Soft segmentation is more flexible than hard segmentation. But the membership functions are usually sensitive to noise. In this paper, we propose a multiphase soft segmentation model for nearly piecewise constant images based on stochastic principle, where pixel intensities are modeled as random variables with mixed Gaussian distribution. The novelty of this paper lies in three aspects. First, unlike some existing models where the mean of each phase is modeled as a constant and the variances for different phases are assumed to be the same, the mean for each phase in the Gaussian distribution in this paper is modeled as a product of a constant and a bias field, and different phases are assumed to have different variances, which makes the model more flexible. Second, we develop a bidirection projected primal dual hybrid gradient (PDHG) algorithm for iterations of membership functions. Third, we also develop a novel algorithm for explicitly computing the projection from to simplex for any dimension using dual theory, which is more efficient in both coding and implementation than existing projection methods. 1. Introduction Image segmentation has played an important role in image processing and computer vision. Recently, variational segmentation models have attracted increasing interest [1–8]. With level set technique [9], variational models can be solved efficiently. The technique was originally developed for two-phase segmentation, and then extended to multiphase segmentation [10–13]. With carefully choosing the initial values, these methods can achieve an ideal solution for multiphase segmentation. However, the nonconvexity of the energy functional in the level set formulation is an inherent drawback of level set-method. As a result, many level set-based variational segmentation models are sensitive to initial values, especially for multiphase segmentation, and may lead to an inferior local minimum. One way to overcome the drawback is to use fuzzy membership function to replace the Heaviside function of a level set function in level set formulation for modeling a characteristic function, called soft segmentation. With this relaxation (characteristic function can be viewed as a special case of membership function), Chan and Bresson et al. [1, 2, 5] proved that global minimum can be achieved in these models due to the convexity of the energy functional. Unfortunately, this method cannot be easily applied to multi-phase segmentation. Very recently, based on a variational convexification technique developed by Pock et al. [14], Brown et al. [15], and Bae et
References
[1]
X. Bresson, S. Esedoglu, P. Vandergheynst, J.-P. Thiran, and S. Osher, “Fast global minimization of the active contour/snake model,” Journal of Mathematical Imaging and Vision, vol. 28, no. 2, pp. 151–167, 2007.
[2]
X. Bresson and T. F. Chan, “Non-local unsupervised variational image segmentation models,” UCLA CAM Report, 2008, http://www.math.ucla.edu/.
[3]
V. Caselles, R. Kimmel, and G. Sapiro, “Geodesic active contours,” International Journal of Computer Vision, vol. 22, no. 1, pp. 61–79, 1997.
[4]
T. F. Chan and L. A. Vese, “Active contours without edges,” IEEE Transactions on Image Processing, vol. 10, no. 2, pp. 266–277, 2001.
[5]
T. F. Chan, S. Esedoglu, and M. Nikolova, “Algorithms for finding global minimizers of image segmentation and denoising models,” SIAM Journal on Applied Mathematics, vol. 66, no. 5, pp. 1632–1648, 2006.
[6]
D. Mumford and J. Shah, “Optimal approximations by piecewise smooth functions and associated variational problems,” Communications on Pure and Applied Mathematics, vol. 42, no. 5, pp. 577–685, 1989.
[7]
N. Paragios and R. Deriche, “Geodesic active regions and level set methods for supervised texture segmentation,” International Journal of Computer Vision, vol. 46, no. 3, pp. 223–247, 2002.
[8]
S. C. Zhu, “Region competition: unifying snakes, region growing, and bayes/mdl for multiband image segmentation,” IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 18, no. 9, pp. 884–900, 1996.
[9]
S. Osher and J. A. Sethian, “Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations,” Journal of Computational Physics, vol. 79, no. 1, pp. 12–49, 1988.
[10]
G. Chung and L. A. Vese, “Energy minimization based segmentation and denoising using a multilayer level set approach,” Energy Minimization Methods in Computer Vision and Pattern Recognition, vol. 3757, pp. 439–455, 2005.
[11]
J. Lie, M. Lysaker, and X. C. Tai, “A variant of the level set method and applications to image segmentation,” Mathematics of Computation, vol. 75, no. 255, pp. 1155–1174, 2006.
[12]
L. A. Vese and T. F. Chan, “A multiphase level set framework for image segmentation using the Mumford and Shah model,” International Journal of Computer Vision, vol. 50, no. 3, pp. 271–293, 2002.
[13]
H.-K. Zhao, T. Chan, B. Merriman, and S. Osher, “A variational level set approach to multiphase motion,” Journal of Computational Physics, vol. 127, no. 1, pp. 179–195, 1996.
[14]
T. Pock, T. Schoenemann, G. Graber, H. Bischof, and D. Cremers, “A convex formulation of continuous multi-label problems,” in Proceedings of the European Conference on Computer Vision (ECCV '08), Marseille, France, October 2008.
[15]
E. S. Brown, T. F. Chan, and X. Bresson, “Convex formulation and exact global solutions for multi-phase piecewise constant Mumford-Shah image segmentation,” UCLA CAM Report cam09-66, 2009.
