Soft Image Segmentation Based on the Mixture of Gaussians and the Phase-Transition Theory

DOI: 10.4236/am.2014.518275   PDF   HTML   XML   2,568 Downloads   3,118 Views   Citations


In this paper, we propose a new soft multi-phase segmentation model where it is assumed that the pixel intensities are distributed as a Gaussian mixture. The model is formulated as a minimization problem through the use of the maximum likelihood estimator and phase-transition theory. The mixture coefficients, which are estimated using a spatially varying mean and variance procedure, are used for image segmentation. The experimental results indicate the effectiveness of the method.

Share and Cite:

Barcelos, C. , Chen, Y. and Chen, F. (2014) Soft Image Segmentation Based on the Mixture of Gaussians and the Phase-Transition Theory. Applied Mathematics, 5, 2888-2898. doi: 10.4236/am.2014.518275.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] Mumford, D. and Shah, J. (1989) Optimal Approximations by Piecewise Smooth Functions and Associated Variational Problems. Communications on Pure and Applied Mathematics, 42, 577-685.
[2] Caselles, V., Kimmel, R. and Sapiro, G. (1997) Geodesic Active Contours. International Journal of Computer Vision, 1, 61-79.
[3] Paragios, N. and Deriche, R. (2002) Geodesic Active Regions and Level Set Methods for Supervised Texture Segmentation. International Journal of Computer Vision, 46, 223-247.
[4] Zhu, S.C. and Yuille, A. (1996) Region Competition: Unifying Snakes, Region Growing, and Bayes/mdl for Multiband Image Segmentation. IEEE Transaction on Pattern Analysis and Machine Intelligence, 18, 884-900.
[5] Osher, S. and Sethian, J.A. (1988) Fronts Propagation with Curvature Dependent Speed: Algorithms Based on Hamilton Jacobi Formulations. Journal of Computational Physics, 79, 12-49.
[6] Vese, L. and Chan. T. (2002) A Multiphase Level Set Framework for Image Segmentation Using the Mumford and Shah Model. International Journal of Computer Vision, 50, 271-293.
[7] Zhao, B.M.H.-K., Chan, T. and Osher, S. (1996) A Variational Level Set Approach to Multiphase Motion. Journal of Computational Physics, 127, 179-195.
[8] Chung, G. and Vese, L. (2005) Energy Minimization Based Segmentation and Denoising Using a Multilayer Level Set Approach. Energy Minimization Methods in Computer Vision and Pattern Recognition, 3757, 439-455.
[9] Lie, M.L.J. and Tai, X.C. (2006) A Variant of the Level Set Method and Applications to Image Segmentation. AMS Mathematics of Computations, 75, 1155-1174.
[10] Chan, T.F., Esedoglu, S. and Nikolova, M. (2006) Algorithms for Finding Global Minimizers of Image Segmentation and Denoising Models. SIAM Journal on Applied Mathematics, 66, 1632-1648.
[11] Bresson, X., Esedoglu, S., Vandergheynst, P., Thiran, J.P. and Osher, S. (2007) Fast Global Minimization of the Active Contour/Snake Model. Journal of Mathematical Imaging and Vision, 28, 151-167.
[12] Aujol, J.F. and Chambolle, A. (2005) Dual Norms and Image Decomposition Models. International Journal of Computer Vision, 63, 85-104.
[13] Aujol, J.F., Gilboa, G., Chan, T. and Osher, S. (2006) Structure-Texture Image Decomposition Modeling Algorithms and Parameter Selection. International Journal of Computer Vision, 67, 111-136.
[14] Carter, J. (2001) Dual Methods for Total Variation-Based Image Restoration. Ph.D. Thesis, University of California, Los Angeles.
[15] Chambolle, A. (2004) An Algorithm for Total Variation Minimization and Applications. Journal of Mathematical Imaging and Vision, 20, 89-97.
[16] Chan, T., Golub, G. and Mulet, P. (1999) A Nonlinear Primal-Dual Method for Total Variation-Based Image Restoration. SIAM Journal on Scientific Computing, 20, 1964-1977.
