Fast Segmentation for the Piecewise Smooth Mumford-Shah Functional

Authors: Yingjie Zhang

Abstract:

This paper is concerned with an improved algorithm 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 - . In our scheme, furthermore, 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. Therefore, some drawbacks of the original algorithm, such as smaller objects generated by noise and local minimal problem also are eliminated or improved. The resulting algorithm has been implemented in Matlab and Visual Cµ, and demonstrated efficiently by several cases.

Keywords: Active contours, energy minimization, image segmentation, level sets.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1055094

References:

[1] D. Mumford and J. Shah, "Optimal approximation by piecewise smooth functions and associated variational problems," Comm. Pure Appl. Math., vol. 42, pp. 577-685, 1989.
[2] M. Kass, A. Witkin, and D. Terzopoulos, "Snakes: Active contour models," International Journal of Computer Vision, vol. 1, no. 4, 1988, pp. 321-331.
[3] P. Perona and J. Malik, "Scale space and edge detection using anisotropic diffusion," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 12, no. 8, 1990, pp. 629-639.
[4] S. Osher and J. A. Sethian, "Fronts propagating with curvature dependent speed: Algorithms based on hamilton-jacobi formulation," Journal of Computational Physics, vol. 79, 1988, pp. 12-49.
[5] T. Chan and L. Vese, "Active Contours without edges," IEEE Transactions on Image Processing, vol. 10, no. 2, 2001, pp. 266-277.
[6] T. Chan and L. Vese, "A level set algorithm for minimizing the Mumford-Shah functional in image processing," IEEE Computer Society, Proceeding of the First IEEE Workshop on Variational and Level Set Methods in Computer Vision, 2001, pp. 161-168.
[7] L. Vese and T. Chan, "A multiphase level set framework for image segmentation using the Mumford and Shah model," International Journal of Computer Vision, vol. 50, no. 3, 2002, pp. 271-293.
[8] Choi, G. Kim, P. Park, G. Wang, and S. Kim, "Efficient PDE-based segmentation algorithms and their application to CT images," Journal. Korean Institute of Plant Engineering, 2003, pp. 1-17.
[9] S. Teboul, L. B. Féraud, G. Aubert, and M. Barlaud, "Variational approach for edge-preserving regularization using coupled PDE-s," IEEE Transaction Image Processing, vol.7, no.3, 1998, pp. 387-397.
[10] S. Geman and D. E. Mcclure, "Bayesian image analysis: An application to single photon emission tomography," Proc. Stat. Comput. Sect. Washington, DC: Amer. Stat. Assoc. 1985, pp. 12-18.
[11] S. Ji and H. Park, "Image segmentation of color image based on region coherency," in Proceedings 1998 International Conference on Image Processing, vol. 1, Chicago, IL, USA, 1998, pp. 80-83.
[12] C. Xu and J. Prince, "Snakes, shapes, and gradient vector flow," IEEE Transactions on Image Processing, vol. 7, 1998, pp. 359-369.
[13] J. Haddon and J. Boyce, "Image segmentation by unifying region and boundary information," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 12, no. 10, 1990, pp. 929-948.
[14] K. Haris, S. Efstratiadis, N. Maglaveras, and A.Katsaggelos, "Hybeid image segmentation using watersheds and fast region merging," IEEE Transactions on Image Processing, vol. 7, no. 12, 1998, pp. 1684-1699.
[15] S. Kichenassamy, A. Kumar, P. Oliver, A. Tannenbaum, and A. Yezzi, "Conformal curvature flows: from phase transitions to active vision," Archive for Rational Mech. and Anal., vol. 134, 1996, pp. 275-301.
[16] D. Marr and E. Hildreth, "Theory of edge detection," Proc. R. Soc. Lond., vol. B207, 1980, pp. 187-217.
[17] P. Hill, C. Canagarajah, and D. Bull, "Texture gradient based watershed segmentation," IEEE International Conference on Acoustics, Speech, and Signal Processing, vol. 4, Orlando, FL, USA, 2002, pp. 3381-3384.
[18] S. Osher and R. Fedkiw, Level set methods and dynamic implicit surfaces, New York, Springer-Verlag, 2003.