Automated Segmentation Using a Fast Implementation of the Chan-Vese Models
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1 Automated Segmentation Using a Fast Implementation of the Chan-Vese Models Huan Xu, and Xiao-Feng Wang,,3 Intelligent Computation Lab, Hefei Institute of Intelligent Machines, Chinese Academy of Science, P.O. Box 3, Hefei, Anhui 33, China Department of Automation, University of Science and Technology of China, Hefei, 37, China 3 Department of Computer Science and Technology, Hefei University, Hefei 3, China xuhuan@mail.ustc.edu.cn, xfwang@iim.ac.cn Abstract. In this paper, a fast implementation method of Chan-Vese models is proposed, which does not require numerical solutions of PDEs. The advantages of traditional level set methods, such as automatic handling of topological changes, are also preserved. The whole process is described as follows: First, the Otsu thresholding method is adopted to obtain the initial contours for the following level set evolution. Then, the initial curves are evolved to approach the true boundaries of objects by using the proposed fast implementation method of Chan-Vese model. Experimental results on some real and synthetic images show that our proposed approach is capable of automatically segmenting images with a low time-consumption. Keywords: Image Segmentation; Otsu Thresholding; Active Contours; Chan- Vese Models. Introduction Image segmentation is a fundamental problem in image processing and computer vision. It is a process of partitioning the image into some non-intersecting regions such that each region is homogeneous and none of two adjacent regions are homogeneous. Various categories of segmentation techniques have been proposed in recent years, and they all face the following two challenging issues: () the accuracy of the segmentation; () the real-time acquirement of the segmentation. Thresholding methods [], which make decisions based on local pixel information, are effective when the intensity levels of the objects fall outside the range of levels in the background. Due to spatial structural information being ignored, they are effective for simple images that display only small amounts of structure. Regarding segmentation as an energy minimization problem, active contour model has became an effective segmentation technique. Active contour models can be broadly categorized into two kinds: parametric active contours [] and geometric active contours [3]. The level set method is a successful improvement to active contour model, which has become increasingly popular and has been proved to be a useful tool for image segmentation D.-S. Huang et al. (Eds.): ICIC 8, LNAI 57, pp. 35 4, 8. Springer-Verlag Berlin Heidelberg 8
2 36 H. Xu and X.-F. Wang since it was proposed by Osher and Sethian [4]. Level set methods are the ones based on active contours particularly designed to handle the segmentation of deformable structure, which display interesting elastic behaviors, and can handle topological changes. Generally, a classical level set method is to consist of an implicit data representation of a hypersurface, a set of PDEs that govern how the curve moves, and the corresponding solutions for implementing this method on computers [5]. Based on the Mumford-Shad functional [6] for segmentation, Chan and Vese [7] proposed an easily handle model, or bimodal Chan-Vese model. This method can partition a given image into two regions, one representing the objects to be detected, and the other one representing the background. However, the method is time-consuming and fails to segment images with intensity inhomogeneity. In this paper, we proposed a modified fast Chan-Vese approach to overcome above drawbacks. The whole process is described as follows: First, the thresholding method is adopted to obtain the initial contours for the following level set evolution. Then, the initial curves are evolved to approach the true boundaries of objects by using the proposed fast implementation method of Chan-Vese model. Compared with classical Chan-Vese model, our approach could automatically get much more accurate results in a fast way. The rest of the paper is organized as follows: Section introduces the level set method based on Chan-Vese (CV) model. Section 3 presents the proposed modified CV method in detail. Experiment results are demonstrated in Section 4. Finally, Section 5 concludes the paper and discusses the future work Chan-Vese Models The Chan-Vese models are curve evolution implementations of a well-posed case of the Mumford-Shah model [6]. The Mumford-Shah method is an energy-based method introduced by Mumford and Shah via an energy function in 989. Consider a possibly noisy or blurry two-dimensional image u with image domain and the segmenting curve C, the general form of the Mumford-Shah function can be written as: μ ν () \ C MS E ( u, C) = u ( x, y) u( x, y) + u( x, y) + Length( C) where μ and ν are parameters. The Length term is a constraint on the curve s length and controls its smoothness. The other two terms can divide the image into different regions while allowing for the discontinuities along the edges. The removal of any of the above three terms in Eqn.() will result in trivial solutions for u and C [8]. Chan and Vese[7] proposed an algorithm for decomposing the image into two regions with piecewise constant approximations. Minimizing Eqn() becomes the minimization of the following energy functional: CV E ( c, c, C) = μ Length( C) + ν Area( inside( C)) (, ) λ (, ) inside( C ) outside( C ) + λ u x y c + u x y c ()