[16]
E. Bae, J. Yuan, and X.-C. Tai, “Global minimization for continuous multiphase partitioning problems using a dual approach,” UCLA CAM Report, 2009, http://www.math.ucla.edu/.
[17]
S. Chen and D. Zhang, “Robust image segmentation using FCM with spatial constraints based on new kernel-induced distance measure,” IEEE Transactions on Systems, Man, and Cybernetics B, vol. 34, no. 4, pp. 1907–1916, 2004.
[18]
X. Li, L. Li, H. Lu, and Z. Liang, “Partial volume segmentation of brain magnetic resonance images based on maximum a posteriori probability,” Medical Physics, vol. 32, no. 7, pp. 2337–2345, 2005.
[19]
B. Mory and R. Ardon, “Fuzzy region competition: a convex two-phase segmentation framework,” in Proceedings of the International Conference on Scale-Space and Variational Methods in Computer Vision, pp. 214–226, 2007.
[20]
B. Mory, R. Ardon, and J. P. Thiran, “Variational segmentation using fuzzy region competition and local non-parametric probability density functions,” in Proceedings of the IEEE 11th International Conference on Computer Vision (ICCV '07), pp. 1–8, October 2007.
[21]
D. L. Pham and J. L. Prince, “An adaptive fuzzy C-means algorithm for image segmentation in the presence of intensity inhomogeneities,” Pattern Recognition Letters, vol. 20, no. 1, pp. 57–68, 1999.
[22]
J. Shen, “A stochastic-variational model for soft Mumford-Shah segmentation,” International Journal of Biomedical Imaging, vol. 2006, Article ID 92329, 14 pages, 2006.
[23]
J. C. Bezdek, “A convergence theorem for the fuzzy ISODATA clustering algorithm,” IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 2, no. 1, pp. 1–8, 1980.
[24]
J. C. Dunn, “A fuzzy relative of the ISODATA process and its use in detecting compact wellseparated clusters,” Jounal of Cybernetics, vol. 3, no. 3, pp. 32–57, 1973.
[25]
F. Chen, Y. Chen, and H. D. Tagare, “An extension of sine-sinc model based on logarithm of likelihood,” in Proceedings of the International Conference on Image Processing, Computer Vision, and Pattern Recognition (IPCV '08), pp. 222–227, July 2008.
[26]
A. Dempster, N. Laird, and D. Rubin, “Maximum likelihood from incomplete data via the EM algorithm,” Journal Royal Statistical Society B, vol. 39, no. 1, pp. 1–8, 1977.
[27]
M. N. Ahmed, S. M. Yamany, N. Mohamed, A. A. Farag, and T. Moriarty, “A modified fuzzy C-means algorithm for bias field estimation and segmentation of MRI data,” IEEE Transactions on Medical Imaging, vol. 21, no. 3, pp. 193–199, 2002.
[28]
C. Li, R. Huang, Z. Ding, C. Gatenby, D. Metaxas, and J. Gore, “A variational level set approach to segmentation and bias correction of images with intensity inhomogeneity,” Medical Image Computing and Computer Assisted Intervention, vol. 11, part 2, pp. 1083–1091, 2008.
[29]
W. M. Wells III, W. E. L. Crimson, R. Kikinis, and F. A. Jolesz, “Adaptive segmentation of mri data,” IEEE Transactions on Medical Imaging, vol. 15, no. 4, pp. 429–442, 1996.
[30]
Y. Boykov and G. Funka-Lea, “Graph cuts and efficient N-D image segmentation,” International Journal of Computer Vision, vol. 70, no. 2, pp. 109–131, 2006.
[31]
J.-B. Hiriart-Urruty and C. Lemarechal, “Convex analysis and minimization algorithms,” in Grundlehren der Mathematischen Wissenschaften, pp. 305–306, Springer, New York, NY, USA, 1993.
[32]
A. Chambolle, “An algorithm for total variation minimization and applications,” Journal of Mathematical Imaging and Vision, vol. 20, no. 1-2, pp. 89–97, 2004.
[33]
M. Zhu and T. Chan, “An efficient primal-dual hybrid gradient algorithm for total variation image restoration,” CAM Report cam08-34, 2008.
[34]
E. S. Brown, T. F. Chan, and X. Bresson, “Convex relaxation method for a class of vector-valued minimization problems with applications to Mumford-Shah segmentation,” UCLA CAM Report cam10-43, 2010.
[35]
E. Esser, X. Zhang, and T. Chan, “A general framework for a class of first order primal-dual algorithms for TV minimization,” Cam Report cam09-67, 2009.
[36]
T. Pock, D. Cremers, H. Bischof, and A. Chambolle, “An algorithm for minimizing the Mumford-Shah functional,” in Proceedings of the 12th International Conference on Computer Vision (ICCV '09), pp. 1133–1140, Kyoto, Japan, October 2009.
[37]
T. Pock, D. Cremers, H. Bischof, and A. Chambolle, “An algorithm for minimizing the Mumford-Shah functional,” in Proceedings of the 12th International Conference on Computer Vision (ICCV '09), pp. 1133–1140, October 2009.
