Image restoration with discrete constrained total variation part I: Fast and exact optimization

Journal of Mathematical Imaging and Vision

Volumn 26, Issue 3, 2006, Pages 261-276

  Darbon, Jérôme ^a,b   Sigelle, Marc ^c  

^a EPITA RESEARCH AND DEVELOPMENT LABORATORY LRDE   (France)

^b TELECOM PARISTECH   (France)

^c INSTITUT TELECOM   (France)

Author keywords

Graph cuts; Level sets; Markov random fields; Restoration; Total variation

Indexed keywords

CONSTRAINT THEORY; GRAPH THEORY; MARKOV PROCESSES; OPTIMIZATION; PROBLEM SOLVING; SET THEORY;

BINARY PROBLEMS; CONVEX DATA FIDELITY FUNCTIONALS; GRAPH CUTS; LEVEL SETS; MARKOV RANDOM FIELD;

IMAGE RECONSTRUCTION;

EID: 33845482689     PISSN: 09249907     EISSN: None     Source Type: Journal    
DOI: 10.1007/s10851-006-8803-0     Document Type: Article

References (47)

1. 2 norms with Application to Bayesian Estimation," IEEE Transactions on Signal Processing, Vol. 42, No. 3, pp. 618-627, 1994.
2. S. Alliney, "A Property of the Minimum Vectors of a Regularizing Functional defined by Means of the Absolute Norm," IEEE Transactions on Signal Processing, Vol. 45, No. 4, pp. 913-917, 1997.
3. J.-R Aujol, G. Aubert, L. Blanc-Feraud, and A. Chambolle, "Image decomposition into a bounded variation component and oscillating component," Journal of Mathematical Imaging and Vision, Vol. 22, No. 1, pp. 71-88, 2005.
4. J.-R Aujol, G. Gilboa, T. Chan, and S. Osher, "Structure-Texture Image Decomposition-Modeling, Algorithms, and Parameter Selection," Technical Report 10, UCLA, 2005.
5. Y. Boykov and V. Kolmogorov, "Computing Geodesic and Minimal Surfaces via Graph Cuts," In: International Conference on Computer Vision, Vol. 1, pp. 26-33, 2003.
6. Y. Boykov and V. Kolmogorov, An Experimental Comparison of Min-Cut/Max-Flow Algorithms for Energy Minimization in Vision, IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol. 26, No. 9, pp. 1124-1137, 2004.
7. Y. Boykov, O. Veksler, and R. Zabih, "Fast Approximate Energy Minimization via Graph Cuts," IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol. 23, No. 11, pp. 1222-1239, 2001.
8. O. Catoni and I. Gaudron, "Détection de contours par seuillage adaptatif et restauration stochastique d'images binaires," In: André Gagalowicz (ed.), Second Annual Confeience on Computer Graphics in Paris, Pixim 89 ACM SIGGRAPH FRANCE Hermes, pp. 341-355, 1989.
9. A. Chambolle, "An Algorithm for Total Variation Minimization and Applications," Journal of Mathematical Imaging and Vision, Vol. 20, pp. 89-97, 2004.
10. A. Chambolle, "Total Variation minimization and a class of binary MRF models," In Springer-Verlag, (ed.), 5th International Workshop on Energy Minimization Methods in Computer Vision and Pattern Recognition(EMMCVPR), Vol. LNCS 3757, pp. 136-152, 2005.
11. T. Chan, S. Esedoglu, and M. Nikolova, "Algorithms for Finding Global Minimizers of Image Segmentation and Denoising Models," Technical Report 54, UCLA, 2004.
12. 1 Function Approximation," SIAM Journal of Applied Mathematics, Vol. 65, No. 5, pp. 1817-1837, 2005.
13. T. Chan, A. Marquina, and P. Mulet, "High-Order Total Variation-Based Image Restoration," SIAM J. on Scientific Computing, Vol. 22, No. 2, pp. 503-506, 2000.
14. T. Cormen, C. Leiserson, R. Rivest, and C. Stein, Introduction to Algorithms. The MIT Press, 2001.
15. J. Darbon and S. Peyronnet, "A Vectorial Self-Dual Morphological Filter based on Total Variation Minimization," In: Springer-Verlag, (ed.), International Symposium on Visual Computing (ISVC), Vol. LNCS 3804, pp. 388-395, 2005.
16. J. Darbon and M. Sigelle, "Exact Optimization of Discrete Constrained Total Variation Minimization Problems," In: Springer-Verlag, (ed.), Tenth International Workshop on Combinatorial Image Analysis (IWCIA 2004), Vol. LNCS 3322, pp. 540-549, 2004.
17. J. Darbon and M. Sigelle, "A Fast and Exact Algorithm for Total Variation Minimization," In: Springer-Verlag, (ed.), 2nd Iberian Conference on Pattern Recognition and Image Analysis (IbPria), Vol. LNCS 3522, pp. 351-359, 2005.
18. 1 Data Fidelity as a Contrast Invariant Filter," In Proceedings of the 4th IEEE International Symposium on Image and Signal Processing and Analysis (ISPA 2005), Zagreb, Croatia, 2005.
19. P. Djurić, Y. Huang, and T. Ghirmai, "Perfect Sampling : A Review and Applications to Signal Processing," IEEE Transactions on Signal Processing, Vol. 50, No. 2, pp. 345-256, 2002.
