multimedia 2009 (mmsp09) paper number 239 sub-pixel accuracy edge fitting by means of b-spline.pdf

8/9/2019 Multimedia 2009 (MMSP09) Paper Number 239 Sub-Pixel Accuracy Edge Fitting by Means of B-Spline.pdf

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Sub-Pixel Accuracy Edge Fitting by Means of B-

SplineRicardo Lucas Bastos Breder 1, Vania Vieira Estrela *2, Joaquim Teixeira de Assis 3

State University of Rio de Janeiro (UERJ), Polytechnic Institute of Rio de Janeiro (IPRJ), CP 972825, CEP 28601-970,

Nova Friburgo, RJ, [email protected]

[email protected]

[email protected]

Abstract Local perturbations around contours strongly

disturb the final result of computer vision tasks. It is common tointroduce a priori information in the estimation process.Improvement can be achieved via a deformable model such asthe snake model. In recent works, the deformable contour ismodeled by means of B-spline snakes which allows local control,concise representation, and the use of fewer parameters. Theestimation of the sub-pixel edges using a global B-spline model

relies on the contour global determination according to aMaximum Likelihood framework and using the observed datalikelihood. This procedure guarantees that the noisiest data willbe filtered out. The data likelihood is computed as a consequence

of the observation model which includes both orientation andposition information. Comparative experiments of this algorithmand the classical spline interpolation have shown that theproposed algorithm outperforms the classical approach forGaussian and Salt & Pepper noise.

I. INTRODUCTIONOne of the greatest challenges of modeling remotely sensed

images is the mixed pixel problem, that is, the level of spatial

detail captured is less than the amount of expected details.

This sub-pixel heterogeneity is important, but not readily

capable of being known. In a traditional manner, each pixel is

classified into one of many land cover types (hardclassification), implying that land cover exactly fits within the

bounds of one or multiple pixels. However, several pixels

consist of a mixture of different classes. The solution to the

mixed pixel problem typically centers on soft classification,

which allows proportions of each pixel to be partitionedbetween classes. An important edge extraction approach is

concerned with improving the detection accuracy and

different sub-pixel edge detectors [10].

Popular approaches used to compute sub-pixel location of

edges are based on the image moments [6, 11, 14, 15, 16, 18].

Image interpolation is another way to find sub-pixel edge

coordinates. [13] presents a technique where the first

derivative perpendicular to the edge orientation is

approximated by a normal function. The edge location is this

function maximum. Other works are based on linear [7],

quadratic interpolation [1], B-spline [17], non-linear functions

[8], and as a segmentation by-product [22, 23].

In all these methods, the estimation is local and does not

include a noise model. Hence, the local perturbations heavily

disturb the final result. A usual filtering approach requires

introducing a priori information in the estimation processsuch as the snake model suggested by [9]. In newer works, the

deformable contour is modeled using piecewise polynomialfunctions (B-spline snakes) [4, 12] allowing local control,

concise representation, and it employs few parameters.

This work introduces a model for estimating sub-pixel

edges using a global procedure based on a B-spline model [3,

18], but the statistical properties of the observations are

computed and used in a Maximum Likelihood (ML)

estimation framework which insures an efficient filtering of

the noisy data. Section II briefly describes the classical B-

spline formulation. Section III introduces the proposedextension to the sub-pixel case. Section IV discusses

experimental results of this algorithm. Finally, Section Vpresents some conclusions.

Fig. 1 Illustration of the mixed-pixel problem [20].

COPYRIGHT NOTICE:

MMSP09, October 5-7, 2009, Rio de Janeiro, Brazil.978-1-4244-4464-9/09/$25.00 2009 IEEE.


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II. B-SPLINE CONTOUR FORMULATIONLet 0 1 1, ,..., kt t t be the set of so-called knots. Spline

functions are polynomial inside each interval 1,i it t andexhibit a certain degree of continuity at the knots (see [3]).

The set , ( ) , 0,..., 1i n t B i k n of the B-splines, constitutesa basis for the linear space of all the splines contained in

,n k nt t . Hence, a spline curvef(t) of degree n is given by :

1

,

0

( ) ( )k n

i i ni

f t p B t

,

where pi is the weight applied to the corresponding basisfunctions Bi,n. The B-Spline functions are defined by therecursion formulas in [2]:

1,0 0 1( )i i i B t t t t ifelse , and

1

, , 1 1, 1

1 1

( ) ( ) ( )i i ni n i n i n

i n i i n i

t t t t B t B t B t

t t t t

.

