research article unsupervised joint image denoising and...

Research ArticleUnsupervised Joint Image Denoising and Active ContourSegmentation in Multidimensional Feature Space

Qi Ge1 Xiao-Yuan Jing2 Fei Wu2 Jingjie Yan1 and Hai-Bo Li13

1College of Telecommunications and Information Engineering Nanjing University of Posts and TelecommunicationsNanjing 210003 China2College of Automation Nanjing University of Posts and Telecommunications Nanjing 210003 China3KTH Royal Institute of Technology 10044 Stockholm Sweden

Correspondence should be addressed to Qi Ge geqinjupteducn

Received 28 April 2016 Accepted 26 June 2016

Academic Editor Giuseppina Colicchio

Copyright copy 2016 Qi Ge et al This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We describe a new method for simultaneous image denoising and level set-based active contour segmentation using multidimen-sional features We consider an image to be a surface embedded in a Riemannian manifold By defining a metric in the embeddedspace which in our case includes multidimensional image features as well as a level set-based active contourmodel a minimizationproblem in the image space can be obtained through the Polyakov action frameworkThe resultingminimization problem is solvedwith a dual algorithm for efficiency Benefits of this new method include the fact that it is independent of any artificial ldquorunningrdquoparameters and experiments using both synthetic and real images show that the method is robust with respect to noise and blurryobject boundaries

1 Introduction

Unsupervised image segmentation is an important problemwith many applications in science including medical imag-ing Image segmentation is a postprocessing problem inmanycomputer vision tasks its aim is to divide an image into finitenumber of subregionsThe features of different subregions areutilized as the segmentation criteria The statistical methodssuch as expectation-maximization (EM) algorithm [1] andfuzzy C-means clustering (FCM) algorithm [2] are applied inclassifying the pixels based on some particular image featuressegmentation criteria In general the statistical methodsachieve the classification based on only one segmentationcriterion However there is various kinds of features in animage and the features may vary spatially Therefore it will benot precise to use one kind of these methods How to extractthe features of an image and how to utilize these features asthe segmentation criterion are significant for segmentation

Many works utilize the difference between invariablepixel intensities as well as their spatial connectivity inassessing whether two pixels belong to the same object

These active contour models based on the level set method[3] classify the pixels by only one image feature that isthe image intensity based on uniform distribution [4ndash6]Nevertheless the image intensity varies spatially thus theimage intensity is not necessarily described by one kind ofspecific distribution For improving the precision the worksof [7 8] extract the multifeature to deal with more complexinformation content Simultaneously the additional artificialparameters are introduced thus it needs the experience to setthe parameters

The Polyakov action was introduced in image processingby Sochen et al in [9] This segmentation model is differentfrom the other segmentation methods in two ways Firstimages are represented as Riemannian manifolds embeddedin a higher dimensional spatial-feature manifold Secondthe Polyakov action provides an efficient mathematicalframework to embed the multifeature of images in higher-dimensional Riemannianmanifolds by harmonicmaps Bres-son et al [10] propose active contour models based on thePolyakov actionThese models map several kinds of featuresfor example color and texture into higher dimensional

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2016 Article ID 3909645 9 pageshttpdxdoiorg10115520163909645

2 Mathematical Problems in Engineering

space Because these models choose a metric with artificialparameters on the feature space it requires careful manualparameter-tuning

In this paper the proposed active contour model isformulated in the framework of the Polyakov action [9]Unlike the other related works [7ndash9] a metric on the featurespace manifold is defined by the invariant geometry ofimages Consequently the proposed method is purely basedon the geometrical features of images without any artificialparameters We implement the segmentation through twosteps First an approximated image removing the noisewhile preserving the main structures is found in the featurespace built on geometrical features of the original imageSecond the active contour is embedded into the feature spacebuilt on both the statistical and geometrical features of theapproximated image For efficiency we solve the proposedmodel via the improved Chambolle dual formulation [10] ofthe minimization problem

The paper is organized as follows In Section 2 we intro-duce the mathematical framework based on the Polyakovaction In Section 3 we introduce the proposed model andthe numerical algorithm of the proposed method is alsosummarized In Section 4 we validate our model by someexperiments on medical images In Section 5 we end thepaper by a brief conclusion

2 Geometrical Framework Based on WeightedPolyakov Action

Sochen et al introduce a general geometrical framework forlow-level vision based on the Polyakov action [9] In thisframework images are represented as the surfaces on a Rie-mannian manifold The Polyakov action is a functional thatmeasures the weight of a mapping X = (119883

1(120590) 119883

119898(120590))

between an 119899-dimensional embedded manifold (eg theimage manifold) Σ with coordinates 120590 = (120590

1 120590

119899)

and the 119898-dimensional manifold 119872 with the coordinates(119883

1(120590) 119883

119898(120590)) 119898 gt 119899 A Riemannian structure metric

119892119906V can be introduced to measure the local distances on

the embedded manifold Σ whereas we use the metric ℎ119894119895to

measure the distance on the manifold 119872 To measure theweight of the mapping X Σ 997891rarr 119872 the Polyakov actionis used as a generalization of the 119871

2-norm on the embedded

image to space feature manifold119872

119878 [119883

119894 119892119906V ℎ119894119895] = intradic119892119892

120583120592120597120583119883

119894120597120592119883

119895ℎ119894119895119889

119899120590 (1)

where 119892 is the determinant of the image metric tensor 119892119906V

and 119892

120583120592 is its inverse The metric 119892 is chosen as the inducedmetric obtained by the pullback relation 119892

119906V = ℎ119894119895120597120583119883

119894120597120592119883

119895the Polyakov energy is shortened to

119878 (119883

119894 ℎ119894119895) = intradic119892119889

119899120590 (2)

In the relevant works [7 8] the authors get the denoisedimage and the segmentation results byminimizing the energyfunctional (2) with respect to denoising and segmentationrespectively In seminal work [9] they embed grey images

in the feature (119909 119910 119868(119909 119910)) where 119868(119909 119910) is the grey inten-sity value for pixel (119909 119910) They choose a metric [ℎ

119894119895] =

diag(1 1 1205732) 120573 gt 0 is a constant Based on this metricon feature space and the Polyakov energy the regularizationterm on the intensity values is given by intradic1 + 120573

2|nabla119868|

2Although it allows setting the scale of the feature dimensionindependently of the spatial dimensions the accuracy of thescale is subject to the artificial parameter 120573

3 The Active Contour Model inMultifeature Space

In this work we utilize an improved geometrical frameworkbased on the weighted Polyakov action without any artificialparameter First we get an approximated image by embed-ding it into the feature space constituted by the features ofthe original image Second given the approximated imageactive contour is driven by embedding the level set functioninto the higher dimensional feature space composed ofthe geometrical and statistical features of the approximatedimage

