Chan-Vese Deformable Model for Different Domain … · Chan-Vese Deformable Model for Different...

Chan-Vese Deformable Model for

Different Domain Images

Harita Singh Shabnam Parveen M.Tech.Student at JMIT Radaur Asstt. Prof., JMIT Radaur

[email protected] [email protected]

Abstract A contour in the image plane defining an initial segmentation is initialized and then evolved using Chan-Vese Model[10] which has four terms and each term has its own significance in evolving the curve. The segmentation of objects is based on techniques of curve evolution based on Mumford-Shah functional[7] which is then solved using the level set method. Level set function is Φ(i,j,t) where (i,j) are coordinates in the image plane and t is an artificial time and it simultaneously defines an edge contour and segmentation of the image. In the current work we have studied the effects of parameters on various images and experimental results are shown keeping number of iterations fixed and size of image as 256x256 . 1.Introduction In [3] a precise relationship between the geometric and parametric active contours has been developed which includes spatially-varying coefficients, both tension and rigidity, and non-conservative external forces. Geometric deformable models implemented using level set methods have advantages over parametric models due to their intrinsic behavior, parameterization independence, and ease of implementation. In[2] a new class of geometric deformable models designed using a novel topology-preserving level set method, is presented which achieves topology preservation by applying the simple point concept from digital topology. This Chan-Vese geometric active contour model (based upon Mumford–Shah Functional) works even for images without edges. The model begins with a contour in the image plane defining an initial

segmentation and then contour is evolved according to evolution equation.

In general form it is written as follows:

F(Φ)= µ(∫|

(H(Φ)dx)p + υ∫ H(Φ)dx + λ1 ∫|I-C1|2

Ω Ω Ω

H(Φ)dx + λ2 ∫|I-C2|2 (1-H(Φ))dx

Ω µ,υ,λ1,λ2 and p are parameters selected to fit a particular class of images. Here, H=Heaviside function I is the image to be segmented Ω=domain of that image C1 and C2 are averages of the image I in the regions where Φ>=0 and Φ<0 respectively. In the above equation the first term can be thought as a penalty on the total length of the edge contour for a given segmentation. The second term is a penalty on the total area of the foreground region found by the segmentation. The third term is proportional to the variance of the image gray level in the in the foreground region. The fourth term does the same for background region. Usually, we take λ1=λ2=1 but if we set λ1=2, λ2=1 then our final segmentation will have a more uniform foreground region at the expense of loss of uniformity in background . In applications where we expect an approximately black background and foreground objects with varying gray

Harita Singh et al ,Int.J.Computer Technology & Applications,Vol 3 (3), 1273-1278

IJCTA | MAY-JUNE 2012 Available [email protected]

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levels, we would set λ1 < λ2.

The Basis of Chan-Vese algorithm is a “Fitting Energy Functional”. The goal of algorithm is to minimize this fitting energy for a given image and corresponding will define segmentation.Chan-Vese algorithm evolves the contour via a level set method denoted as Φ(i,j,t).The edge contour is taken to be the zero level set Φ>=0) and Φ<0 and segmentation is given by the two regions. The level set function will be evolved according to some Partial Differential equation and hopefully will reach a steady state limit that gives a useful segmentation of the image. The most important step is to actually determine a level set function that segments the image in a meaningful way. The simplest example would be to define the level set function to be the value of a gray level image at each pixel minus some threshold i.e. set Φ(i,j)=I(I,j)-t. Then the level set function is positive in regions where the gray level is above the threshold and negative in regions where the gray level is below the threshold. Active contour models are also used for 2D and 3D biomedical images formulated using the level set method[9].These models can also be generalized to segmentation of images with more than two segments. Snakes are also active contour models[6]-they lock onto nearby edges,localizing them accurately. Snakes were used extensively in image processing applications, particularly to locate object boundaries.An improved region-based active contour/surface model for 2D/3D brain MR image segmentation is introduced in [5].The model combines the advantages of both local and global intensity information, which enable the model to cope with intensity inhomogeneity. In [14] a graph cut based active contour without edges segmentation model has been discussed to track pedestrian in thermal images. The deformable model is based on the Mumford- Shah piecewise constant energy formulation. A novel 3D brain cortex segmentation procedure[11] has been stated utilizing dual front active contours which minimize image-based energies in a manner that yields flexibly global minimizers based on active regions. Region-based information and boundary-based information may be combined flexibly in the evolution potentials for accurate segmentation results.

