Variable exponent, linear growth functionals in image restoration

SIAM Journal on Applied Mathematics

Volumn 66, Issue 4, 2006, Pages 1383-1406

  Chen, Yunmei ^a   Levine, Stacey ^b   Rao, Murali ^a  

^a UNIVERSITY OF FLORIDA   (United States)

^b DUQUESNE UNIVERSITY   (United States)

Author keywords

BV space; Dirichlet boundary condition; Image restoration; Linear growth functional; Variable exponent

Indexed keywords

BV-SPACE; DIRICHLET BOUNDARY CONDITION; LINEAR GROWTH FUNCTIONAL; VARIABLE EXPONENT;

BOUNDARY CONDITIONS; DIFFUSION; MATHEMATICAL MODELS; MATHEMATICAL TECHNIQUES;

IMAGE RECONSTRUCTION;

EID: 33747163537     PISSN: 00361399     EISSN: None     Source Type: Journal    
DOI: 10.1897/05-356R.1     Document Type: Article

References (37)

1. R. ACAR AND C. R. VOGEL, Analysis of bounded variation penalty methods for ill-posed problems, Inverse Problems, 10 (1994), pp. 1217-1229.
2. F. ANDREU, C. BALLESTER, V. CASELLES, AND J. M. MAZÓN, Minimizing total variation flow, C. R. Acad. Sci. Paris Sér. I Math., 331 (2000), pp. 867-872.
3. F. ANDREU, C. BALLESTER, V. CASELLES, AND J. M. MAZÓN, The Dirichlet problem for the total variation flow, J. Funct. Anal., 180 (2001), pp. 347-403.
4. F. ANDREU, J. M. MAZÓN, J. S. MOLL, AND V. CASELLES, The minimizing total variation flow with measure initial conditions, Commun. Contemp. Math., 6 (2004), pp. 431-494.
5. F. ANDREU-VAILLO, V. CASELLES, AND J. M. MAZÓN, Parabolic Quasilinear Equations Minimizing Linear Growth Functionals, Progr. Math. 223, Birkhäuser Verlag, Basel, 2004.
6. G. AUBERT AND L. VESE, A variational method in image recovery, SIAM J. Numer. Anal., 34 (1997), pp. 1948-1979.
7. N, J. Differential Equations, 184 (2002), pp. 475-525.
8. M. BERTALMÍO, G. SAPIRO, V. CASELLES, AND C. BALLESTER, Image inpainting, in Proceedings of the 27th Annual Conference on Computer Graphics and Interactive Techniques (SIGGRAPH), ACM Press/Addison Wesley, New York, 2000, pp. 417-424.
9. P. BLOMGREN, T. F. CHAN, P. MULET, AND C. WONG, Total variation image restoration: Numerical methods and extensions, in Proceedings of the IEEE International Conference on Image Processing, Vol. III, IEEE, Los Alamitos, CA, 1997, pp. 384-387.
10. G. BOUCHITTÉ AND G. DAL MASO, Integral representation and relaxation of convex local functionals on BV(Ω), Ann, Scuola Norm. Sup. Pisa Cl. Sci. (4), 20 (1993), pp. 483-533.
11. G. BOUCHITTÉ AND M. VALADIER, Integral representation of 'convex functionals on a space of measures, J. Funct. Anal., 80 (1988), pp. 398-420.
12. H. BRÉZIS, Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert, North-Holland Mathematics Studies 5, Notas de Matemática (50), 'North-Holland, Amsterdam, 1973.
13. V. CASELLES, J.-M. MOREL, AND C. SBERT, An axiomatic approach to image interpolation, IEEE Trans. Image Process., 7 (1998), pp. 376-386.
14. V. CASELLES, AND L. RUDIN, Multiscale total variation, in Proceedings of the Second European Conference on Image Processing (Palma, Spain, September, 1995), pp. 376-386.
15. A. CHAMBOLLE AND P.-L. LIONS, Image recovery via total variation minimization and related problems, Numer. Math., 76 (1997), pp. 167-188.
16. T. CHAN, A. MARQUINA, AND P. MULET, High-order total variation-based image restoration, SIAM J. Sci. Comput., 22 (2000), pp. 603-516.
