Image decomposition using total variation and div(BMO)

Multiscale Modeling and Simulation

Volumn 4, Issue 2, 2005, Pages 390-423

  Le, Triet M ^a   Vese, Luminita A ^a  

^a UNIVERSITY OF CALIFORNIA   (United States)

Author keywords

Bounded mean oscillation; Bounded variation; Cartoon and texture; Denoising; Functional minimization; Image decomposition; Nonlinear partial differential equation; Oscillatory function

Indexed keywords

COMPUTER SIMULATION; MATHEMATICAL MODELS; OPTIMIZATION; PARTIAL DIFFERENTIAL EQUATIONS; PROBABILITY DISTRIBUTIONS;

BOUNDED MEAN OSCILLATION; DENOISING; FUNCTIONAL MINIMIZATION; IMAGE DECOMPOSITION;

IMAGE ANALYSIS;

EID: 33645696337     PISSN: 15403459     EISSN: 15403467     Source Type: Journal    
DOI: 10.1137/040610052     Document Type: Article

References (47)

1. R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975.
2. L. Alvarez, Y. Gousseau, and J.-M. Morel, Scales in natural images and a consequence on their bounded variation norm, in Scale-Space Theories in Computer Vision, Lecture Notes in Comput. Sci. 1682, Springer, New York, 1999, pp. 247-258.
3. L. Ambrosio, Variational problems in SBV and image segmentation, Acta Appl. Math., 17 (1989), pp. 1-40.
4. L. Ambrosio, N. Fusco, and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford University Press, New York, 2000.
5. F. Andreu-Vaillo, V. Caselles, and J. M. Mazon, Parabolic Quasilinear Equations Minimizing Linear Growth Functionals, Progr. Math. 223, Birkhäuser Verlag, Basel, 2004.
6. G. Aubert and J.-F. Aujol, Modeling very oscillating signals. Application to image processing, Appl. Math. Optim., 51 (2005), pp. 163-182.
7. J.-F. Aujol, G. Aubert, L. Blanc-Féraud, and A. Chambolle, Image decomposition application to SAR images, in Scale-Space Methods in Computer Vision, Lecture Notes in Comput. Sci. 2695, Springer, New York, 2003, pp. 297-312.
8. J.-F. Aujol and A. Chambolle, Dual norms and image decomposition models, Internat. J. Comput. Vision, 63 (2005), pp. 85-104.
9. J. Bourgain and H. Brezis, On the equation divY = f and application to control of phases, J. Amer. Math. Soc., 16 (2002), pp. 393-426.
10. A. Chambolle and P.-L. Lions, Image recovery via total variation minimization and related problems, Numer. Math., 76 (1997), pp. 167-188.
11. 1 Function Approximation, UCLA CAM Report 04-07, Los Angeles, CA, 2004.
12. T. F. Chan, S. Esedoglu, and M. Nikolova, Algorithms for Finding Global Minimizers of Denoising and Segmentation Models, UCLA CAM Report 04-54, Los Angeles, CA, 2004.
13. T. F. Chan and L. Vese, An active contour model without edges, in Scale-Space Theories in Computer Vision, Lecture Notes in Comput. Sci. 1682, Springer, New York, 1999, pp. 141-151.
14. T. F. Chan and L. Vese, Active contours without edges, IEEE Trans. Image Process., 10 (2001), pp. 266-277.
15. E. Cheon, A. Paranjpye, L. Vese, and S. Osher, Noise Removal Project by Total Variation Minimization, Math 199 REU project, UCLA Department of Mathematics, Los Angeles, CA, 2002.
16. I. Daubechies and G. Teschke, Wavelet-based image decomposition by variational functionals, in Wavelet Applications in Industrial Processing, Proc. SPIE 5266, F. Truchetet, ed., SPIE, Bellingham, WA, 2004, pp. 94-105.
17. T. de Pauw, On SBV dual, Indiana Univ. Math. J., 47 (1998), pp. 99-121.
18. L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions, CRC Press, Boca Raton, FL, 1991.
19. S. Esedoglu and S. Osher, Decomposition of images by the anisotropic Rudin-Osher-Fatemi model, Comm. Pure Appl. Math., 57 (2004), pp. 1609-1626.
20. J. B. Garnett, Bounded Analytic Functions, Pure Appl. Math. 96, Academic Press, New York, London, 1981.