The , ( )i n tB are nonnegative, i yi xiJ G G and obey the

relationship: , ( ) 1i n tiB for all values of t. Without loss ofgenerality, the index n for the cubic B-spline can be dropped.Eq. (1) can be re-written in matrix notation as:

( ) ( ).Tf t B t .

( t ) x t , y t v is the vector of B-spline functions

0 2 N 1( t ) B ( t ),B ( t ),...,B ( t )B and is the weight vector:

T

0 N 1p , ...,p . The 2D version of Eq. (1) describes an

open parametric snake on the plane: ( t ) x t , y t v . The

pixels ,i xi yip p p are now 2D vectors and are called thecontrol points.

III.SUBPIXEL B-SPLINE FITTINGThis sub-section aims to obtain a likelihood expression for

the observation, supposing additive Gaussian noise b with

standard deviation b.

Let ( . ) ( . ), ( . ) , 1,...,e e e ev v i h x i h y i h i M be a set ofedge pixels with integer coordinatesh computed by a classical

algorithm. To simplify notation, ( . )ev i h = ev for all variablesrelated to the pixel i.

For each pixel i belonging to this set, an observation vectorOi= (Xi , Hi) is assumed, where vector Hi= (Hxi , Hyi) is the

local gradient of the image andXi corresponds to a sub-pixel

position estimation of the edge along Hi. Xi is estimated in

the pixel local coordinate system by means of a quadratic

interpolation [5]. These variables are supposed to be the

observation version of the true variables ,i iY G , where

,oi oix y are the cartesian coordinates of the observation iO .

( ) ( ) ( ). , ( ) ( ).s s x s yv t x t B t y t B t is the edge B-

spline model, where ,x y is the vector of k control

points M k . The likelihood becomes

/ / , / i i i iP O P X H P H .

To derive P / , ,i iX H these assumptions are made: iX

depends only on ( ) ( ) , ( )s s si s siv t i X t i x y t i y , thecorresponding edge point in the direction iH has polar

coordinates ,i iY H and / ,i i iP X H Y is Gaussian. Then,

2

2

1/ , / , exp

22

oi si oi si

i i i i i

XiXi

X X Y Y P X H P X H Y

The computation of Xi is based on a modified version

of the method described in [3]. Now, P /iH requires the

components of iH . They can be computed by convolving the

original image with the derivative kernels xK and yK .

Assuming an additive Gaussian noise, it is straightforward to

show that /i iP H G is also Gaussian. The cross-correlationand auto-correlation functions of the vertical and horizontal

components ofH are given by:

2, .

, , * , .

xy i j i j xi yj xi yj

b i j i j x y

C x x y y E H H E H E H

x x y y K x y K x y

xK

and yK

are the complex conjugate of the matrices of xK

and yK , respectively. This relation leads to

. / / / 0xi yi xi yi xi xi yi yi xi yiE H H G G E H G G E H G G ,

and

2 2,

. / / / ( , )xi xi xi xi xi xi xi b xi j

E H H G E H G E H G K i j 2 2 2

,

( , )b y Hi j

K i j .

P /iH represents the conditional expected value of w

assuming z. Let Ji be the vector defined by i yi xiJ G G

and letRi be the unit vector collinear to sv in siv . Hi and Ji

are clearly related by a normal distribution. To find


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P /iH , the well known property of the edge beingorthogonal to the local gradient is used:

P( / )P( )P( / ) P( / ( ))

P( ( )) i i ii i i

i

H J J H H R

R ,

for all the values of Ji collinear toRi, that isJi = a Ri. It can

be rewritten as:

i

P / P / s i sO v O v .

The gradient does not have a privileged direction. Uniform

regions are supposed to be more frequent than the regions

with a high gradient magnitude. Thus, the a priori probabilityofJi can be modelled by

2 2

2 2

|| ||1 1P( ) exp exp

i

i

J aJ

.

Combining the two previous expressions results in

2 2yi xi

2 2H

2 2

xi xi yi yi

2 2 2H H

H HP( / ) .exp

2

( H R H R )exp .

2 ( 2 )

iH K

K is a normalization constant involving ( )iP R , and H .

If H , then:

2 2 2yi xi xi xi yi yi

2 2H

H H ( H R H R )P( / ) .exp exp .