31 Approximating Image under an Improved GeometricalFramework The original image 119868(119909 119910) is defined on theimage manifold Σ with coordinates (119909 119910) The approximatedimage 119906 is defined on the image manifold Σ and denotedby 119906(119909 119910) To preserve the main edges of the originalimage we extract the geometrical features of edgesnabla119878119868 = ((119868

119909)

3+ 119868119909(119868119910)

2 (119868119910)

3+ 119868119910(119868119909)

2) derived from the

anisotropic diffusion equation [2] Considering the intensityvalue 119868(119909 119910) as another feature we build the feature space(119909 119910 119906 (nabla

119878119868(119909 119910) minus nabla

119878119906(119909 119910)) (119868(119909 119910) minus 119906(119909 119910))) denoted

by (119909 119910 119906 1198911 1198912) for the sake of simplicity To avoid the

influence of the artificial parameter we choose a metrictensor [ℎ

119894119895] on the feature space 119872 which is defined by the

invariant geometry of the original image 119868 Consider [ℎ119894119895] =

diag(1 1 1 radic1 + 119868

2

119909+ 119868

2

119910+ 2119868119909119868119910 1radic1 + 119868

119909119909+ 119868119910119910

+ 2119868119909119910)

The pullback relation yields the determinant of metric tensor[119892120583120592] on manifold Σ

119892 = 1 + |nabla119906|

2+ (radic1 + 119868

2

119909+ 119868

2

119910+ 2119868119909119868119910)

2

sdot

1003816100381610038161003816

nabla119878119868 (119909 119910) minus nabla

119878119906 (119909 119910)

1003816100381610038161003816

2

+(

1

radic1 + 119868119909119909

+ 119868119910119910

+ 2119868119909119910

)

2

sdot

1003816100381610038161003816

119868 (119909 119910) minus 119906 (119909 119910)

1003816100381610038161003816

2

(3)

Analogizing based on the Polyakov energy (2) we get theapproximated image 119906 by minimizing the energy functionalas follows

1198641= int

Σ

|nabla119906| + radic1 + 119868

2

119909+ 119868

2

119910+ 2119868119909119868119910

1003816100381610038161003816

nabla119878119868 (119909 119910)

minus nabla119878119906 (119909 119910)

1003816100381610038161003816

119889119909 119889119910

Mathematical Problems in Engineering 3

+ int

Σ

1

radic1 + 119868119909119909

+ 119868119910119910

+ 2119868119909119910

1003816100381610038161003816

119868 (119909 119910)

minus 119906 (119909 119910)

1003816100381610038161003816

119889119909 119889119910

(4)

where theweight coefficient of second term corresponding tothe third element of the metric [ℎ

119894119895] denotes the coefficients

of first fundamental form in differential geometry When thisweight coefficient is larger the edge structure is enhancedin the vicinity of the edges otherwise smoothing the imageis strengthened The weight coefficient of the third termcorresponding to the last element of the metric [ℎ

119894119895] denotes

the coefficients of second fundamental form in differentialgeometry Approximating the intensity 119868(119909 119910) is strength-ened when this coefficient is larger whereas smoothing theimage is strengthened when the weight is smaller

32 Active Contour Evolution under the Improved GeometricalFramework The active contour is represented as the zerolevel set function 120601(119909 119910) = 0 on the image manifold ΣFor avoiding the effects of the intensity nonuniformity weextract the statistical features (119888

1 1198882) on the local region of

size 3 times 3 where 1198881 1198882denote the mean intensity in the local

region inside and outside the zero level set The feature spaceis (119909 119910 120601(119909 119910) (119906(119909 119910) minus 119888

1)

2minus (119906(119909 119910) minus 119888

2)

2) denoted by

(119909 119910 120601(119909 119910) 1198913) The metric tensor defined on this feature

space is [ℎ119894119895] = diag(1 1 1radic1 + 119868

2

119909+ 119868

2

119910+ 2119868119909119868119910 120601) The

pullback relation yields the determinant of metric tensor[

120583120592] on manifold Σ

= 1 +(

1

radic1 + 119868

2

119909+ 119868

2

119910+ 2119868119909119868119910

)

2

1003816100381610038161003816

nabla120601

1003816100381610038161003816

2

+ ((119906 (119909 119910) minus 1198881)

2

minus (119906 (119909 119910) minus 1198882)

2

)

2

120601

2

(5)

According to the Polyakov energy (2) we drive the curveevolution by minimizing the energy functional as follows

1198642= int

Σ

(

1

radic1 + 119868

2

119909+ 119868

2

119910+ 2119868119909119868119910

)

1003816100381610038161003816

nabla120601

1003816100381610038161003816

+ int

Σ

10038161003816100381610038161003816

(119906 (119909 119910) minus 1198881)

2

minus (119906 (119909 119910) minus 1198882)

210038161003816100381610038161003816

120601

(6)

where the weight of first term is actually an edge detectorThe curve evolution tends to stop when it decreases to zerowhereas the evolution goes on

33 Dual Algorithm To apply the dual gradient algorithmwe introduce the dual variable 119901 The total variation term in(4) and (6) can be formulated as follows

int

Ω

|nabla119906| 119889119909 = max119901isin1198621

⟨119906 div 1199011⟩

1198621fl 1199011| 1199011isin 119862

1

119888(Ω 119877

2)

1003816100381610038161003816

1199011

1003816100381610038161003816

le 1 forall119909 isin Ω

int

Ω

1

radic1 + 119868

2

119909+ 119868

2

119910+ 2119868119909119868119910

1003816100381610038161003816

nabla120601

1003816100381610038161003816

119889119909 = max119901isin119862119892

⟨120601 div 1199012⟩

1198622fl

1199012| 1199012isin 119862

1

119888(Ω 119877

2)

1003816100381610038161003816

1199012

1003816100381610038161003816

le

1

radic1 + 119868

2

119909+ 119868

2

119910+ 2119868119909119868119910

forall119909 isin Ω

(7)

The approximation formulation of the energy of our modelcan be rewritten as

min0le120601le1119906

max1199011isin1198621

1199012isin1198622

(1198641+ 1198642)

= ⟨120601 div 1199012⟩ + ⟨119906 div 119901

1⟩

+ radic1 + 119868

2

119909+ 119868

2

119910+ 2119868119909119868119910

1003817100381710038171003817

nabla119878119868 minus nabla119878119906

10038171003817100381710038171198711

+

1

radic1 + 119868119909119909

+ 119868119910119910

+ 2119868119909119910

119868 minus 1199061198711

+ ⟨120601 119877 (119906 1198881 1198882)⟩

(8)

where 119877(1198881 1198882 119906) = |(119906 minus 119888

1)

2minus (119906 minus 119888

2)