21. J. B. Garnett and P. W. Jones, BMO from dyadic BMO, Pacific J. Math., 99 (1982), pp. 351-371.
22. D. Goldfarb and W. Yin, Second-Order Cone Programming Methods for Total Variation-Based Image Restoration, CORC Report TR-2004-05, Columbia University, New York, NY, 2004.
23. Y. Gousseau and J.-M. Morel, Are natural images of bounded variation?, SIAM J. Math. Anal., 33 (2001), pp. 634-648.
24. J.-L. Journe, Calderon-Zygmund Operators, Pseudo-differential Operators and the Cauchy Integral of Calderon, Lecture Notes in Math. 994, Springer-Verlag, Berlin, New York, 1983.
25. H. Koch and D. Tataru, Well-posedness for the Navier-Stokes equations, Adv. in Math., 157 (2001), pp. 22-35.
26. T. Mei, BMO is the intersection of two translates of dyadic BMO, C. R. Acad. Sci. Paris, 336 (2003), pp. 1003-1006.
27. Y. Meyer, Oscillating Patterns in Image Processing and Nonlinear Evolution Equations, Univ. Lecture Ser. 22, AMS, Providence, RI, 2001.
28. J.-M. Morel and S. Solimini, Variational Methods in Image Segmentation: With Seven Image Processing Experiments, Progr. Nonlinear Differential Equations Appl., Birkhäuser Boston, Boston, MA, 1994.
29. D. Mumford and B. Gidas, Stochastic models for generic images, Quart. Appl. Math., 59 (2001), pp. 85-111.
30. D. Mumford and J. Shah, Optimal approximations by piecewise smooth functions and associated variational problems, Comm. Pure Appl. Math., 42 (1989), pp. 577-685.
31. A. Obereder, S. Osher, and O. Scherzer, On the Use of Dual Norms in Bounded Variation Type Regularization, UCLA CAM Report 04-35, Los Angeles, CA, 2004.
32. S. Osher and O. Scherzer, G-norm properties of bounded variation regularization, Commun. Math. Sci., 2 (2004), pp. 237-254.
33. -1 norm, Multiscale Model. Simul., 1 (2003), pp. 349-370.
34. L. Rudin, S. Osher, and E. Fatemi, Nonlinear total variation based noise removal algorithms, Phys. D, 60 (1992), pp. 259-268.
35. J.-L. Starck, M. Elad, and D. L. Donoho, Image decomposition: Separation of texture from piecewise smooth content, in Proceedings of the SPIE Conference on Signal and Image Processing: Wavelet Applications in Signal and Image Processing X, San Diego, 2003.
36. E. M. Stein, Harmonic Analysis: Real Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton University Press, Princeton, NJ, 1993.
37. P Sobolev spaces, Colloq. Math., 60/61 (1990), pp. 129-139.
38. D. Strong, Adaptive Total Variation Minimizing Image Restoration, Ph.D. dissertation, UCLA Mathematics Department CAM Report 97-38, Los Angeles, CA, 1997.
39. D. Strong and T. Chan, Edge-preserving and scale-dependent properties of total variation regularization, Inverse Problems, 19 (2003), pp. S165-S187.
40. 2) decompositions, Multiscale Model. Simul., 2 (2004), pp. 554-579.
41. A. Torchinsky, Real Variable Methods in Harmonic Analysis, Pure Appl. Math. 123, Academic Press, Orlando, FL, 1986.
42. H. Triebel, The Structure of Functions, Monogr. Math. 97, Birkhäuser Verlag, Basel, 2001.
43. L. Vese, A study in the BV space of a denoising-deblurring variational problem, Appl. Math. Optim., 44 (2001), pp. 131-161.
44. L. Vese and S. Osher, Modeling textures with total variation minimization and oscillating patterns in image processing, J. Sci. Comput., 19 (2003), pp. 553-572.
45. L. A. Vese and S. J. Osher, Image denoising and decomposition with total variation minimization and oscillatory functions, J. Math. Imaging Vision, 20 (2004), pp. 7-18.
46. H.-K. Zhao T. Chan, B. Merriman, and S. Osher, A variational level set approach to multiphase motion, J. Comput. Phys., 127 (1996), pp. 179-195.
47. S. C. Zhu and D. Mumford, Prior learning and Gibbs reaction-diffusion, IEEE Transactions on Pattern Analysis and Machine Intelligence, 19 (1997), pp. 1236-1250.