol 207 no 1 pp 3ndash17 2007

[21] S-L Mei C-J Du and S-W Zhang ldquoAsymptotic numericalmethod for multi-degree-of-freedom nonlinear dynamic sys-temsrdquo Chaos Solitons and Fractals vol 35 no 3 pp 536ndash5422008

[22] S-L Mei and S-W Zhang ldquoCoupling technique of variationaliteration and homotopy perturbation methods for nonlinearmatrix differential equationsrdquoComputers andMathematics withApplications vol 54 no 7-8 pp 1092ndash1100 2007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 9

300

250

200

150

100

50

00 50 100 150 200 250 300

Figure 3 Adaptive wavelet collocation points on level set

The experimental procedure is described as follows

(1) convert the infested onion image to grayscale(Figure 2(a)) and solve for the gradient map(Figure 2(b))

(2) use the grayscale image to test the Sobel operatorCanny operator Qtdecomp algorithms and waveletprecise integration method Use the gradient mat totest the OSTU method and watershed algorithm

(3) applying the watershed method to segment the imagewhich has been processed by OSTU method in orderto avoid the oversegmentation from the watershedalgorithm

The segmentation results are shown in Figure 2 Thewatershed algorithm segmentation result shows serious over-segmentation (Figure 2(d)) and cannot recognize the infectedpart Although OSTUmethod separated part infected regionof the onion the partition boundary is discontinuity andis difficult to measure infection specific gravity To avoidoversegmentation OSTUwas overlapped with the watershedsegmentationThe result is shown in Figure 2(e) in which thepartition boundary is clear but is unable to distinguish thevirus infected part The Sobel operator recognition on thepart of the infection is also not good (Figure 2(g)) Cannyoperator and Qtdecomp algorithm identified the area ofinfection but the boundary points of segmentation regionare disorder and cannot be measuredThe precise integrationmethod presented in this paper can identify the infected areaclearly So it is helpful to onion evaluation and classification

In fact it is impossible to project image segmentationby a single algorithm The important reason that partialdifferential equations are effective for image segmentationis that the method integrated many image segmentationprinciples to the model of partial differential equations Inthis paper the C-V image segmentation model is a globalconvex optimization variational model which is establishedon image piecewise smooth (119888

1and 1198882are the average gray

values inside (Ω1) and outside (Ω

2) of the object contour

respectively) To ensure the accuracy of image segmentation

the curvature of the image the border gradient and levelset function evolution were taken into account in imagesegmentation It means that the global convex optimizationmodel of image segmentation has been built based onintegration of a variety of image segmentation theories andhas obvious advantages In addition the method of iterativesolution of the self-adaptive method can also be integratedinto the segmentation process to ensure the accuracy ofsegmentation method further However the speed of thealgorithm will be affected Therefore it is important to findefficient and accurate numerical method

42 Efficiency Comparison of Multiscale Adaptive WaveletNumerical Method and Difference Method The C-V modelwas used for 256 times 302 images segmentation and divideddifference method was used to disperse partial differentialequations So discrete 7312 (256 times 302) ordinary differentialequations are huge solvingworkload But the adaptivewaveletprecise integration method can reduce the scale to 9576equations It can improve solution efficiency greatly due toless workload and low memory demand Of course the useof adaptive wavelet precise integration method for solvingthe number of distribution points will dynamically changeas the solution process In addition as shown in Figure 3distribution points are relatively dense within the ellipsering and another location was sparse The evident grayscaledifference between the infected and the healthy parts leadto this special points distribution Furthermore distributionpoints also exhibit regular matrix form which result fromblock solving method of wavelet transform to improve theefficiency The matrix-like distribution is from the boundaryeffect among the different blocks The interval wavelet caneffectively reduce the range effect but it will also increase thecomputation work of the wavelet transform

In this paper difference method was tested in MATLABThewavelet interpolation operatorwas implementedwithVCprogramming and other parts with MATLAB programmingOn the same computer difference method takes 03 secondsthe adaptive wavelet precise integration method takes 018seconds The results also show that the wavelet transform