2| We then apply the

split Chambolle dual algorithm [10] to solve the optimizationproblem

Introducing the auxiliary variables V1 V2 solving the

energy functional (8) is equivalent to minimizing the prob-lem as follows

min119906120601

max1199011isin1198621

1199012isin1198622

(1198641+ 1198642) = ⟨119906 div 119901

1⟩ +

1

2120579

1003817100381710038171003817

119868 minus 119906 minus V1

1003817100381710038171003817

2

1198712

+ 1205721

1003817100381710038171003817

1003817100381710038171003817


10038171003817100381710038171198711 minus V1

1003817100381710038171003817

2

1198712

+ 1205722

1003817100381710038171003817

V1

10038171003817100381710038171198711 + ⟨120601 div 119901

2⟩

+

1

2120579

1003817100381710038171003817

120601 minus V2

1003817100381710038171003817

2

1198712

+ ⟨119877 (1198881 1198882 119906) V

2⟩ + 120574120578 (V

2)

(9)

where the parameter 120579 gt 0 is chosen to be small for avoidingsmearing the edges (in this paper we choose 120579 = 015)V(119911) = max0 2|119911 minus 12| minus 1 is an exact penalty functionprovided that the constant 120574 is chosen large enough comparedto 120582 such as 120574 gt (1205822)119877

119871infin(Ω)

1205721= radic1 + 119868

2

119909+ 119868

2

119910+ 2119868119909119868119910

and1205722= 1radic1 + 119868

119909119909+ 119868119910119910

+ 2119868119909119910Theminimization problem

(9) can be divided into four subproblems as follows and canbe solved alternatively

(a) Given image 119868 V1 update 119906 we search for 119906 as the

solution of

min119906

⟨119906 div 1199011⟩ +

1

2120579

1003817100381710038171003817

119868 minus 119906 minus V1

1003817100381710038171003817

2

1198712 (10)


(a) Original image (b) CV model 200th (c) SLM 50th

(d) The proposed model 50th (e) The approximated image

Figure 1 Segmentation for synthetic noisy image

The solution of (10) is given by

119906 = 119868 minus V1minus 120579 div 119901

1 (11)

where 1199011= (119901

1 119901

2) can be updated by fixed point method

initializing 1199011= 0 and updating

(1199011)

119896+1

=

(1199011)

119896

+ 120575nabla (div (1199011)

119896

minus (119868 minus V1) 120579)

1 + 120575

100381610038161003816100381610038161003816

nabla (div (1199011)

119896

minus (119868 minus V1) 120579)

100381610038161003816100381610038161003816

(12)

In this paper we choose 120575 le 18 to ensure convergence(b) Given 119906 V

2 we search for 120601 by solving the minimiza-

tion problem as follows

min120601V2

⟨120601 div 1199012⟩ +

1

2120579

1003817100381710038171003817

120601 minus V2

1003817100381710038171003817

2

1198712 (13)


120601 = V2minus 120579 div 119901

2 (14)

Equation (14) is solved by a fixed point method

(1199012)

119896+1

=

(1199012)

119896

+ 120575nabla (div (1199012)

119896

minus V2

119896120579)

1 + 120575

100381610038161003816100381610038161003816


119896

minus V2

119896120579)

100381610038161003816100381610038161003816

119896 = 0 1 2

(15)

(c) Given the solution of 120601 we search V2by solving

min120601V2

1

2120579

1003817100381710038171003817

120601 minus V2

1003817100381710038171003817

2

1198712 + ⟨119877 (119888

1 1198882 119906) V

2⟩ + 120574120592 (V

2) (16)


V2= min max 120601 minus 120579119877 (119888

1 1198882 119906) 0 1 (17)

(d) Given 119868 119906 we search for V1as the solution of

min120601119906

1

2120579

1003817100381710038171003817

119868 minus 119906 minus V1

1003817100381710038171003817

2

1198712 + 1205721

1003817100381710038171003817

1003817100381710038171003817


10038171003817100381710038171198711 minus V1

1003817100381710038171003817

2

1198712

+ 1205722

1003817100381710038171003817

V1

10038171003817100381710038171198711

(18)


V1

=

119868 minus 119906 + 21205721120579

1003817100381710038171003817


10038171003817100381710038171198711 minus 120579120572

1 if 119868 minus 119906 ge 120579120572

1

119868 minus 119906 + 21205721120579

1003817100381710038171003817


10038171003817100381710038171198711 minus 120579120572

1 if 119868 minus 119906 le 120579120572

1

0 if |119868 minus 119906| ge 1205791205721

(19)

After V1is solved it is utilized in (12) for next iteration

The algorithm of minimizing our model is described inthe following

Step 1 Initialize 1199060 = 120601

0= V1

0= V2

0= 119901

0

1= 119901

0

2= 0

Step 2 Given the fixed threshold of iterations 119879iteration gt 0 if119896 = 119879iteration then stop else go to Step 3


(a) (b) (c)

(d) (e) (f)

Figure 2 Segmentation for brain MRI (a) Original image and initial level set contour (b) Segmentation result of the CV model with200 iterations (c) Segmentation result of the SLM with 100 iterations (d) Segmentation result of the RSF model with 200 iterations (e)Segmentation result of the proposed model with 30 iterations (f) The approximated result of the proposed model

Step 3 Do the iteration for solving subproblem

Step 31 While |119906119896 minus 119906

119896minus1| gt 120576 do

Step 32 Update the dual variable 1199011198961rarr 119901

119896+1

1by (14) and then

update 119906119896 rarr 119906

119896+1 by (13) go to Step 33

Step 33 Given V1198962 update the dual variable 119901

119896

2rarr 119901

119896+1

2

according to (15) and then update 120601119896 rarr 120601

119896+1 by (16) go toStep 34

Step 34 Given 120601

119896+1 119906119896+1 compute 119877(1198881 1198882 119906) and update

V1198962rarr V119896+12

by (17) Go to Step 35

Step 35 Given 119868 119906 update V1198961rarr V119896+11

by (19) then go toStep 31 otherwise go to Step 4Step 4 End while

4 Experimental Results

All the experiments are run with Matlab code on the PC ofCPU 32GHz RAM 728M we show the experiments resultsfor medical image segmentation of Chan-Vese model (CV)

[11] the structure-based level set method (SLM) [12] andthe region-scale fitting model (RSF) [4] Figure 1 shows theexperiments on the synthesized noisy images This image isof size 266 times 313 with 10 white Gaussian noise

As shown in Figures 1(b) and 1(c) the CV model and theSLM models are sensitive to noise As shown in Figure 1(d)the proposed model is robust to noise and obtain the correctboundary The CV model generates the unwanted contoursbecause of the strong noise Based on the edge detectorfunction the SLMmodel is more robust to the noise than theCV model The result of the proposed model shows that it isable to extract the real object boundary even when the noiseis strong

Figure 2 is brain magnetic resonance image (MRI) ofsize 397 times 397 with 2 noise and 10 level intensitynonuniformityThe brainMRI mainly consists of three partsthe cerebrospinal fluid the gray matter and the white matterThe cerebrospinal fluid is the dark matter which exists in twoplaces the middle of the brain surrounded by the gray matterand the gap between the cranium and the brain The task ofsegmenting the brain MRI is to extract the contour profilebetween the white matter and the gray matter Since the