3 Automated Segmentation Using a Fast Implementation of the Chan-Vese Models 37 where μ, ν, λ and λ are positive constants, usually fixing λ = λ = and ν =. c and c are the averages of u inside C and outside C, respectively. To solve this minimization problem, we use the level set method [4] to represent C, i.e., C is the zero level set of a Lipschitz functionϕ : R R. Then, we can replace the unknown variable C by the unknown variableϕ, and the energy functional () can be written as: E ( c, c, ϕ) = μ δ ( ϕ( x, y)) ϕ( x, y) + ν H ( ϕ( x, y)) + CV ε ε ε λ u ( x, y) c H ( ϕ( x, y)) + λ u ( x, y) c ( H ( ϕ( x, y))) ε ε where Hε ( z) and δ ε ( z) are the regularized approximation of Heaviside function and Dirac delta function, respectively. This minimization problem is solved by taking the Euler-Lagrange equations and updating the level set function ϕ ( xy, ) by the gradient descent method: ϕ ϕ = δε ( ϕ)[ μdiv( ) v λ( u c) + λ( u c) ] = t ϕ c and c can be updated at each iteration by: u ( x, y) H( ϕ( x, y)) u ( x, y)( H ( ϕ( x, y))) c ( ϕ) = c( ϕ) = H( ϕ( x, y)) ( H( ϕ( x, y))) The main advantage for this model is that it can automatically detect interior contours, where the initial curve can be placed anywhere in the image, and detect both contours with, or without gradient [7]. Since this model assumes that an image consists of statistically homogeneous regions, with intensities in each region being a constant up to a certain noise level. Therefore, the method fails to segment images with intensity inhomogeneity and is time-consuming. 3 Our Modified Fast Chan-Vese Approach 3. Initial Contour Using Thresholding Segmentation Manual definition of initial contour hinders the automation of Chan-Vese model. However, in computer vision and image processing, Otsu method [9] is used to perform thresholding, or, the transformation of a gray-level image to a binary image. So we could use the method to get an initial contour for the following level set evolution. Otsu proposed an algorithm for automatic threshold selection from a histogram of image. Let the pixels of a given image be represented in L gray levels[,,... L ]. The (3) (4) (5)
4 38 H. Xu and X.-F. Wang number of pixels at level i is denoted by n i, and the total number of pixels by N = n + n n i... Then, suppose that the pixels were dichotomized into two classes C and C, which denote pixels with levels [,... k ] and [ k +,..., L], respectively. This method is based on a discriminant criterion, which is the ration of between-class variance and total variance of gray levels. The optimal threshold of an image depends on maximizing between-class variance to maximize the separability of the resultant classes in gray levels. 3. Fast implementation of the Chan-Vese Model The Chan-Vese models are usually implemented by solving PDEs, such as the level set equations [4, ] and Poisson equations []. These methods are computationally intense, although they are theoretically sound. In this section, we would discuss the proposed fast implementation method for Chan-Vese models. Assuming that the evolving curve C in is the boundary of an open set of. inside( C ) denotes the region C and outside( C ) denotes the region \ C, ϕ is the level set function. c And c are the averages of u inside C and outside C, respectively. They could obtain from Eqn.(5). Furthermore, let s define: cmax = max( c ( ϕ), c ( ϕ)) cmin = min( c ( ϕ), c ( ϕ)) Then we give our evolution equation: ϕ cmax + cmin = ( u )( cmax cmin) (6) t For numerical implementation, the Heaviside function H ( ϕ ) here is regularized as: z Hε ( z) = ( + arctan( )) π ε Our approach can easily implemented use the difference scheme and iterations: k k ϕ + i, j ϕi, j cmax + cmin = ( u )( cmax cmin) τ k+ k cmax + cmin i, j i, j u cmax cmin ϕ = ϕ + τ( )( ) (7) where τ is the time step. Here the whole process of our fast Chan-Vese method is described in Fig.. Input image Otsu Thresholding Fast CV Final Segment Result Fig.. The process of our method
5 Automated Segmentation Using a Fast Implementation of the Chan-Vese Models 39 4 Experimental Results The proposed method is implemented on a computer which has two Intel(R) Pentium(R).9GHz CPUs, G bytes RAM, and runs the Microsoft Windows operating system. The CPU times given in this paper are the sums of system CPU times and user CPU times. The system CPU time is usually very small, typically.-.8 seconds. In the following experiment, the parameters for all the images areτ =., ε =. (a) (b) (c) (d) (e) Fig.. Comparison of the proposed fast CV approach to the classical Chan-Vese method. (a) Original image (image size is x) (b) Segmentation result using Fast CV (Elapsed time is seconds) (c) The level set function of fast CV (d) Segmentation result using Chan- Vese model (Elapsed time is.586 seconds) (e) The level set function of Chan-Vese model. The test images are segmented using the proposed method and the classical method with PDEs solution. Fig. shows that the proposed method is times faster than the classical method with PDEs solution and achieves the same segmentation results, Fig. (b)-(c) demonstrate that the proposed method can also efficiently handle the topological changes. The effects of intensity inhomogeneity on the proposed method are illustrated in Fig.3. The classical Chan-Vese model fails to segment the image in Fig.3(a) (as shown in Fig.3(d)). The reason is that the motion of the contours in the Chan-Vese model is guided by global image information only, which can lead to wrong segmentation result for the images with intensity inhomogeneity. In contrast, Fig.3 (b) and (c) show that the proposed fast CV method works on the image in Fig.3 (a). We also test our approach on real image (as shown in fig.4 (a)). It can be seen that our approach can get a clear boundary with elapsed time being.397 seconds, while the Chan-Vese model needs seconds.
