Denoising by BV-duality

Journal of Scientific Computing

Volumn 28, Issue 2-3, 2006, Pages 411-444

  Kindermann, Stefan ^a   Osher, Stanley ^a   Xu, Jinjun ^a  

^a UNIVERSITY OF CALIFORNIA   (United States)

Author keywords

Bounded variation; Denoising; G norm; Image processing; Level sets; Rudin Osher Fateni

Indexed keywords

BOUNDED VARIATION; DENOISING; G-NORM; LEVEL SETS; RUDIN-OSHER-FATENI;

COMPUTATION THEORY; LINEAR EQUATIONS; NUMERICAL ANALYSIS; PROBLEM SOLVING;

IMAGE PROCESSING;

EID: 33747882794     PISSN: 08857474     EISSN: None     Source Type: Journal    
DOI: 10.1007/s10915-006-9074-z     Document Type: Article

References (33)

1. Adams, R. A. (1975). Sobolev Spaces, Academic Press, New York San Francisco, London.
2. Ambrosio, L., Fusco, N., and Pallara, D. (2000). Functions of Bounded Variation and Free Discontinuity Problems, Oxford University Press, New York.
3. Aubert, G., and Aujol, J.-F. (2005). Modeling very oscillating signals. application to image processing. Appl. Math. Optim. 51, 163-182.
4. Aujol, F., Aubert, G., Blanc-Féraud, L., and Chambolle, A. (2005). Image decomposition into a bounded variation component and an oscillating component. J. Math. Imag. Vis. 22, 71-88.
5. Chan, T., Esedoḡlu, S., and Nikolova, M. (2004). Algorithms for finding global minimizers of image segmentation and denoising models, preprint, CAM-report, UCLA, 2004.
6. Chan, T. F., and Vese, L. A. (2001). Active contours without edges. IEEE Trans. Image Proc. 10, 266-277.
7. Crandall, M. G., and Lions, P.-L. (1984). Two approximations of solutions of Hamilton-Jacobi equations. Math. Comp. 43, 1-19.
8. Ekeland, I., and Témam, R. (1999). Convex analysis and variational problems, vol. 28, SIAM, Philadelphia, PA.
9. Engl, H. W., Hanke, M., and Neubauer, A. (1996). Regularization of Inverse Problems, Kluwer, Dordrecht.
10. Enquist, B., Tornberg, A.-K., and Tsai, R. (2005). Discretization of dirac delta functions in level set methods. J. Comp. Phys. 207, 28-51.
11. Esedoḡlu, S., and Osher, S. J. (2004). Decomposition of images by the anisotropic Rudin-Osher-Fatemi model. Comm. Pure Appl. Math. 57, 1609-1626.
12. Esedoḡlu, S., and Tsai, R. (2006). Threshold dynamics for the piecewise constant mumford-shah functional. J. Comput. Phys. 211, 367-384.
13. Gilbarg, D., and Trudinger, N. S. (1997). Elliptic Partial Differential Equation of Second-Order. Springer, Berlin, Heidelberg.
14. Giusti, E.(1984). Minimal Surfaces and Functions of Bounded Variation, vol. 80 of Monographs in Mathematics, Birkhäuser Verlag, Basel.
15. Haddard, A., and Meyer, Y. (2004). Variational methods in image processing, preprint, CAM-report, UCLA, 2004.
16. Jiang, G.-S., and Peng, D. (2000). Weighted ENO schemes for Hamilton-Jacobi equations. SIAM J. Sci. Comput. 21, 2126-2143.
17. Jiang, G.-S., and Shu, C.-W. (1996). Efficient implementation of weighted ENO schemes. J. Comput. Phys. 126, 202-228.
18. Kindermann. S., and Osher, S. (2005). Saddle point formulation for a cartoon-texture decomposition, preprint, CAM-report, UCLA, 2005.
19. Le, T. M., and Vese, L. (2005). Image decomposition using total variation and div(BMO). Multisc. Model. and Simul. 4, 390-423.
20. Liu, X.-D., Osher, S., and Chan, T. (1994). Weighted essentially non-oscillatory schemes. J. Comput. Phys. 115, 200-212.
21. Meyer, Y. (2001). Oscillating Patterns in Image Processing and Nonlinear Evolution Equations, vol. 22 of University Lecture Series, American Mathematical Society, Providence, RI.
22. Mumford, D., and Shah, J. (1989). Optimal approximations by piecewise smooth functions and associated variational problems. Comm. Pure Appl. Math. 42, 577-685.
23. Osher, S., Burger, M., Goldfarb, D., Xu, J., and Yin, W. (2005). An iterative regularization method for total variation based image restauration. Multisc. Model. Simul. 4, 460-489.
24. Osher, S., and Fedkiw, R. (2003). Level Set Methods and Dynamic Implicit Surfaces, vol. 153 of Applied Mathematical Sciences, Springer-Verlag, New York.
25. Osher, S., and Scherzer, O. (2004). G-norm properties of bounded variation regularization. Commun. Math. Sci. 2, 237-254.
26. Osher, S., and Sethian, J. A. (1988). Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations. J. Comput. Phys. 79, 12-49.
27. Rudin, L. I., Osher, S., and Fatemi, E. (1992). Nonlinear total variation based noise removal algorithms. Phys. D 60, 259-268.
28. Shu, C.-W., and Osher, S. (1988). Efficient implementation of essentially nonoscillatory shock-capturing schemes. J. Comput. Phys. 11, 439-471.
29. Shu, C.-W., and Osher, S. (1989). Efficient implementation of essentially nonoscillatory shock-capturing schemes. II. J. Comput. Phys. 83, 32-78.
30. ∞ approximation of vector fields in the plane. In Nonlinear Partial Differential Equations in Applied Science (Tokyo, 1982), vol. 81 of North-Holland Math. Stud., North-Holland, Amsterdam, 1983, pp. 273-288.
31. Sussman, M., Smereka, P., and Osher, S. (1994). A level set approach for computing solutions to incompressible two-phase flow. J. Comput. Phys. 114, 146-159.
32. Vese, L. A., and Osher, S. J. (2004). Image denoising and decomposition with total variation minimization and oscillatory functions. J. Math. Imaging Vision. 20, 7-18.
33. Zeidler, E. (1985). Nonlinear Functional Analysis and its Applications. III, Springer-Verlag, New York.