(a) (b) (c)

(d) (e) (f)


contrast of boundary between the white matter and the graymatter is lower than the boundary between the cerebrospinalfluid and the gray matter the latter is always extractedwrongly as the object boundary As shown in Figures 2(b)and 2(c) the CV model and the SLM model cannot extractthe object boundary with low-contrast Although the RSFmodel can extract the boundaries with low-contrast it alsoextracts some unwanted objects The segmentation results ofthe proposed model show that it is robust to noise and canextract more boundaries with low-contrast

Figure 3 is a brain MRI of size 258 times 258 with 5 noiseand 40 intensity nonuniformity As shown in Figures 3(b)and 3(c) the drawbacks of the RSFmodel and the SLMmodelstill exist Compared with the other methods the proposedmethod clearly extracts more object boundaries with low-contrast between the gray matter and the white matter

Figure 4 show the segmentation results of the activecontour methods based on multilayer level set functions Bythe multilayer level set functions the cerebrospinal fluid thegray matter and the white matter can be extracted simulta-neously In Figures 4(b) and 4(f) the results of CV modelshow that the cerebrospinal fluid is not extracted completely

As shown in Figures 4(c) and 4(g) the boundaries betweenthe white and gray matters are not extracted completely bySLM model We can observe that in Figures 4(d) and 4(h)the results of the proposed method based on multilayer levelset functions show that the completed boundaries of thecerebrospinal fluid the gray matter and the white matter areextracted simultaneously To show the convergence speed ofthe compared methods Table 1 shows the iteration numbersand processing time in each iteration for the CV model theSLM model based on the steepest descent method and theproposed model with both single- and multilayer level setfunctions Table 2 shows the segmentation accuracy of thecompared active contourmodels based onmultilayer level setfunctions

The data in Figure 5 is download from the website [13]We also show the segmentation accuracy in Table 3 by theDICE metric [14] compared with the ground truth given inthis website We can see from Figures 5(c) and 5(d) that theCV model and the SLM model only extract the boundarieswith high contrast while the object boundaries between thegray matter and white matter are not extracted And someCSF of the image is not extracted The proposed method can


Table 1 Iteration numbers and processing time in each iteration

CV model SLM model The proposed modelSingle level set function 035 sec 023 sec 018 secMultilayer functions 147 sec 062 sec 039 secIteration number for convergence 100 iterations 50 iterations 30 iterations

(a) (b) (c) (d)

(e) (f) (g) (h)

Figure 4 Segmentation for brain MRIs by the active contour methods based on multilayer level set functions (a) and (e) are original imagesand initial level set contours (b) and (f) are segmentation results of the CV model with 100 iterations (c) and (g) are segmentation results ofthe SLM with 50 iterations (d) and (h) are the results of the proposed method with 30 iterations

Table 2 Quantitative evaluation for brain MR-data in Figure 4

The CVmodel

The SLMmodel

The proposedmodel

Overall accuracy 652 346 915GM (dice metric) 783 824 972WM (dice metric) 690 135 947

extract themore completedCSF and can preserve the cerebralcortex more efficiently

5 Conclusion

In this paper we propose a new variational model forimage segmentation and image denoising simultaneouslyWeobtain the approximated image by embedding the approxi-mating criteria into a specific multifeature space And then

the segmentation result is obtained by embedding the activecontour into another multifeature space which is composedby the segmentation criteria depending on the approximatedimage The segmentation and the denoising problems aresolved by the split Chambolle dual algorithm alternatelyThe comparisons of the other popular segmentation modelsdemonstrate the accuracy and efficiency of the proposedmodel

Additional Points

The following are the research highlights of this paper Theproposed variational model incorporates segmentation anddenoising together Segmentation and denoising processingare achieved alternately by the Polyakov action frameworkAn improved Polyakov action framework is purely basedon the geometric features of the image without any manual



The CV model The SLMmodel The RSF model The proposed modelOverall accuracy 154 157 826 887GM (dice metric) 135 141 796 904WM (dice metric) 162 168 837 879

(a) (b) (c)

(d) (e) (f)

120

100

80

60

40

20

180

160140

120

10080

60

100

150

200

80

160140

120 150

(g)

120

100

80

60

40

20

180

160140

120

10080

60

100

150

200

0

160140

120 150

2

(h)

120

100

80

60

40

20

180

160140

120

10080

60

100

150

200

0

160140

120 150

(i)

Figure 5 Segmentation for 3D brain image (a) original image (b) segmentation result of the CVmodel with 200 iterations (c) segmentationresult of the SLM model with 100 iterations (d) segmentation result of the RSF model with 200 iterations (e) segmentation result of theproposed model with 30 iterations (f) the denoising result of the proposed model (g) the 3D original image (h) the 3D segmentation resultof the proposed method (i) the approximated 3D image


parameters Minimizing the variational model is achieved bythe improved Chambolle algorithm

Competing Interests

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

Acknowledgments

This work was supported in part by the Natural ScienceFoundation Science Foundation of China under Grant nos61502244 61402239 and 71301081 the Science Foundation ofJiangsu Province underGrant nos BK20150859 BK20130868and BK20130877 the Science Foundation of Jiangsu ProvinceUniversity (15KJB520028)NJUPTTalent Introduction Foun-dation (NY213007) NJUPT Advanced Institute Open Foun-dation (XJKY14012) China Postdoctoral Science Founda-tion (2015M580433 2014M551637) and Postdoctoral ScienceFoundation of Jiangsu Province (1401046C)

References

[1] A P Dempster N M Laird and D B Rubin ldquoMaximumlikelihood from incomplete data via the EM algorithmrdquo Journalof the Royal Statistical Society Series B vol 39 no 1 pp 1ndash381977

[2] J C Bezdek Pattern RecognitionWith Fuzzy Objective FunctionAlgorithms Plenum Press New York NY USA 1981

[3] V Caselles R Kimmel and G Sapiro ldquoGeodesic active con-toursrdquo International Journal of Computer Vision vol 22 no 1pp 61ndash79 1997

[4] C Li C-Y Kao J C Gore and Z Ding ldquoMinimization ofregion-scalable fitting energy for image segmentationrdquo IEEETransactions on Image Processing vol 17 no 10 pp 1940ndash19492008

[5] A KMishra PW Fieguth andD A Clausi ldquoDecoupled activecontour (DAC) for boundary detectionrdquo IEEE Transactions onPattern Analysis andMachine Intelligence vol 33 no 2 pp 310ndash324 2011

[6] P ArbelaezMMaire C Fowlkes and JMalik ldquoContour detec-tion and hierarchical image segmentationrdquo IEEE Transactionson Pattern Analysis and Machine Intelligence vol 33 no 5 pp898ndash916 2011

