A study in the BV space of a denoising-deblurring variational problem

Applied Mathematics and Optimization

Volumn 44, Issue 2, 2001, Pages 131-161

  Vese, L ^a  

^a UNIVERSITY OF CALIFORNIA   (United States)

Author keywords

convergence; Convex functions of measures; Duality; Elliptic parabolic PDEs; Finite differences scheme; Functions of bounded variation; Maximal monotone operators; Relaxation; Signal and image processing; Variational methods

Indexed keywords

BOUNDARY CONDITIONS; CONVERGENCE OF NUMERICAL METHODS; FINITE DIFFERENCE METHOD; FUNCTIONS; IMAGE RECONSTRUCTION; IMAGE RETRIEVAL; MATHEMATICAL OPERATORS; NONLINEAR EQUATIONS; PARTIAL DIFFERENTIAL EQUATIONS; PROBLEM SOLVING; RELAXATION PROCESSES;

CONVEX FUNCTIONS OF MEASURES; DEBLURRING; DENOISING; DUALITY; FUNCTIONS OF BOUNDED VARIATION; MAXIMAL MONOTONE OPERATORS;

VARIATIONAL TECHNIQUES;

EID: 0011021531     PISSN: 00954616     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00245-001-0017-7     Document Type: Article

References (34)

1. Acart R, Vogel CR (1994) Analysis of bounded variation penalty methods for ill-posed problems, Inverse Problem 10:1217-1229
2. Alvarez L, Mazorra L (1994) Signal and image restoration using shock filters and anisotropic diffusion. SIAM J Numer Anal 31(2):590-605
3. Aubert G, Vese, L (1997) A variational method in image recovery. SIAM J Numer Anal 34(5):1948-1979
4. Bouchitté G, Braides A, Buttazzo G (1995) Relaxation results for some free discontinuity problems. J Reine Angew Math 458: 1-18
5. Bouchitté G, Buttazzo G (1990) New lower semicontinuity results for nonconvex functionals defined on measures. Nonlinear Anal TMA 15(7):679-692
6. Bouchitté G, Buttazzo G (1992) Integral-representation of nonconvex functionals defined on measures. Ann Inst H Poincaré 9(1):101-117
7. Bouchitté G, Buttazzo G (1993) Relaxation for a class of nonconvex functionals defined on measures. Ann Inst H Poincaré 10(3):345-361
8. Brezis H (1973) Operateurs maximaux monotones. Mathematics Studies. North-Holland, Amsterdam
9. Brezis H (1992) Analyse fonctionnelle. Masson, Paris
10. Bruck RE (1975) Asymptotic convergence of nonlinear contraction semi-groups in Hilbert spaces. J Funct Anal 18:15-26
11. Chambolle A, Lions PL (1997) Image recovery via total variation minimization and related problems. Numer Math 76:167-188
12. Charbonnier P (1994) Reconstruction d'image: regularisation avec prise en compte des discontinuités. PhD Thesis, University of Nice - Sophia Antipolis
13. Charbonnier P, Blanc-Féraud L, Aubert G, Barlaud M (1997) Deterministic edge-preserving regularization in computed imaging. IEEE Trans Image Process 6(2):298-311
14. Dal Maso G (1993) An Introduction to Γ-Convergence. Birkhäuser, Boston
15. Dal Maso G, Morel JM Solimni S (1992) A variational method in image segmentation: existence and approximation results. Acta Mat 168:89-151.
16. De Giorgi E, Ambrosio L (1988) Un nuovo tipo di funzionale del calcolo delle variazioni. Atti Accad Naz Lincei Rend Cl Sci Fis Mat Natur 82:199-210
17. Demengel F, Temam R (1984) Convex functions of a measure and applications. Indiana Univ Math J 33:673-709
18. Ekeland I, Temam R (1974) Analyse convexe et problèmes variationnels. Dunod-Gauthier-Villars, Paris
19. Evans LC, Gariepy RF (1992) Measure Theory and Fine Properties of Functions. CRC Press, London
20. Giusti E (1984) Minimal Surfaces and Functions of Bounded Variation. Birkhäuser, Boston
21. Goffman C, Serrin J (1964) Sublinear functions of measures and variational integrals. Duke Math J 31:159-178
22. Kohn R, Temam R (1983) Dual spaces of stresses and strains, with applications to Hencky plasticity. Appl Math Optim 10(1):1-35
23. Lichnevsky A, Temam R (1978) Pseudo-solutions of time-dependent minimal surface problems. J Differential Equations 30:340-364
24. Lions JL, Magenes E (1968) Problèmes aux limites non homogènes. Dunod, Paris
25. Mumford D, Shah J (1989) Optimal Approximations by Piecewise Smooth Functions and Associated Variational Problems. Comm Pure Appl Math XLII:577-685
26. Osher S, Rudin L (1990) Feature-oriented image enhancement using shock filters. SIAM J Numer Anal 27(4):919-940
27. Rockafellar PT (1970) Convex Analysis. Princeton University Press, Princeton, NJ
28. Rudin L, Osher S (1994) Total variation based image restoration with free local constraints. Proc IEEE ICIP, Austin, TX, pp I:31-35
29. Rudin L, Osher S, Fatemi E (1992) Nonlinear total variation based noise removal algorithms. Phys D 60:259-268
30. Temam R (1983) Dual variational principles in mechanics and physics. In: Semi-Infinite Programming and Applications (AV Fiacco and KO Kortanec, eds). Lecture Notes in Economics and Mathematical Systems 215. Springer-Verlag, Berlin
31. Temam R, Strang G (1980/1981) Functions of Bounded Deformation. Arch Rational Mech Anal 75(1):7-21
32. Vese L (1996) Variational problems and PDEs for image analysis and curve evolution. PhD Thesis, University of Nice - Sophia Antipolis
33. Zeidler E (1990) Nonlinear Functional Analysis and its Applications, IIB: Nonlinear Monotone Operators. Springer-Verlag, Berlin
34. Ziemer WP (1989) Weakly Differentiable Functions. Springer-Verlag, Berlin