20. R. L. Dobrushin, "The Problem of Uniqueness of a Gibbsian Random Field and the Problem of Phase Transition," Functional Analysis and Applications, Vol. 2, pp. 302-312, 1968.
21. D. Dobson and C. Vogel, "Recovery of Blocky Images from Noisy and Blurred Data," SIAM Journal on Applied Mathematics, Vol. 56, No. 4, pp. 1181-1199, 1996.
22. S. Durand, E Malgouyres, and B. Rougé, "Image Deblurring, Spectrum Interpolation and Application to Satellite Imaging," SIAM Journal on Applied Mathematics, Vol. 56, No. 4, pp. 1181-1199, 1996.
23. L. Evans and R. Gariepy, Measure Theory and. Fine Properties of Functions. CRC Press, 1992.
24. G. Gallo, M. Grigoriadis, and R. Tarjan, "A fast parametric maximum flow algorithm and applications," Siam Journal on Computing, Vol. 18, No. 1, pp. 30-55, 1989.
25. S. Geman and D. Geman, "Stochastic Relaxation, Gibbs Distributions, and the Bayesian Restoration of Images," IEEE Pattern Analysis and Machine Intelligence, Vol. 6, No. 6, pp. 721-741, 1984.
26. H-O. Georgii, Gibbs Measures and Phase Transitions, de Gruyter - Studies in Mathematics, Vol. 9. 1988.
27. D. Greig, B. Porteous, and A. Seheult, "Exact maximum a posteriori estimation for binary images," Journal of the Royal Statistics Society, Vol. 51, No. 2, pp. 271-279, 1989.
28. F. Guichard and J. Morel, "Mathematical Morphology, Almost Everywhere", In: Prceedings of ISMM, pp. 293-303. (2002).
29. D. Hochbaum, "An efficient algorithm for image segmentation, Markov Random Fields and related problems," Journal of the ACM, Vol. 48, No. 2, pp. 686-701, 2001.
30. H. Ishikawa, "Exact optimization for Markov random fields with convex priors," IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol. 25, No. 10, pp. 1333-1336, 2003.
31. V. Kolmogorov and R. Zabih, "What Energy can be Minimized via Graph Cuts?" IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol. 26, No. 2, pp. 147-159, 2004.
32. Y. Meyer. Osculating patterns in image processing and nonlinear evolution equations, Vol. 22 of University Lecture Series, American Mathematical Society, Providence, RI, 2001, The fifteenth Dean Jacqueline B. Lewis memorial lectures.
33. H. Nguyen, M. Worring, and R. van den Boomgaard, "Watersnakes: Energy-Driven Watershed Segmentation," IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol. 23, No. 3, pp. 330-342, 2003.
34. M. Nikolova, "Minimizers of Cost-Functions Involving Nonsmooth Data-Fidelity Terms," SIAM J. Num. Anal., Vol. 40, No. 3, pp. 965-994, 2002.
35. M. Nikolova, "A Variational Approach to Remove Outliers and Impulse Noise," Journal of Mathematical Imaging and Vision, Vol. 20, pp. 99-120, 2004.
36. -1 Norm," J. Mult. Model. and Simul., Vol. 1, No. 3, 2003.
37. E. Pechersky, A. Maruani, and M. Sigelle, "On Gibbs Fields in Image Processing," Markov Processes and Related Fields, Vol. 1, No. 3, pp. 419-442, 1995.
38. I. Pollak, A. Willsky, and Y. Huang, "Nonlinear Evolution Equations as Fast and Exact Solvers of Estimation Problems," IEEE Transactions on Signal Processing, Vol. 53, No. 2, pp. 484-498, 2005.
39. J.G. Propp and D.B. Wilson, "Exact sampling with coupled Markov chains and statistical mechanics," Random Structures and Algorithms, Vol. 9, No. 1, pp. 223-252, 1996.
40. L. Rudin, S. Osher, and E. Fatemi, "Nonlinear Total Variation Based Noise Removal Algorithms," Physica D., Vol. 60, pp. 259-268, 1992.
41. K. Sauer and C. Bouman, "Bayesian Estimation Of Transmission Tomograms Using Segmentation Based Optimization," IEEE Transactions on Nuclear Science, Vol. 39, No. 4, pp. 1144-1152, 1992.
42. M. Sigelle, Champs de Markov en Traitement d'Images et Modèles de la Physique Statistique : Application à la Relaxation d'Images de Classification http://www.tsi.enst.fr/sigelle/tsi-these.html. PhD thesis, ENST, 1993.
43. L. Vese and S. Osher, "Image Denoising and Decomposition with Total Variation and Oscillatory Functions," Journal of Mathematical Imaging and Vision, Vol. 20, No. 1-2, pp. 7-18, 2004.
44. C. Vogel and M. Oman, "Iterative Method for Total Variation Denoising," STAM J. Sci. Comput., Vol. 17, pp. 227-238, 1996.
45. G. Winkler, Image Analysis, Random Fields and Dynamic Monte Carlo Methods," Applications of mathematics. Springer-Verlag, 2003.
46. W. Yin, D. Goldfarb, and S. Osher, "Total Variation Based Image Cartoon-Texture Decomposition," In 3nd IEEE Workshop on Variational, Geometric and Level Set Methods in Computer Vision, Beijing, China, 2005.
47. B. Zalesky, "Network Flow Optimization for Restoration of Images," Journal of Applied Mathematics, Vol. 2, No. 4, pp. 199-218, 2002.