[17] Chen, S. and Zhang, D. (2004) Robust Image Segmentation Using FCM with Spatial Constraints Based on New Kernel-Induced Distance Measure. IEEE Transactions on Systems Man and Cybernetics, 34, 1907-1916.
[18] Mory, B. and Ardon, R. (2007) Fuzzy Region Competition: A Convex Two-Phase Segmentation Framework. International Conference on Scale-Space and Variational Methods in Computer Vision, Ischia, 30 May-2 June 2007, 214-226.
[19] Mory, B., Ardon, R. and Thiran, J.P. (2007) Variational Segmentation Using Fuzzy Region Competition and Local Non-Parametric Probability Density Functions. ICCV 2007, 1-8.
[20] Li, L., Li, X., Lu, H. and Liang, Z. (2005) Partial Volume Segmentation of Brain Magnetic Resonance Images Based on Maximum a Posteriori Probability. Medical Physics, 32, 2337-2345.
[21] Pham, D.L. and Prince, J.L. (1998) An Adaptive Fuzzy C-Means Algorithm for the Image Segmentation in the Presence of Intensity Inhomogeneities. Pattern Recognition Letters, 20, 57-68.
[22] Bezdek, J.C. (1980) A Convergence Theorem for the Fuzzy ISODATA Clustering Algorithm. IEEE Transactions on Pattern Analysis and Machine Intelligence, PAMI-2, 1-8.
[23] Dunn, J.C. (1973) A Fuzzy Relative of the ISODATA Process and Its Use in Detecting Compact Well-Separated Clusters. Journal of Cybernetics, 3, 32-57.
[24] Dempster, A., Laird, N. and Rubin, D. (1977) Maximum Likelihood from Incomplete Data via the EM Algorithm. Journal of the Royal Statistical Society, Series B, 39, 1-8.
[25] Wells, W., Grimson, E., Kikinis, R. and Jolesz, F. (1996) Adaptive Segmentation of MRI Data. IEEE Transactions on Medical Imaging, 15, 429-442.
[26] Cahn, J. and Hilliard, J. (1958) Free Anergy of a Non-Uniform System. I. Interfacial Free Energy. The Journal of Chemical Physics, 28, 258-267.
[27] Modica, L. (1987) The Gradient Theory of Phase Transitions and the Minimal Interface Criterion. Archive for Rational Mechanics and Analysis, 98, 123-142.
[28] Modica, L. and Mortola, S. (1977) Un esempio di Gamma-convergenza. Bollettino della Unione Matematica Italiana B, 14, 285-299.
[29] Bourdin, B. and Chambolle, A. (2003) Design-Dependent Loads in Topology Optimization. ESAIM: Control, Optimisation and Calculus of Variations, 9, 19-48.
[30] Burger, M. and Stainko, R. (2006) Phase-Field Relaxation of Topology Optimization with Local Stress Constraints. SIAM Journal on Control and Optimization, 45, 1447-1466.
[31] Wang, M.Y. and Zhou, S. (2004) A Variational Method for Structural Topology Optimization. CMES, 6, 547-566.
[32] Rumpf, M. and Wirth, B. (2009) A Nonlinear Elastic Shape Averaging Approach. SIAM Journal on Imaging Sciences, 2, 800-833.
[33] Jung, Y., Kang, S. and Shen, J. (2007) Multiphase Image Segmentation via Modica-Mortola Phase Transition. SIAM Applied Mathematics, 67, 1213-1232.
[34] Li, Y. and Kim, J. (2011) Multiphase Image Segmentation Using a Phase-Field Model. Computers and Mathematics with Applications, 62, 737-745.
[35] Shen, J. (2006) A Stochastic Variational Model for Soft Mumford-Shah Segmentation. International Journal of Biomedical Imaging. Published Online.
[36] Posirca, I., Chen, Y. and Barcelos, C.Z. (2011) A New Stochastic Variational PDE Model for Soft Mumford-Shah Segmentation. Journal of Mathematical Analysis and Applications, 384, 104-114.
[37] Ambrosio, L. and Tortorelli, V.M. (1990) Approximation of Functional Depending on Jumps by Elliptic Functional via t-Convergence. Communications on Pure and Applied Mathematics, 43, 999-1036.
[38] Ambrosio, L. and Tortorelli, V.M. (1992) On the Approximation of Free Discontinuity Problems. Bollettino della Unione Matematica Italiana, 6, 105-123.

comments powered by Disqus

Copyright © 2020 by authors and Scientific Research Publishing Inc.

Creative Commons License

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.