[7] Q Ge L Xiao J Zhang and Z H Wei ldquoAn improved region-based model with local statistical features for image segmenta-tionrdquo Pattern Recognition vol 45 no 4 pp 1578ndash1590 2012

[8] H Wu V Appia and A Yezzi ldquoNumerical conditioning prob-lems and solutions for nonparametric iid statistical activecontoursrdquo IEEE Transactions on Pattern Analysis and MachineIntelligence vol 35 no 6 pp 1298ndash1311 2013

[9] N Sochen R Kimmel and R Malladi ldquoA general frameworkfor low level visionrdquo IEEE Transactions on Image Processing vol7 no 3 pp 310ndash318 1998

[10] X Bresson S Esedoglu P Vandergheynst J-P Thiran and SOsher ldquoFast global minimization of the active contoursnakemodelrdquo Journal of Mathematical Imaging and Vision vol 28 no2 pp 151ndash167 2007

[11] DMumford and J Shah ldquoOptimal approximations by piecewisesmooth functions and associated variational problemsrdquo Com-munications on Pure and Applied Mathematics vol 42 no 5pp 577ndash685 1989

[12] B Dizdaroglu E Ataer-Cansizoglu J Kalpathy-Cramer KKeck M F Chiang and D Erdogmus ldquoStructure-based levelset method for automatic retinal vasculature segmentationrdquoEURASIP Journal on Image and Video Processing vol 2014 no1 article 39 26 pages 2014

[13] httpwwwcmamghharvardeduibsr[14] L R Dice ldquoMeasures of the amount of ecologic association

between speciesrdquo Ecology vol 26 no 3 pp 297ndash302 1945

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


space Because these models choose a metric with artificialparameters on the feature space it requires careful manualparameter-tuning

In this paper the proposed active contour model isformulated in the framework of the Polyakov action [9]Unlike the other related works [7ndash9] a metric on the featurespace manifold is defined by the invariant geometry ofimages Consequently the proposed method is purely basedon the geometrical features of images without any artificialparameters We implement the segmentation through twosteps First an approximated image removing the noisewhile preserving the main structures is found in the featurespace built on geometrical features of the original imageSecond the active contour is embedded into the feature spacebuilt on both the statistical and geometrical features of theapproximated image For efficiency we solve the proposedmodel via the improved Chambolle dual formulation [10] ofthe minimization problem

The paper is organized as follows In Section 2 we intro-duce the mathematical framework based on the Polyakovaction In Section 3 we introduce the proposed model andthe numerical algorithm of the proposed method is alsosummarized In Section 4 we validate our model by someexperiments on medical images In Section 5 we end thepaper by a brief conclusion

2 Geometrical Framework Based on WeightedPolyakov Action

Sochen et al introduce a general geometrical framework forlow-level vision based on the Polyakov action [9] In thisframework images are represented as the surfaces on a Rie-mannian manifold The Polyakov action is a functional thatmeasures the weight of a mapping X = (119883

1(120590) 119883

119898(120590))

between an 119899-dimensional embedded manifold (eg theimage manifold) Σ with coordinates 120590 = (120590

1 120590

119899)

and the 119898-dimensional manifold 119872 with the coordinates(119883

1(120590) 119883

119898(120590)) 119898 gt 119899 A Riemannian structure metric

119892119906V can be introduced to measure the local distances on

the embedded manifold Σ whereas we use the metric ℎ119894119895to

measure the distance on the manifold 119872 To measure theweight of the mapping X Σ 997891rarr 119872 the Polyakov actionis used as a generalization of the 119871

2-norm on the embedded

image to space feature manifold119872

119878 [119883

119894 119892119906V ℎ119894119895] = intradic119892119892

120583120592120597120583119883

119894120597120592119883

119895ℎ119894119895119889

119899120590 (1)

where 119892 is the determinant of the image metric tensor 119892119906V

and 119892

120583120592 is its inverse The metric 119892 is chosen as the inducedmetric obtained by the pullback relation 119892

119906V = ℎ119894119895120597120583119883

119894120597120592119883

119895the Polyakov energy is shortened to

119878 (119883

119894 ℎ119894119895) = intradic119892119889

119899120590 (2)

In the relevant works [7 8] the authors get the denoisedimage and the segmentation results byminimizing the energyfunctional (2) with respect to denoising and segmentationrespectively In seminal work [9] they embed grey images

in the feature (119909 119910 119868(119909 119910)) where 119868(119909 119910) is the grey inten-sity value for pixel (119909 119910) They choose a metric [ℎ

119894119895] =

diag(1 1 1205732) 120573 gt 0 is a constant Based on this metricon feature space and the Polyakov energy the regularizationterm on the intensity values is given by intradic1 + 120573

2|nabla119868|

2Although it allows setting the scale of the feature dimensionindependently of the spatial dimensions the accuracy of thescale is subject to the artificial parameter 120573

3 The Active Contour Model inMultifeature Space

In this work we utilize an improved geometrical frameworkbased on the weighted Polyakov action without any artificialparameter First we get an approximated image by embed-ding it into the feature space constituted by the features ofthe original image Second given the approximated imageactive contour is driven by embedding the level set functioninto the higher dimensional feature space composed ofthe geometrical and statistical features of the approximatedimage

31 Approximating Image under an Improved GeometricalFramework The original image 119868(119909 119910) is defined on theimage manifold Σ with coordinates (119909 119910) The approximatedimage 119906 is defined on the image manifold Σ and denotedby 119906(119909 119910) To preserve the main edges of the originalimage we extract the geometrical features of edgesnabla119878119868 = ((119868

119909)

3+ 119868119909(119868119910)

2 (119868119910)

3+ 119868119910(119868119909)

2) derived from the

anisotropic diffusion equation [2] Considering the intensityvalue 119868(119909 119910) as another feature we build the feature space(119909 119910 119906 (nabla

119878119868(119909 119910) minus nabla

119878119906(119909 119910)) (119868(119909 119910) minus 119906(119909 119910))) denoted

by (119909 119910 119906 1198911 1198912) for the sake of simplicity To avoid the

influence of the artificial parameter we choose a metrictensor [ℎ

119894119895] on the feature space 119872 which is defined by the

invariant geometry of the original image 119868 Consider [ℎ119894119895] =

diag(1 1 1 radic1 + 119868

2

119909+ 119868

2

119910+ 2119868119909119868119910 1radic1 + 119868

119909119909+ 119868119910119910

+ 2119868119909119910)

The pullback relation yields the determinant of metric tensor[119892120583120592] on manifold Σ

119892 = 1 + |nabla119906|

2+ (radic1 + 119868

2

119909+ 119868

2

119910+ 2119868119909119868119910)

2

sdot

1003816100381610038161003816

nabla119878119868 (119909 119910) minus nabla

119878119906 (119909 119910)

1003816100381610038161003816

2

+(

1

radic1 + 119868119909119909

+ 119868119910119910

+ 2119868119909119910

)