10 Mathematical Problems in Engineering

of the iterative process reduces the overall computationalefficiency of the algorithm

5 Conclusions

Shannon wavelet precise integration method is a new imagesegmentation method based on the C-V model which wasused to construct adaptive wavelet interpolation operator dueto multiscale characteristics of wavelet transform combinedwith the time precise integration technology The methodmakes full use of the multiscale characteristics and thehigh precise performance of precise integration methodCompared to the gradient method and wavelet transformmethod of image segmentation object boundary obtained byWPIM segmentation method is clear and closed comparedto the watershed method the WPIM method avoids over-and undersegmentation problems and is very suitable formeasurement of image segmentation such as onion qualityassessment

The adaptive interpolation operator in the Shannonwavelet precision integration method can reduce the amountof the collocation points and improve the calculation effi-ciency As the interpolation operator contains a wavelettransform process the corresponding algorithm needs to doa wavelet transform between each two iteration time stepsSo the cost of the wavelet transformation is an importantpart of the calculation amount of the algorithm Compactlysupported orthogonal wavelet function can be expected tosolve the problem

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors appreciate the funding support from NationalNatural Science Foundation of China (Award nos 41171184and 41171337) The software tools were provided by the Foodand Fiber Sensing Lab of University of Georgia and theComputer Center of China Agricultural University

References

[1] Y Chen Y Xia Y Bian and Z-P Zhong ldquoImage measurementof precision aluminum alloy forgingsrdquo Journal of PlasticityEngineering vol 17 no 6 pp 77ndash81 2010

[2] S-L Mei Q-S Lu S-W Zhang and L Jin ldquoAdaptive intervalwavelet precise integrationmethod for partial differential equa-tionsrdquo Applied Mathematics and Mechanics vol 26 no 3 pp364ndash371 2005

[3] H-H Yan ldquoAdaptive wavelet precise integration method fornonlinear black-scholes model based on variational iterationmethodrdquo Abstract and Applied Analysis vol 2013 Article ID735919 6 pages 2013

[4] S-L Pang ldquoWavelet numerical method for nonlinear randomsystemrdquo Transactions of the Chinese Society of AgriculturalMachinery vol 38 no 3 pp 168ndash170 2007

[5] Y Wang ldquoWavelet precise time-integration method for heatconduction equationrdquo Journal of Chongqing Institute of Technol-ogy vol 21 no 8 pp 130ndash132 2007

[6] L X Zhang Y Yang and S L Mei ldquoWavelet precise integrationmethod on image denoisingrdquoTransactions of the Chinese Societyof Agricultural Machinery vol 37 no 7 pp 109ndash112 2006

[7] W N Xu S L Mei P X Wang and Y Yang ldquoAdaptivewavelet precise integration method on remote sensing imagedenoisingrdquo Transactions of the Chinese Society of AgriculturalMachinery vol 42 no 4 pp 148ndash152 2011

[8] R-Y Xing ldquoWavelet-based homotopy analysismethod for non-linear matrix system and its application in burgers equationrdquoMathematical Problems in Engineering vol 2013 Article ID982810 7 pages 2013

[9] S-L Mei ldquoConstruction of target controllable image segmen-tation model based on homotopy perturbation technologyrdquoAbstract and Applied Analysis vol 2013 Article ID 131207 8pages 2013

[10] L Liu ldquoConstruction of interval shannon wavelet and itsapplication in solving nonlinear black-scholes equationrdquoMath-ematical Problems in Engineering vol 2014 Article ID 541023 8pages 2014

[11] C Cattani ldquoShannon wavelets theoryrdquo Mathematical Problemsin Engineering vol 2008 Article ID 164808 24 pages 2008

[12] C Cattani ldquoSecond order Shannon wavelet approximationof C2-functionsrdquo UPB Scientific Bulletin Series A AppliedMathematics and Physics vol 73 no 3 pp 73ndash84 2011

