Multiphase image segmentation via Modica-Mortola phase transition

SIAM Journal on Applied Mathematics

Volumn 67, Issue 5, 2007, Pages 1213-1232

  Jung, Yoon Mo ^a   Kang, Sung Ha ^b   Shen, Jianhong ^a  

^a UNIVERSITY OF MINNESOTA   (United States)

^b UNIVERSITY OF KENTUCKY   (United States)

Author keywords

convergence; Convex splitting; Convex concave procedure; Modica Mortola model; Multiphase; Partial differential equation; Phase transition; Segmentation; Variational

Indexed keywords

CONVEX CONCAVE PROCEDURE; CONVEX SPLITTING; MODICA MORTOLA MODEL; NEURAL COMPUTATION;

COMPUTATION THEORY; CONVERGENCE OF NUMERICAL METHODS; MATHEMATICAL MODELS; PARTIAL DIFFERENTIAL EQUATIONS; PHASE TRANSITIONS;

IMAGE SEGMENTATION;

EID: 35348937115     PISSN: 00361399     EISSN: None     Source Type: Journal    
DOI: 10.1137/060662708     Document Type: Article

References (55)

1. G. ALBERTI, Variational models for phase transitions, an approach via Γ-convergence, in Calculus of Variations and Partial Differential Equations (Pisa, 1996), Springer-Verlag, Berlin, 2000, pp. 95-114.
2. L. AMBROSIO AND V. M. TORTORELLI, Approximation of functionals depending on jumps by elliptic functional via Γ- convergence, Comm. Pure Appl. Math., 43 (1990), pp. 999-1036.
3. L. AMBROSIO AND V. M. TORTORELLI, On the approximation of free discontinuity problems, Boll. Un. Mat. Ital. B(7), 6 (1992), pp. 105-123.
4. A. BERTOZZI, S. ESEDOGLU, AND A. GILETTE, Inpainting of binary images using the CahnHilliard equation, IEEE Trans. Image Process., 16 (2007), pp. 285-291.
5. B. BOURDIN AND A. CHAMBOLLE, Implementation of an adaptive finite-element approximation of the Mumford-Shah functional, Numer. Math., 85 (2000), pp. 609-646.
6. A. BRAIDES, Γ-Convergence for Beginners, Oxford Lecture Ser. Math. Appl. 22, Oxford Uni-versity Press, Oxford, UK, 2002.
7. J. W. CAHN AND J. E. HILLIARD, Free energy of a non-uniform system I. Interfacial free energy, J. Chem. Phys., 28 (1958), pp. 258-267.
8. A. CHAMBOLLE, Image segmentation by variational methods: Mumford and Shah functional and the discrete approximations, SIAM J. Appl. Math., 55 (1995), pp. 827-863.
9. A. CHAMBOLLE, Finite-differences discretizations of the Mumford-Shah functional, M2AN Math. Model. Numer. Anal., 33 (1999), pp. 261-288.
10. T. F. CHAN, S.-H. KANG, AND J. SHEN, Total variation denoising and enhancement of color images based on the CB and HSV color models, J. Visual Comm. Image Rep., 12 (2001), pp. 422-435.
11. T. F. CHAN, J. SHEN, AND L. VESE, Variational PDE models in image processing, Amer. Math. Soc. Notice, 50 (2003), pp. 14-26.
12. T. CHAN AND J. SHEN, Variational restoration of nonflat image features: Models and algorithms, SIAM J. Appl. Math., 61 (2000), pp. 1338-1361.
13. T. F. CHAN AND L. VESE, A level set algorithm for minimizing the Mumford-Shah functional in image processing, in Proceedings of the 1st IEEE Workshop on Variational and Level Set Methods in Computer Vision, 2001, pp. 161-168.
14. T. F. CHAN AND J. SHEN, Image Processing and Analysis: Variational, PDE, Wavelet, and Stochastic Methods, SIAM, Philadelphia, 2005.
15. T. F. CHAN AND L. A. VESE, Active contours without edges, IEEE Trans. Image Process., 10 (2001), pp. 266-277.
16. T. F. CHAN AND L. A. VESE, A multiphase level set framework for image segmentation using the Mumford and Shah model, Int. J. Comp. Vision, 50 (2002), pp. 271-293.
17. J. T. CHUNG AND L. A. VESE, Image segmentation using a multilayer level-set approach, UCLA CAM report 03-53, UCLA, Los Angeles, CA, 2003.
18. F. CUCKER AND S. SMALE, On the mathematical foundations of learning, Bull. Amer. Math. Soc. (N.S.), 39 (2001), pp. 1-49.
19. I. DAUBECHIES, Ten Lectures on Wavelets, SIAM, Philadelphia, 1992.
20. S. ESEDOGLU AND R. MARCH, Segmentation with depth but without detecting junctions, J. Math. Imaging Vision, 18 (2003), pp. 7-15.
21. S. ESEDOGLU AND J. SHEN, Digital inpainting based on the Mumford-Shah-Euler image model, European J. Appl. Math., 13 (2002), pp. 353-370.
22. D. EYRE, An Unconditionally Stable One-Step Scheme for Gradient Systems, http://www.math.utah.edu/~eyre/research/methods/stable.ps (1998).
23. S. GEMAN AND D. GEM AN, Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images, IEEE Trans. Pattern Anal. Machine Intell., 6 (1984), pp. 721-741.
