Enhanced and restored signals as a generalized solution for shock filter models. Part I - Existence and uniqueness result of the Cauchy problem

Journal of Mathematical Analysis and Applications

Volumn 279, Issue 1, 2003, Pages 189-209

  Remaki, L ^a   Cheriet, M ^a  

^a ÉCOLE DE TECHNOLOGIE SUPÉRIEURE   (Canada)

Author keywords

Cauchy problem; Generalized functions; Numerical schemes; Partial differential equations (PDEs); Shock filters; Signals enhancement and restoration

Indexed keywords

EID: 0037996660     PISSN: 0022247X     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0022-247X(02)00713-8     Document Type: Article

References (19)

1. L. Alvarez, P.L. Lions, J.M. Morel, Image selective smoothing and edge detection by nonlinear diffusion (ii), SIAM J. Numer. Anal. 29 (1992) 845-866.
2. L. Alvarez, L. Mazorra, Signal and image restoration using shock filters and anisotropic diffusion, SIAM J. Numer. Anal. 31 (1994) 590-605.
3. H.C. Andrews, B.R. Hunt, Digital Image Restoration, Prentice Hall, Englewood Cliffs, NJ, 1977, pp. 339-370.
4. S. Bernard, J.F. Colombeau, A. Meul, L. Remaki, Conservation laws with discontinuous coefficients, J. Math. Anal. Appl. 258 (2001) 63-86.
5. F. Catte, T. Coll, P.L. Lions, J.M. Morel, Image selective smoothing and edge detection by nonlinear diffusion (i), SIAM J. Numer. Anal. 29 (1992) 182-193.
6. J.F. Colombeau, Multiplication of Distribution, in: Lecture Notes in Mathematics, Vol. 1532, Springer-Verlag, 1992.
7. J.F. Colombeau, A. Heibig, Generalized solution to Cauchy problems, Monatsh. Math. 117 (1994) 33-49.
8. J.F. Colombeau, A. Heibig, A. Oberguggenberger, Generalized solution to partial differential equations of evolution type, Acta Appl. Math. 45 (1996) 115-145.
9. B.R. Frieden, Restoring with maximum likelihood and maximum entropy, J. Opt. Soc. Amer. 62 (1972) 511-518.
10. A. Heibig, M. Moussaoui, Generalized and classical solutions of nonlinear parabolic equations, Nonlinear Anal. 24 (1995) 789-794.
11. B.R. Hunt, The application of constrained least squares estimation to image restoration by digital computers, IEEE Trans. Comput. C-22 (1973) 805-8012.
12. J.J. Koenderink, The structure of images, Biol. Cybernet. 53 (1984) 363-370.
13. D. Marr, Vision, Freeman, San Francisco, CA, 1982.
14. M. Oberguggenberger, Hyperbolic systems with discontinuous coefficients: generalized solutions and a transmission problem in acoustics, J. Math. Anal. Appl. 142 (1989) 452-467.
15. S. Osher, L. Rudin, Feature oriented image enhancement using shock filters, SIAM J. Numer. Anal. 27 (1990) 919-940.
16. P. Perona, J. Malik, Scale-space and edge detection using anisotropic diffusion, IEEE Trans. Pattern Anal. Mach. Intelligence 12 (1990) 629-639.
17. L. Remaki, Étude théorique et numérique des équations quasi-linéaires à coefficients discontinus et acoustique linéaire 2D, Ph.D. thesis, Université Claude Bernard Lyon 1, Lyon, France, 1997.
18. L. Rudin, Shock filters, in: Rockwell International Science Center Annual DARPA T.R., 1984.
19. L. Schwartz, Sur l'impossibilité de la multiplication des distributions, C. R. Acad. Sci. Paris 239 (1954) 847-848.