[13] C Cattani and L M S Ruiz ldquoDiscrete differential operators inmultidimensional haar wavelet spacesrdquo International Journal ofMathematics and Mathematical Sciences vol 2004 no 44 pp2347ndash2355 2004

[14] C Cattani A Ciancio and B Lods ldquoOn a mathematical modelof immune competitionrdquo Applied Mathematics Letters vol 19no 7 pp 678ndash683 2006

[15] C Cattani ldquoConnection coefficients of Shannon waveletsrdquoMathematical Modelling and Analysis vol 11 no 2 pp 117ndash1322006

[16] S-L Mei and D-H Zhu ldquoInterval shannon wavelet collocationmethod for fractional fokker-planck equationrdquo Advances inMathematical Physics vol 2013 Article ID 821820 12 pages2013

[17] L-W Liu ldquoInterval wavelet numerical method on fokker-planck equations for nonlinear random systemrdquo Advances inMathematical Physics vol 2013 Article ID 651357 7 pages 2013

[18] J-H He ldquoHomotopy perturbation method for bifurcation ofnonlinear problemsrdquo International Journal of Nonlinear Sciencesand Numerical Simulation vol 6 no 2 pp 207ndash208 2005

[19] J-H He ldquoSome asymptotic methods for strongly nonlinearequationsrdquo International Journal of Modern Physics B vol 20no 10 pp 1141ndash1199 2006

[20] J-H He ldquoVariational iteration method-Some recent resultsand new interpretationsrdquo Journal of Computational and AppliedMathematics vol 207 no 1 pp 3ndash17 2007

[21] S-L Mei C-J Du and S-W Zhang ldquoAsymptotic numericalmethod for multi-degree-of-freedom nonlinear dynamic sys-temsrdquo Chaos Solitons and Fractals vol 35 no 3 pp 536ndash5422008

[22] S-L Mei and S-W Zhang ldquoCoupling technique of variationaliteration and homotopy perturbation methods for nonlinearmatrix differential equationsrdquoComputers andMathematics withApplications vol 54 no 7-8 pp 1092ndash1100 2007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Mathematical Problems in Engineering

of the iterative process reduces the overall computationalefficiency of the algorithm

5 Conclusions

Shannon wavelet precise integration method is a new imagesegmentation method based on the C-V model which wasused to construct adaptive wavelet interpolation operator dueto multiscale characteristics of wavelet transform combinedwith the time precise integration technology The methodmakes full use of the multiscale characteristics and thehigh precise performance of precise integration methodCompared to the gradient method and wavelet transformmethod of image segmentation object boundary obtained byWPIM segmentation method is clear and closed comparedto the watershed method the WPIM method avoids over-and undersegmentation problems and is very suitable formeasurement of image segmentation such as onion qualityassessment

The adaptive interpolation operator in the Shannonwavelet precision integration method can reduce the amountof the collocation points and improve the calculation effi-ciency As the interpolation operator contains a wavelettransform process the corresponding algorithm needs to doa wavelet transform between each two iteration time stepsSo the cost of the wavelet transformation is an importantpart of the calculation amount of the algorithm Compactlysupported orthogonal wavelet function can be expected tosolve the problem

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors appreciate the funding support from NationalNatural Science Foundation of China (Award nos 41171184and 41171337) The software tools were provided by the Foodand Fiber Sensing Lab of University of Georgia and theComputer Center of China Agricultural University

References

[1] Y Chen Y Xia Y Bian and Z-P Zhong ldquoImage measurementof precision aluminum alloy forgingsrdquo Journal of PlasticityEngineering vol 17 no 6 pp 77ndash81 2010

[2] S-L Mei Q-S Lu S-W Zhang and L Jin ldquoAdaptive intervalwavelet precise integrationmethod for partial differential equa-tionsrdquo Applied Mathematics and Mechanics vol 26 no 3 pp364ndash371 2005

[3] H-H Yan ldquoAdaptive wavelet precise integration method fornonlinear black-scholes model based on variational iterationmethodrdquo Abstract and Applied Analysis vol 2013 Article ID735919 6 pages 2013