2

iH K

Vectors i || ||i iR T T and

T

s si

x ( t( i )) y ( t( i )),

t t

T T

are constant along an edge. iTcomponents are also B-spline

functions having splinebasis as follows:

n,i

i ,n 1 i 1,n 1

i n i i n 1 i 1

( t ) n n( t ) ( t ).

t t t t t

BB B

If the iO s are conditionally independent, then the global

likelihood becomes i

P P s i sO / v O / v , and

i

arg .max P( / , )P( ) i i i

X H H / , leading to the

minimization of the quadratic energy function:

( ) ( ) ( )TE d B W d B .

1 1( ,..., , ,..., ,0,...,0)T

o oM o oM x x y yd is a (3MI) vector, Wis a

(3M3M) diagonal matrix with 2, , 1/i i i i M i M XiW W W ,

and 21 / 2i+2M,i+2MW T H .

B is a (3M2N ) matrix whose components are ( )ij jB B t i . Currently, the points i are uniformly

distributed: ( ) /t i i M . xB and yB are given by

,2 1,2( / ) ( / )xij xi i iB NH B i M B i M , and

,2 1,2( / ) ( / )yij xij yj j jB B NH B i M B i M .

Fig. 2. Test image.

Fig. 3. Results for the suggested algorithm (red), the true edge (white) and the

CSI (blue). (a) 10% of noise, (b) 20% of noise, (c) 40% of noise and (d) 50%of noise.

The solution arg min E( )*

is given by theweighted

leastsquare relation: arg min E( )*

.


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IV.EXPERIMENTALRESULTSSeveral tests were performed with the squared area of a

synthetic image (Fig. 2). For all the tests, the gradient was

computed by means of the derivative of a Gaussian kernel

with a standard deviation equal to 1. The number of knots Nwas set toM/4.

Fig. 4. Results for the suggested algorithm (red), the true edge (white) and

the CSI (blue). (a)p0 = 0.2, = 0.3; (b)p0 = 0.2, = 0.6;(c)p0 = 0.4, = 0.3;and (d)p0 = 0.4, = 0.6.

First, this algorithm was tested for an additive normal noise

with standard deviations equal to 0.1, 0.2, 0.4 and 0.5 whichrepresent, respectively, 10%, 20%, 40% and 50% of themaximum amplitude. Fig. 3 shows the results obtained with

this algorithm (red), the true edge (white) and the classical

spline interpolation (CSI; blue). The suggested model hasbeen developed for Gaussian noise, but it remains efficient for

other types of noise as well. In Fig. 4, the results obtained

with Salt & Pepper noise are shown, with 02n y p , for low

rate noise 0( 20)p and high rate noise 0( 40)p with two

different values of the amplitude . The Salt & Pepper noise

pdf is

( ) 0 ( ) (1 2. 0) ( ) 0 ( )P x p x p x p x .

(a) (b)Fig. 5. Results for standard cubic B-spline fitting (red) and for stochastic

cubic B-Sline (blue): (a) No noise; and (b) Gaussian noise with SNR=20 dB.

(a) (b)

Fig. 6. Results for standard cubic B-spline fitting (a) and for stochastic cubic

B-Sline (b) for Gaussian noise with SNR=20 dB.

Figures 5 and 6 show the performance of the proposed

algorithm in real images. The rectangles mark regions whereone can check the results of standard cubic B-spline and its

stochastic counterpart for noisy images with SNR=20 dB. In

Fig. 6, the performance of our algorithm seems to be slightly

better.

V. CONCLUSIONIn this paper, a robust algorithm for edge estimation with

sub-pixel accuracy based on a stochastic cubic B-spline is

introduced. The model parameters are calculated according to

an ML estimation framework relying on a Gaussian noise

model. The data likelihood is computed from the observation

model which includes both orientation and position

information.Comparative numerical experiments between this algorithm

and the CSI have been carried out for two different types ofnoise: Gaussian and Salt & Pepper. The experiments have

shown that our algorithm outperforms the other two, for both

kinds of noise. It remains accurate even for high noise levels

while the usual methods are generally sensitive to local

perturbations due to the global computation of a given

boundary using a ML rule. The likelihood of the observations

is explicitly computed.

ACKNOWLEDGMENTProf. Estrela and Prof. de Assis are grateful for the support

received by CAPES, FAPERJ and CNPq.

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