17. T. F. CHAN, S. H. KANG, AND J. SHEN, Euler's elastica and curvature-based inpainting, SIAM J. Appl. Math., 63 (2002), pp. 564-592.
18. T. F. CHAN AND J. SHEN, Mathematical models for local nontexture inpaintings, SIAM J. Appl. Math., 62 (2002), pp. 1019-1043.
19. Y. CHEN AND M. RAO, Minimization problems and associated flows related to weighted p energy and total variation, SIAM J. Math. Anal., 34 (2003), pp. 1084-1104.
20. D. C. DOBSON AND F. SANTOSA, Recovery of blocky images from noisy and blurred data, SIAM J. Appl. Math., 56 (1996), pp. 1181-1198.
21. L. C. EVANS AND R. F. GARIEPY, Measure Theory and Fine Properties of Functions, Stud. Adv. Math., CRC Press, Boca Raton, FL, 1992.
22. E. GIUSTI, Minimal Surfaces and Functions of Bounded Variation, Monogr. Math. 80, Birkhäuser Verlag, Basel, 1984.
23. R. HARDT AND X. ZHOU, An evolution problem for linear growth functional, Comm. Partial Differential Equations, 19 (1994), pp. 1879-1907.
24. O. A. LADYZ̈ENSKAJA, V. A. SOLONNIKOV, AND N. N. URALCEVA, Linear and Quasilinear Equations of Parabolic Type, Transl. Math. Monogr. 23, AMS, Providence, RI, 1967.
25. A. LICHNEWSKY AND R. TEMAM, Pseudosolutions of the time-dependent minimal surface problem, 3. Differential Equations, 30 (1978), pp. 340-364.
26. M. LYSAKER, A. LUNDERVOLD, AND X.-C. TAI, Noise removal using fourth-order partial differential equation with applications to medical magnetic resonance images in space and time, IEEE Trans. Image Process., 12 (2003), pp. 1579-1590.
27. S. MASNOU AND J. M. MOREL, Level lines based disocclusion, in Proceedings of the IEEE International Conference on Image Processing (Chicago, IL, October 4-7, 1998), Vol. III, IEEE, Los Alamitos, CA, pp. 259-263.
28. M. NIKOLOVA, Weakly constrained minimization: Application to the estimation of images and signals involving constant regions, J. Math. Imaging Vision, 21 (2004), pp. 155-175.
29. S. OSHER AND J. A. SETHIAN, Propagating with curvature-dependent speed: Algoritms based on Hamilton-Jacobi formulations, J. Comput. Phys., 79 (1988), pp. 12-49.
30. W. RING, Structural properties of solutions to total variation regularization problems, M2AN Math. Model. Numer. Anal., 34 (2000), pp. 799-810.
31. L. RUDIN, S. OSHER, AND E. FATEMI, Nonlinear total variation based noise removal algorithms, Phys. D, 60 (1992), pp. 259-268.
32. D. M. STRONG AND T. F. CHAN, Spatially and Scale Adaptive Total Variation Based Régularisation and Anisotropic Diffusion in Image Processing, Technical Report CAM96-46, University of California, Los Angeles, CA, 1996. Available online at http://www.math. ucla.edu/applied/cam/index.html.
33. L. VESE, A study in the BV space of a denoising-deblurring variational problem, Appl. Math. Optim., 44 (2001), pp. 131-161.
34. R. T. WHITAKER AND S. M. PIZER, A mulit-scale approach to nonuniform diffusion, Comput. Vision Graph. Image Process. Image Understand., 57 (1993), pp. 99-110.
35. Y.-L. YOU AND M. KAVEH, Fourth-order partial differential equations for noise removal, IEEE Trans. Image Process., 9 (2000), pp. 1723-1730.
36. Y.-L. YOU, W. XU, A. TANNENBAUM, AND M. KAVEH, Behavioral analysis of anisotropic diffusion in image processing, IEEE Trans. Image Process., 5 (1996), pp. 1539-1553.
37. X. ZHOU, An evolution problem for plastic antiplanar shear, Appl. Math. Optim., 25 (1992), pp. 263-285.