6 4 H. Xu and X.-F. Wang (a) (b) (c) (d) (e) Fig. 3. Comparison of the proposed fast CV approach to the classical Chan-Vese method. (a) Original image (image size is 3x3) (b) Segmentation result using fast CV result (Elapsed time is.397 seconds) (c) The level set function of fast CV (d) Segmentation result using Chan-Vese model (Elapsed time is seconds) (e) The level set function of Chan-Vese model. (a) (b) (c) (d) (e) Fig. 4. Comparison of the proposed fast CV approach to the classical Chan-Vese method. (a) Original image (image size is 99x4) (b) Segmentation result using fast CV result (Elapsed time is.89 seconds) (c) The level set function of fast CV (d) Segmentation result using Chan-Vese model (Elapsed time is seconds ) (e) The level set function of Chan-Vese model. 5 Conclusion In this paper, a fast implementation method for the Chan-Vese models is proposed, which does not require solutions of the PDEs. Otsu thresholding method is first used
7 Automated Segmentation Using a Fast Implementation of the Chan-Vese Models 4 to obtain the initial contour for the following level set evolution, and then the proposed modified fast Chan-Vese approach is used to evolve the initial contour with an automatic segmentation solution scheme. We have applied our approach on some synthetic and real images, experimental results show that our approach is fast and can achieve good results. Our future works will focus on extending the fast scheme to multiphase Chan-Vese model to get more satisfactory results on multiphase images. Acknowledgements. This work was supported by the grants of the National Science Foundation of China, Nos & 6773, the grant of the Graduate Students Scientific Innovative Project Foundation of CAS (Xiao-Feng Wang), the grant of the Scientific Research Foundation of Education Department of Anhui Province, No. KJ7B33, the grant of the Young Teachers Scientific Research Foundation of Education Department of Anhui Province, No. 7JQ5. References. Lim, Y.W., Lee, S.U.: On the Color Image Segmentation Algorithm based on the Thresholding and the Fuzzy C-Means Techniques. Pattern Recognition 3(9), (99). Kass, M., Witkin, A., Terzopoulos, D.: Snakes: Active contour models. Int. J. Comput. Vis., 3 33 (987) 3. Caselles, V., Kimmel, R., Sapiro, G.: Geodesic Active Contours. In: Proc. 5th International conf. on Computer Vision, Boston, pp (995) 4. Osher, S., Sethian, J.A.: Fronts Propagating with Curvature-Dependent Speed: Algorithms based on Hamilton Jacobi Formulation. Journal of Computational Physics 79, 49 (988) 5. Tsai, Y.H.S., Osher, S.: Total Variation and Level Set Based Methods in Image Science. Acta Numerica, 6. (5) 6. Mumford, D., Shah, J.: Optimal Approximation by Piecewise Smooth Functions and Associated Variational Problems. Commun. Pure Appl. Math. 4, (989) 7. Chan, T.F., Vese, L.A.: Active Contours without Edges. IEEE Trans. Image Processing (), () 8. Tsai, A., Yezzi, A., Willsky, A.S.: Curve Evolution Implementation of the Mumford Shah Functional for Image Segmentation, Denoising, Interpolation, and Magnification. IEEE Trans. Image Process (8), () 9. Otsu, N.: A Threshold Selection Method from Gray Level Histogram. IEEE Transactions on System, Man and Cybernetics 8, 6 66 (978). Sethian, J.: Level Set Methods and Fast Marching Methods. Cambridge Monograph on Applied and Computational Mathematics. Cambridge University Press, Cambridge (999). Vese, L., Chan, T.: A Multiphase Level Set Framework for Image Segmentation using the Mumford and Shah Model. Inter. J. Computer Vision 5(3), 7 93 ()
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