2

sdot

1003816100381610038161003816

119868 (119909 119910) minus 119906 (119909 119910)

1003816100381610038161003816

2

(3)

Analogizing based on the Polyakov energy (2) we get theapproximated image 119906 by minimizing the energy functionalas follows

1198641= int

Σ

|nabla119906| + radic1 + 119868

2

119909+ 119868

2

119910+ 2119868119909119868119910

1003816100381610038161003816

nabla119878119868 (119909 119910)

minus nabla119878119906 (119909 119910)

1003816100381610038161003816

119889119909 119889119910


+ int

Σ

1

radic1 + 119868119909119909

+ 119868119910119910

+ 2119868119909119910

1003816100381610038161003816

119868 (119909 119910)

minus 119906 (119909 119910)

1003816100381610038161003816

119889119909 119889119910

(4)




119894119895] denotes






1)

2minus (119906(119909 119910) minus 119888

2)

2) denoted by



2

119909+ 119868

2

119910+ 2119868119909119868119910 120601) The



= 1 +(

1

radic1 + 119868

2

119909+ 119868

2

119910+ 2119868119909119868119910

)

2

1003816100381610038161003816

nabla120601

1003816100381610038161003816

2

+ ((119906 (119909 119910) minus 1198881)

2

minus (119906 (119909 119910) minus 1198882)

2

)

2

120601

2

(5)


1198642= int

Σ

(

1

radic1 + 119868

2

119909+ 119868

2

119910+ 2119868119909119868119910

)

1003816100381610038161003816

nabla120601

1003816100381610038161003816

+ int

Σ

10038161003816100381610038161003816

(119906 (119909 119910) minus 1198881)

2

minus (119906 (119909 119910) minus 1198882)

210038161003816100381610038161003816

120601

(6)



int

Ω

|nabla119906| 119889119909 = max119901isin1198621

⟨119906 div 1199011⟩

1198621fl 1199011| 1199011isin 119862

1

119888(Ω 119877

2)

1003816100381610038161003816

1199011

1003816100381610038161003816


int

Ω

1

radic1 + 119868

2

119909+ 119868

2

119910+ 2119868119909119868119910

1003816100381610038161003816

nabla120601

1003816100381610038161003816

119889119909 = max119901isin119862119892

⟨120601 div 1199012⟩

1198622fl

1199012| 1199012isin 119862

1

119888(Ω 119877

2)

1003816100381610038161003816

1199012

1003816100381610038161003816

le

1

radic1 + 119868

2

119909+ 119868

2

119910+ 2119868119909119868119910


(7)


min0le120601le1119906

max1199011isin1198621

1199012isin1198622

(1198641+ 1198642)

= ⟨120601 div 1199012⟩ + ⟨119906 div 119901

1⟩

+ radic1 + 119868

2

119909+ 119868

2

119910+ 2119868119909119868119910

1003817100381710038171003817


10038171003817100381710038171198711

+

1

radic1 + 119868119909119909

+ 119868119910119910

+ 2119868119909119910

119868 minus 1199061198711

+ ⟨120601 119877 (119906 1198881 1198882)⟩

(8)

where 119877(1198881 1198882 119906) = |(119906 minus 119888

1)

2minus (119906 minus 119888

2)





min119906120601

max1199011isin1198621

1199012isin1198622

(1198641+ 1198642) = ⟨119906 div 119901

1⟩ +

1

2120579

1003817100381710038171003817

119868 minus 119906 minus V1

1003817100381710038171003817

2

1198712

+ 1205721

1003817100381710038171003817

1003817100381710038171003817


10038171003817100381710038171198711 minus V1

1003817100381710038171003817

2

1198712

+ 1205722

1003817100381710038171003817

V1

10038171003817100381710038171198711 + ⟨120601 div 119901

2⟩

+

1

2120579

1003817100381710038171003817

120601 minus V2

1003817100381710038171003817

2

1198712

+ ⟨119877 (1198881 1198882 119906) V

2⟩ + 120574120578 (V

2)

(9)


119871infin(Ω)

1205721= radic1 + 119868

2

119909+ 119868

2

119910+ 2119868119909119868119910

and1205722= 1radic1 + 119868

119909119909+ 119868119910119910




solution of

min119906

⟨119906 div 1199011⟩ +

1

2120579

1003817100381710038171003817

119868 minus 119906 minus V1

1003817100381710038171003817

2

1198712 (10)






119906 = 119868 minus V1minus 120579 div 119901

1 (11)

where 1199011= (119901

1 119901



(1199011)

119896+1

=

(1199011)

119896

+ 120575nabla (div (1199011)

119896

minus (119868 minus V1) 120579)

1 + 120575

100381610038161003816100381610038161003816


119896

minus (119868 minus V1) 120579)

100381610038161003816100381610038161003816

(12)




min120601V2

⟨120601 div 1199012⟩ +

1

2120579

1003817100381710038171003817

120601 minus V2

1003817100381710038171003817

2

1198712 (13)


120601 = V2minus 120579 div 119901

2 (14)


(1199012)

119896+1

=

(1199012)

119896

+ 120575nabla (div (1199012)

119896

minus V2

119896120579)

1 + 120575

100381610038161003816100381610038161003816


119896

minus V2

119896120579)

100381610038161003816100381610038161003816

119896 = 0 1 2

(15)


min120601V2

1

2120579

1003817100381710038171003817

120601 minus V2

1003817100381710038171003817

2

1198712 + ⟨119877 (119888

1 1198882 119906) V

2⟩ + 120574120592 (V

2) (16)


V2= min max 120601 minus 120579119877 (119888

1 1198882 119906) 0 1 (17)


min120601119906

1

2120579

1003817100381710038171003817

119868 minus 119906 minus V1

1003817100381710038171003817

2

1198712 + 1205721

1003817100381710038171003817

1003817100381710038171003817


10038171003817100381710038171198711 minus V1

1003817100381710038171003817

2

1198712

+ 1205722

1003817100381710038171003817

V1

10038171003817100381710038171198711

(18)


V1

=

119868 minus 119906 + 21205721120579

1003817100381710038171003817


10038171003817100381710038171198711 minus 120579120572

1 if 119868 minus 119906 ge 120579120572

1

119868 minus 119906 + 21205721120579

1003817100381710038171003817


10038171003817100381710038171198711 minus 120579120572

1 if 119868 minus 119906 le 120579120572

1

0 if |119868 minus 119906| ge 1205791205721

(19)




0= V1

0= V2

0= 119901

0

1= 119901

0

2= 0



(a) (b) (c)

(d) (e) (f)



Step 31 While |119906119896 minus 119906

119896minus1| gt 120576 do


119896+1

1by (14) and then

update 119906119896 rarr 119906

119896+1 by (13) go to Step 33


119896

2rarr 119901

119896+1

2


119896+1 by (16) go toStep 34



V1198962rarr V119896+12










(a) (b) (c)

(d) (e) (f)