[4] S-L Pang ldquoWavelet numerical method for nonlinear randomsystemrdquo Transactions of the Chinese Society of AgriculturalMachinery vol 38 no 3 pp 168ndash170 2007

[5] Y Wang ldquoWavelet precise time-integration method for heatconduction equationrdquo Journal of Chongqing Institute of Technol-ogy vol 21 no 8 pp 130ndash132 2007

[6] L X Zhang Y Yang and S L Mei ldquoWavelet precise integrationmethod on image denoisingrdquoTransactions of the Chinese Societyof Agricultural Machinery vol 37 no 7 pp 109ndash112 2006

[7] W N Xu S L Mei P X Wang and Y Yang ldquoAdaptivewavelet precise integration method on remote sensing imagedenoisingrdquo Transactions of the Chinese Society of AgriculturalMachinery vol 42 no 4 pp 148ndash152 2011

[8] R-Y Xing ldquoWavelet-based homotopy analysismethod for non-linear matrix system and its application in burgers equationrdquoMathematical Problems in Engineering vol 2013 Article ID982810 7 pages 2013

[9] S-L Mei ldquoConstruction of target controllable image segmen-tation model based on homotopy perturbation technologyrdquoAbstract and Applied Analysis vol 2013 Article ID 131207 8pages 2013

[10] L Liu ldquoConstruction of interval shannon wavelet and itsapplication in solving nonlinear black-scholes equationrdquoMath-ematical Problems in Engineering vol 2014 Article ID 541023 8pages 2014

[11] C Cattani ldquoShannon wavelets theoryrdquo Mathematical Problemsin Engineering vol 2008 Article ID 164808 24 pages 2008

[12] C Cattani ldquoSecond order Shannon wavelet approximationof C2-functionsrdquo UPB Scientific Bulletin Series A AppliedMathematics and Physics vol 73 no 3 pp 73ndash84 2011

[13] C Cattani and L M S Ruiz ldquoDiscrete differential operators inmultidimensional haar wavelet spacesrdquo International Journal ofMathematics and Mathematical Sciences vol 2004 no 44 pp2347ndash2355 2004

[14] C Cattani A Ciancio and B Lods ldquoOn a mathematical modelof immune competitionrdquo Applied Mathematics Letters vol 19no 7 pp 678ndash683 2006

[15] C Cattani ldquoConnection coefficients of Shannon waveletsrdquoMathematical Modelling and Analysis vol 11 no 2 pp 117ndash1322006

[16] S-L Mei and D-H Zhu ldquoInterval shannon wavelet collocationmethod for fractional fokker-planck equationrdquo Advances inMathematical Physics vol 2013 Article ID 821820 12 pages2013

[17] L-W Liu ldquoInterval wavelet numerical method on fokker-planck equations for nonlinear random systemrdquo Advances inMathematical Physics vol 2013 Article ID 651357 7 pages 2013

[18] J-H He ldquoHomotopy perturbation method for bifurcation ofnonlinear problemsrdquo International Journal of Nonlinear Sciencesand Numerical Simulation vol 6 no 2 pp 207ndash208 2005

[19] J-H He ldquoSome asymptotic methods for strongly nonlinearequationsrdquo International Journal of Modern Physics B vol 20no 10 pp 1141ndash1199 2006

[20] J-H He ldquoVariational iteration method-Some recent resultsand new interpretationsrdquo Journal of Computational and AppliedMathematics vol 207 no 1 pp 3ndash17 2007

[21] S-L Mei C-J Du and S-W Zhang ldquoAsymptotic numericalmethod for multi-degree-of-freedom nonlinear dynamic sys-temsrdquo Chaos Solitons and Fractals vol 35 no 3 pp 536ndash5422008

[22] S-L Mei and S-W Zhang ldquoCoupling technique of variationaliteration and homotopy perturbation methods for nonlinearmatrix differential equationsrdquoComputers andMathematics withApplications vol 54 no 7-8 pp 1092ndash1100 2007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of