Another algorithm for automated segmentation of white matter in brain MRI images, was proposed[8] which can be used to create connected representations of the gray matter in the cerebral cortex of the brain. Automating the postmortem identification of deceased individuals based on dental characteristics is receiving increased attention especially with the large number of victims encountered in mass disasters. In [4] the task of teeth contour extraction is accomplished using active contour without edges. This technique is based on the intensity ofthe overall region of the tooth image and, therefore, does not necessitate the presence of a sharp boundary between teeth. In [12] a novel formulation framework of the minimal surface problem, called Active Geometric Functions (AGF), is proposed to reach truly real-time performance in segmenting 4D ultrasound data. Another improved region-based active contour model in a variational level set formulation in[13] is described. An energy functional with a local intensity fitting term, is defined which induces a local force to attract the contour and stops it at object boundaries, and an auxiliary global intensity fitting term, which drives the motion of the contour far away from object boundaries. Therefore, the combination of these two forces allows for flexible initialization of the contours. The dimensional changes induced by P.vivax malaria parasites in erythrocytes[15] in human blood are prominent and varies with the degree of parasitaemia. The changes in the shape of erythrocyte and its cytoplasm have been determined by shape descriptors and gray scale variation of the cytoplasm by microscopic imaging and image processing tools. An improved algorithm is also described[1] based on the piecewise-smooth Mumford and Shah (MS) functional for an efficient and reliable segmentation. In order to speed up convergence, an additional force, at each time step, is introduced further to drive the evolution of the curves instead of only driven by the extensions of the complementary functions u + and u−.The piecewise-constant MS functional is integrated to generate the extra force based on a temporary image that is dynamically created by computing the union of u + and u – during segmenting.



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2.Formulation Relation with the Mumford Shah Functional: The Mumford Shah functional for segmentation is:

F(u,C)= µ.Length (C)+ λ∫ |u0 (x,y)-u(x,y)|2 dxdy+ Ω

∫ | u (x,y)|2 dxdy Ω\C

Where u0 : Ω R is a given image, µ and λ are positive parameters. The solution image u obtained by minimizing this functional is formed by smooth regions Ri and with sharp boundaries, denoted here by C. A reduced form of this problem is simply the restriction of F to piecewise constant functions u, i.e., u=constant ci on each connected component Ri of Ω\C. Therefore ci =average(u0) on each connected component Ri.. The reduced case is called the minimal partition problem. Our active contour model with υ=0 and λ1 = λ2 = λ is a particular case of the minimal partition problem, in which we look for the best approximation u of uo, as a function taking only two values, namely u= average(u0) inside C or average(u0)outside C and with one edge C, represented by the snake or the active contour. Using the Heaviside function,and the one-dimensional Dirac measure δ0,and defined, respectively, by H(z) = 1, if z >=0 H(z) = 0, if z<0 δ0= d H(z) dz in the sense of distributions. But we have used regularized value of H H=1/2 (1+2/π arctan(z/ε)) We express the terms in the energy F in the following way:

LengthΦ=0= ∫ | H(Φ(x,y))|dxdy

Ω

= ∫ δ0(Φ(x,y)) | Φ(x,y)|dxdy,

Ω

Area Φ>=0=∫ H(Φ(x,y))dxdy,

Ω

and

∫ |u0(x,y)-c1|2dxdy

Φ>0

=∫ |u0(x,y)-c1|2 H(Φ(x,y))dxdy,

Ω

∫ |u0(x,y)-c2|2 dxdy

Φ<0

=∫ |u0(x,y)-c2|2(1-H(Φ (x,y)))dxdy.

Ω Then the energy F(c1,c2, Φ)

= µ∫ δ (Φ(x,y))| Φ(x,y)|dxdy,

Ω

+ υ ∫ H(Φ (x,y))dxdy.

Ω

+ λ1 ∫ |u0(x,y)-c1|2H(Φ (x,y))dxdy.

Ω

+ λ2 ∫ |u0(x,y)-c2|2(1-H(Φ (x,y)))dxdy.

Ω Also u can simply be written using level set formulation as: u(x,y)= c1H(Φ(x,y))+c2(1-H(Φ(x,y))),(x,y)є Ω Keeping Φ fixed and minimizing energy F(c1,c2,Φ) with respect to the constants c1 and c2, it is easy to express these constants function of Φ by

c1(Φ)= ∫ u0(x,y)H(Φ (x,y))dxdy. Ω

∫ H(Φ (x,y))dxdy.

If ∫ H(Φ (x,y))dxdy>0 (i.e. if the curve

Ω

has a nonempty interior in Ω) and

c2(Φ)= ∫ u0(x,y)(1-H(Φ (x,y)))dxdy Ω

∫ (1-H(Φ (x,y)))dxdy Ω



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If ∫(1- H(Φ (x,y)))dxdy>0 (i.e. if the curve

Ω

has a nonempty exterior in Ω).For the corresponding degenerate cases there are no constraints on the values of c1 and c2.Then c1 and c2 are infact given by c1(Φ)=average(u0) in Φ>=0 c2(Φ)=average(u0) in Φ<0

3.Algorithm

• Initialize c0 by Φ0 ,n=0 .

• Compute c1 (Φn) and c2(Φ

n) • Solve the PDE in Φ to obtain (Φn+1) . • Check whether n=1000 ,If not,n=n+1 and repeat.

4.Experimental Results

All the images are originally of size 256x256.