(a) (b) (c) (d)

(e) (f) (g) (h)



The CVmodel

The SLMmodel

The proposedmodel



5 Conclusion



Additional Points





(a) (b) (c)

(d) (e) (f)

120

100

80

60

40

20

180

160140

120

10080

60

100

150

200

80

160140

120 150

(g)

120

100

80

60

40

20

180

160140

120

10080

60

100

150

200

0

160140

120 150

2

(h)

120

100

80

60

40

20

180

160140

120

10080

60

100

150

200

0

160140

120 150

(i)




Competing Interests


Acknowledgments


References






















Volume 2014




Journal of











Journal of


Function Spaces






Algebra










+ int

Σ

1

radic1 + 119868119909119909

+ 119868119910119910

+ 2119868119909119910

1003816100381610038161003816

119868 (119909 119910)

minus 119906 (119909 119910)

1003816100381610038161003816

119889119909 119889119910

(4)




119894119895] denotes






1)

2minus (119906(119909 119910) minus 119888

2)

2) denoted by



2

119909+ 119868

2

119910+ 2119868119909119868119910 120601) The



= 1 +(

1

radic1 + 119868

2

119909+ 119868

2

119910+ 2119868119909119868119910

)

2

1003816100381610038161003816

nabla120601

1003816100381610038161003816

2

+ ((119906 (119909 119910) minus 1198881)

2

minus (119906 (119909 119910) minus 1198882)

2

)

2

120601

2

(5)


1198642= int

Σ

(

1

radic1 + 119868

2

119909+ 119868

2

119910+ 2119868119909119868119910

)

1003816100381610038161003816

nabla120601

1003816100381610038161003816

+ int

Σ

10038161003816100381610038161003816

(119906 (119909 119910) minus 1198881)

2

minus (119906 (119909 119910) minus 1198882)

210038161003816100381610038161003816

120601

(6)



int

Ω

|nabla119906| 119889119909 = max119901isin1198621

⟨119906 div 1199011⟩

1198621fl 1199011| 1199011isin 119862

1

119888(Ω 119877

2)

1003816100381610038161003816

1199011

1003816100381610038161003816


int

Ω

1

radic1 + 119868

2

119909+ 119868

2

119910+ 2119868119909119868119910

1003816100381610038161003816

nabla120601

1003816100381610038161003816

119889119909 = max119901isin119862119892

⟨120601 div 1199012⟩

1198622fl

1199012| 1199012isin 119862

1

119888(Ω 119877

2)

1003816100381610038161003816

1199012

1003816100381610038161003816

le

1

radic1 + 119868

2

119909+ 119868

2

119910+ 2119868119909119868119910


(7)


min0le120601le1119906

max1199011isin1198621

1199012isin1198622

(1198641+ 1198642)

= ⟨120601 div 1199012⟩ + ⟨119906 div 119901

1⟩

+ radic1 + 119868

2

119909+ 119868

2

119910+ 2119868119909119868119910

1003817100381710038171003817


10038171003817100381710038171198711

+

1

radic1 + 119868119909119909

+ 119868119910119910

+ 2119868119909119910

119868 minus 1199061198711

+ ⟨120601 119877 (119906 1198881 1198882)⟩

(8)

where 119877(1198881 1198882 119906) = |(119906 minus 119888

1)

2minus (119906 minus 119888

2)





min119906120601

max1199011isin1198621

1199012isin1198622

(1198641+ 1198642) = ⟨119906 div 119901

1⟩ +

1

2120579

1003817100381710038171003817

119868 minus 119906 minus V1

1003817100381710038171003817

2

1198712

+ 1205721

1003817100381710038171003817

1003817100381710038171003817


10038171003817100381710038171198711 minus V1

1003817100381710038171003817

2

1198712

+ 1205722

1003817100381710038171003817

V1

10038171003817100381710038171198711 + ⟨120601 div 119901

2⟩

+

1

2120579

1003817100381710038171003817

120601 minus V2

1003817100381710038171003817

2

1198712

+ ⟨119877 (1198881 1198882 119906) V

2⟩ + 120574120578 (V

2)

(9)


119871infin(Ω)

1205721= radic1 + 119868

2

119909+ 119868

2

119910+ 2119868119909119868119910

and1205722= 1radic1 + 119868

119909119909+ 119868119910119910




solution of

min119906

⟨119906 div 1199011⟩ +

1

2120579

1003817100381710038171003817

119868 minus 119906 minus V1

1003817100381710038171003817

2

1198712 (10)






119906 = 119868 minus V1minus 120579 div 119901

1 (11)

where 1199011= (119901

1 119901



(1199011)

119896+1

=

(1199011)

119896

+ 120575nabla (div (1199011)

119896

minus (119868 minus V1) 120579)

1 + 120575

100381610038161003816100381610038161003816


119896

minus (119868 minus V1) 120579)

100381610038161003816100381610038161003816

(12)




min120601V2

⟨120601 div 1199012⟩ +

1

2120579

1003817100381710038171003817

120601 minus V2

1003817100381710038171003817

2

1198712 (13)


120601 = V2minus 120579 div 119901

2 (14)


(1199012)

119896+1

=

(1199012)

119896

+ 120575nabla (div (1199012)

119896

minus V2

119896120579)

1 + 120575

100381610038161003816100381610038161003816


119896

minus V2

119896120579)

100381610038161003816100381610038161003816

119896 = 0 1 2

(15)


min120601V2

1

2120579

1003817100381710038171003817

120601 minus V2

1003817100381710038171003817

2

1198712 + ⟨119877 (119888

1 1198882 119906) V

2⟩ + 120574120592 (V

2) (16)


V2= min max 120601 minus 120579119877 (119888

1 1198882 119906) 0 1 (17)


min120601119906

1

2120579

1003817100381710038171003817

119868 minus 119906 minus V1

1003817100381710038171003817

2

1198712 + 1205721

1003817100381710038171003817

1003817100381710038171003817


10038171003817100381710038171198711 minus V1

1003817100381710038171003817

2

1198712

+ 1205722

1003817100381710038171003817

V1

10038171003817100381710038171198711

(18)


V1

=

119868 minus 119906 + 21205721120579

1003817100381710038171003817


10038171003817100381710038171198711 minus 120579120572

1 if 119868 minus 119906 ge 120579120572

1

119868 minus 119906 + 21205721120579

1003817100381710038171003817


10038171003817100381710038171198711 minus 120579120572

1 if 119868 minus 119906 le 120579120572

1

0 if |119868 minus 119906| ge 1205791205721

(19)




0= V1

0= V2

0= 119901

0

1= 119901

0

2= 0



(a) (b) (c)

(d) (e) (f)



Step 31 While |119906119896 minus 119906

119896minus1| gt 120576 do


119896+1

1by (14) and then

update 119906119896 rarr 119906

119896+1 by (13) go to Step 33


119896

2rarr 119901

119896+1

2


119896+1 by (16) go toStep 34



V1198962rarr V119896+12










(a) (b) (c)