24. E. DE GIORGI AND T. FRANZONI, SU un tipo di convergenza variazionale, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8), 58 (1975), pp. 842-850.
25. R. V. KOHN AND V. V. SLASTIKOV, Geometrically constrained walls, CaIc. Var. Partial Differential Equations, 28 (2007), pp. 33-57.
26. J. LIE, M. LYSAKER, AND X.-C. TAI, A variant of the level set method and applications to image segmentation, Math. Comp., 75 (2006), pp. 1155-1174.
27. F.-F. LI AND P. PERONA, A Bayesian hierarchical model for learning natural scene categories, IEEE CVPR, 2 (2005), pp. 524-531.
28. F.-F. LI, R. VANRULLEN, C. KOOH, AND P. PERONA, Rapid natural scene categorization in the near absence of attention, Proc. Natl. Acad. Sci. USA, 99 (2002), pp. 9596-9601.
29. R. MARCH AND M. DOZIO, A variational method for the recovery of smooth boundaries, Image Vision Comput., 15 (1997), pp. 705-712.
30. R. MARCH, Visual reconstruction with discontinuities using variational methods, Image Vision Comput., 10 (1992), pp. 30-38.
31. G. DAL MASO, An introduction to Γ-Conuergence, Progr. Nonlinear Differential Equations Appl. 8, Birkhäuser Boston, Boston, MA, 1993.
32. Y. MEYER, Oscillating Patterns in Image Processing and Nonlinear Evolution Equations, Univ. Lecture Ser. 22, AMS, Providence, RI, 2001.
33. -convergenza, Boll. Un. Mat. Ital. B (5), 14 (1977), pp. 285-299.
34. L. MODICA, The gradient theory of phase transitions and the minimal interface criterion, Arch. Rational Mech. Anal., 98 (1987), pp. 123-142.
35. J.-M. MOREL AND S. SOLIMINI, Variational Methods in Image Segmentation, Progr. Nonlinear Differential Equations Appl. 14, Birkhäuser Boston, Boston, MA, 1995.
36. D. MUMFORD AND J. SHAH, Optimal approximations by pieceiuise smooth/unctions and asso-ciated variational problems, Comm. Pure Appl. Math., 42 (1989), pp. 577-685.
37. M. NITZBERG, D. MUMFORD, AND T. SHIOTA, Filtering, Segmentation, and Depth, Lecture Notes in Comput. Sci. 662, Springer-Verlag, Berlin, 1993.
38. S. OSHER AND R. FEDKIW, Level Set Methods and Dynamic Implicit Surfaces, Appl. Math. Sci. 153, Springer-Verlag, New York, 2003.
39. S. OSHER AND J. A. SETHIAN, Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations, J. Comput. Phys., 79 (1988), pp. 12-49.
40. -1 norm, Multiscale Model. Simul., 1 (2003), pp. 349-370.
41. T. POGGIO AND S. SMALE, The mathematics of learning: Dealing with data, Amer. Math. Soc. Notice, 50 (2003), pp. 537-544.
42. B. SANDBERG, T. CHAN, AND L. VESE, A Level-Set and Gabor-Based Active Contour Algorithm for Segmenting Textured Images, UCLA CAM report 02-39, UCLA, Los Angeles, CA, 2002.
43. J. A. SETHIAN, Level Set Methods and Fast Marching Methods, Cambridge Monogr. Appl. Comput. Math. 3, 2nd ed., Cambridge University Press, Cambridge, UK, 1999.
44. J. SHEN, Bayesian video dejittering by the BV image model, SIAM J. Appl. Math., 64 (2004), pp. 1691-1708.
45. 1 images and the Mumford-Shah-Sobolev model for segmented image decomposition, AMRX Appl. Math. Res. Express, 4 (2005), pp. 143-167.
46. J. SHEN, A stochastic-variational model for soft mumford-shah segmentation, International Journal of Biomedical Imaging, Vol. 2006 (2006), article ID92329.
47. J. SHI AND J. MALIK, Normalized cuts and image segmentation, IEEE Trans. Pattern Anal. Machine Intell., 22 (2000), pp. 888-905.
48. S. SMALE AND D.-X. ZHOU, Shannon sampling and function reconstruction from point values, Bull. Amer. Math. Soc. (N.S.), 41 (2004), pp. 279-305.
49. X.-C. TAI AND T. F. CHAN, A survey on multiple level set methods with applications for identifying piecewise constant functions, Int. J. Numer. Anal. Model., 1 (2004), pp. 25-48.
50. X.-C TAI AND C.-H.YAO, Image segmentation by piecewise constant Mumford-Shah model without estimating the constants, J. Comput. Math., 24 (2006), pp. 435-443.
51. L. A. VESE, T. F. CHAN, AND B. Y. SANDBERG, Active contours without edges for vector-valued images, J. Visual Comm. Image Rep., 11 (2000), pp. 130-141.
52. Z. W. TU AND S. C. ZHU, Image segmentation by data-driven Markov Chain Monte Carlo, IEEE Trans. Pattern Anal. Machine Intell., 24 (2002), pp. 657-673.
53. C. R. VOGEL, Computational Methods for Inverse Problems, SIAM, Philadelphia, 2002.
54. B. P. VOLLMAYR-LEE AND A. D. RUTENBERG, Fast and accurate coarsening simulation with an unconditionally stable time step, Phys. Rev. E (3), 68 (2003), 066703.
55. A. L. YUILLE AND A. RANGARAJAN, The concave-convex procedure (CCCP), Neural Comput., 15 (2003), pp. 915-936.