Figure.1Contour evolving on brain image Figure.2 Effect on the brain image on changing the with initial values of parameters value of p h=1.0, epsilon=1.0,dt=0.1, υ=0, h=1.0, epsilon=1.0,dt=0.1, υ=0, µ=255*255*0.05,λ1=1.0,λ2=1.0,p=1, µ=255*255*0.05,λ1=1.0,λ2=1.0, p=0,

eps=0.0001, iterations=1000 eps=0.0001,iterations=1000



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Figure.3 Contour evolving on astronomical Figure.4 Effect on astronomical image on image with initial values of parameters changing the value of p h=1.0, epsilon=1.0,dt=0.1, υ=0, h=1.0, epsilon=1.0,dt=0.1, υ=0, µ=255*255*0.05,λ1=1.0,λ2=1.0,p=1, µ=255*255*0.05,λ1=1.0,λ2=1.0, p=0,


Figure.5 Contour evolving on a noisy image Figure.6 Effect on noisy image on with initial values of parameters changing the value of λ2 h=1.0, epsilon=1.0,dt=0.1, υ=0, h=1.0, epsilon=1.0,dt=0.1, υ=0, µ=255*255*0.05,λ1=1.0,λ2=1.0,p=1, µ=0,λ1=1.0,λ2=2.0, p=0,


5.Conclusion On decreasing the value of p length term in the formulation is affected which is responsible for maintaining smoothness of length while evolution of the curve. So if we decrease value of p it leads to breakage of curve.Same effect can be seen for astronomical image.Also on increasing the value of λ2 there is increase in uniformity in the region(Φ<0) at the expense of decreasing uniformity in the region (Φ>0).More is the smoothness in region faster is the evolution.Another observation is that if we set µ=0 for noisy images then it does not give good results.



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6.References [1]Yingjie Zhang, “Fast Segmentation for the Piecewise Smooth Mumford-Shah Functional,” International Journal of Signal Processing 2;4 © www.waset.org Fall 2006. [2] Xiao Han, Chenyang Xu and Jerry L. Prince, “A Topology Preserving Level Set Method for Geometric Deformable Models,” IEEE Transactions On Pattern Analysis And Machine Intelligence, Vol. 25, No. 6, June 2003. [3]Chenyang Xu, Anthony Yezzi, Jr. and Jerry L. Prince, “On the Relationship between Parametric and Geometric Active Contours,” In Proc. of 34th Asilomar Conference on Signals, Systems, and Computers, pp. 483-489, October 2000. [4] Samir Shah, Ayman Abaza, Arun Ross and Hany Ammar, “Automatic Tooth Segmentation Using Active Contour Without Edges,” IEEE 2006 Biometrics Sympoium. [5] Li Wang, Chunming Li, Quansen Sun, Deshen Xia, and Chiu-Yen Kao, “ Brain MR Image Segmentation Using Local and Global Intensity Fitting Active Contours/Surfaces,” Springer-Verlag Berlin Heidelberg 2008. [6]Michael Kass,Andrew Witkin and D.Terzopoulos, “Snakes:Active Contour Models,” International Journal of Computer Vision,321-331(1998). [7] David Mumford and Jayant Shah, “Optimal Approximations by Piecewise Smooth Functions and Associated Variational Problems,” Commun.Pure Appl. Math, Vol.42,pp 577-685,1989. [8] Gowri Srinivasa, Vivek S. Oak, Siddharth J. Garg, Matthew C. Fickus and Jelena Kovacevic, “Voting-Based Active Contour Segmentation Of FMRI Images Of The Brain,” Proc. IEEE Conf.on Image Proc.,San Diego,USA Oct.2008. [9]Tony F.Chan Luminita A.Vese, “Active Contour and Segmentation Models Using Geometric PDE’s for Medical Imaging,” Geometric Methods in Bio-Medical Image Processing'', Series: Mathematics and Visualization, Springer, 2002, pp. 63-75. [10]Tony F.Chan and Luminita A.Vese, “Active Contours Without Edges,” IEEE Transactions on Image Processing, Vol. 10, No. 2, February2001. [11] Hua Li, Anthony Yezzi and Laurent D. Cohen, “ 3D Brain Segmentation Using Dual-Front Active Contours with Optional User-Interaction,” International Journal of Biomedical ImagingVolume 2006, Article ID 53186. [12] Qi Duan, Elsa D. Angelini, Shunichi Homma and Andrew F. Laine, “Real-Time Segmentation Of 4D Ultrasound By Active Geometric Functions,” IEEE International Symposium on Biomedical Imaging,2008. [13] Li Wang , Chunming Li, Quansen Sun ,Deshen Xia and Chiu-Yen Kao, “Active Contours Driven by Local and Global Intensity Fitting Energy with Application to Brain MR Image Segmentation,” Journal of Mathematical Imaging and Vision 37(2): 98-111 (2010). [14] Noha El-Zehiry and Adel Elmaghraby, “A Graph Cut Based Active Contour without Edges with Relaxed Homogeneity Constraint,” International Conference on Pattern Recognition,2008. [15] R.Renuka Devi , V.Rajagopal, M.Senthil kumar , G.Magesh, “Computerized Shape Analysis Of ErythrocytesAnd Their Formed Aggregates In Patients Infected With P.Vivax Malaria,” Advanced Computing: An International Journal ( ACIJ ), Vol.2, No.2, March 2011



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