(d) (e) (f)










(a) (b) (c) (d)

(e) (f) (g) (h)



The CVmodel

The SLMmodel

The proposedmodel



5 Conclusion



Additional Points





(a) (b) (c)

(d) (e) (f)

120

100

80

60

40

20

180

160140

120

10080

60

100

150

200

80

160140

120 150

(g)

120

100

80

60

40

20

180

160140

120

10080

60

100

150

200

0

160140

120 150

2

(h)

120

100

80

60

40

20

180

160140

120

10080

60

100

150

200

0

160140

120 150

(i)




Competing Interests


Acknowledgments


References






















Volume 2014




Journal of











Journal of


Function Spaces






Algebra














119906 = 119868 minus V1minus 120579 div 119901

1 (11)

where 1199011= (119901

1 119901



(1199011)

119896+1

=

(1199011)

119896

+ 120575nabla (div (1199011)

119896

minus (119868 minus V1) 120579)

1 + 120575

100381610038161003816100381610038161003816


119896

minus (119868 minus V1) 120579)

100381610038161003816100381610038161003816

(12)




min120601V2

⟨120601 div 1199012⟩ +

1

2120579

1003817100381710038171003817

120601 minus V2

1003817100381710038171003817

2

1198712 (13)


120601 = V2minus 120579 div 119901

2 (14)


(1199012)

119896+1

=

(1199012)

119896

+ 120575nabla (div (1199012)

119896

minus V2

119896120579)

1 + 120575

100381610038161003816100381610038161003816


119896

minus V2

119896120579)

100381610038161003816100381610038161003816

119896 = 0 1 2

(15)


min120601V2

1

2120579

1003817100381710038171003817

120601 minus V2

1003817100381710038171003817

2

1198712 + ⟨119877 (119888

1 1198882 119906) V

2⟩ + 120574120592 (V

2) (16)


V2= min max 120601 minus 120579119877 (119888

1 1198882 119906) 0 1 (17)


min120601119906

1

2120579

1003817100381710038171003817

119868 minus 119906 minus V1

1003817100381710038171003817

2

1198712 + 1205721

1003817100381710038171003817

1003817100381710038171003817


10038171003817100381710038171198711 minus V1

1003817100381710038171003817

2

1198712

+ 1205722

1003817100381710038171003817

V1

10038171003817100381710038171198711

(18)


V1

=

119868 minus 119906 + 21205721120579

1003817100381710038171003817


10038171003817100381710038171198711 minus 120579120572

1 if 119868 minus 119906 ge 120579120572

1

119868 minus 119906 + 21205721120579

1003817100381710038171003817


10038171003817100381710038171198711 minus 120579120572

1 if 119868 minus 119906 le 120579120572

1

0 if |119868 minus 119906| ge 1205791205721

(19)




0= V1

0= V2

0= 119901

0

1= 119901

0

2= 0



(a) (b) (c)

(d) (e) (f)



Step 31 While |119906119896 minus 119906

119896minus1| gt 120576 do


119896+1

1by (14) and then

update 119906119896 rarr 119906

119896+1 by (13) go to Step 33


119896

2rarr 119901

119896+1

2


119896+1 by (16) go toStep 34



V1198962rarr V119896+12










(a) (b) (c)

(d) (e) (f)










(a) (b) (c) (d)

(e) (f) (g) (h)



The CVmodel

The SLMmodel

The proposedmodel



5 Conclusion



Additional Points





(a) (b) (c)

(d) (e) (f)

120

100

80

60

40

20

180

160140

120

10080

60

100

150

200

80

160140

120 150

(g)

120

100

80

60

40

20

180

160140

120

10080

60

100

150

200

0

160140

120 150

2

(h)

120

100

80

60

40

20

180

160140

120

10080

60

100

150

200

0

160140

120 150

(i)




Competing Interests


Acknowledgments


References






















Volume 2014




Journal of











Journal of


Function Spaces






Algebra










(a) (b) (c)

(d) (e) (f)



Step 31 While |119906119896 minus 119906

119896minus1| gt 120576 do


119896+1

1by (14) and then

update 119906119896 rarr 119906

119896+1 by (13) go to Step 33


119896

2rarr 119901

119896+1

2


119896+1 by (16) go toStep 34



V1198962rarr V119896+12










(a) (b) (c)

(d) (e) (f)










(a) (b) (c) (d)

(e) (f) (g) (h)



The CVmodel

The SLMmodel

The proposedmodel



5 Conclusion



Additional Points





(a) (b) (c)

(d) (e) (f)

120

100

80

60

40

20

180

160140

120

10080

60

100

150

200

80

160140

120 150

(g)

120

100

80

60

40

20

180

160140

120

10080

60

100

150

200

0

160140

120 150

2

(h)

120

100

80

60

40

20

180

160140

120

10080

60

100

150

200

0

160140

120 150

(i)




Competing Interests


Acknowledgments


References






















Volume 2014




Journal of











Journal of


Function Spaces






Algebra










(a) (b) (c)

(d) (e) (f)










(a) (b) (c) (d)

(e) (f) (g) (h)



The CVmodel

The SLMmodel

The proposedmodel



5 Conclusion



Additional Points





(a) (b) (c)

(d) (e) (f)

120

100

80

60

40

20

180

160140

120

10080

60

100

150

200

80

160140

120 150

(g)

120

100

80

60

40

20

180

160140

120

10080

60

100

150

200

0

160140

120 150

2

(h)

120

100

80

60

40

20

180

160140

120

10080

60

100

150

200

0

160140

120 150

(i)




Competing Interests


Acknowledgments


References






















Volume 2014




Journal of











Journal of


Function Spaces






Algebra












(a) (b) (c) (d)

(e) (f) (g) (h)



The CVmodel

The SLMmodel

The proposedmodel



5 Conclusion



Additional Points





(a) (b) (c)

(d) (e) (f)

120

100

80

60

40

20

180

160140

120

10080

60

100

150

200

80

160140

120 150

(g)

120

100

80

60

40

20

180

160140

120

10080

60

100

150

200

0

160140

120 150

2

(h)

120

100

80

60

40

20

180

160140

120

10080

60

100

150

200

0

160140

120 150

(i)




Competing Interests


Acknowledgments


References






















Volume 2014




Journal of











Journal of


Function Spaces






Algebra












(a) (b) (c)

(d) (e) (f)

120

100

80

60

40

20

180

160140

120

10080

60

100

150

200

80

160140

120 150

(g)

120

100

80

60

40

20

180

160140

120

10080

60

100

150

200

0

160140

120 150

2

(h)

120

100

80

60

40

20

180

160140

120

10080

60

100

150

200

0

160140

120 150

(i)




Competing Interests


Acknowledgments


References






















Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Competing Interests


Acknowledgments


References






















Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article unsupervised joint image denoising and...

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