Young measure solutions and instability of the one-dimensional Perona-Malik equation

Journal of Mathematical Analysis and Applications

Volumn 308, Issue 2, 2005, Pages 467-490

  Taheri, Shahnaz ^a   Tang, Qi ^a   Zhang, Kewei ^a  

^a UNIVERSITY OF SUSSEX   (United Kingdom)

Author keywords

Edge detection; Forward backward diffusion; Image processing; Perona Malik equation; Young measure solutions

Indexed keywords

EID: 20744434267     PISSN: 0022247X     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.jmaa.2004.11.034     Document Type: Article

References (44)

1. R.A. Adams Sobolev Spaces 1975 Academic Press San Diego
2. E. Acerbi N. Fusco Semicontinuity problems in the calculus of variations Arch. Rational Mech. Anal. 86 1984 125-145
3. L. Alvarez F. Guichard P.L. Lions S.M. Morel Axioms and fundamental equations of image processing Arch. Rational Mech. Anal. 123 1993 199-257
4. J.M. Ball Convexity conditions and existence theorems in nonlinear elasticity Arch. Rational Mech. Anal. 1977 337-403
5. J.M. Ball A version of the fundamental theorem of Young measures in: M. Rascle D. Serre M. Slemrod (Eds.), Partial Differential Equations and Continuum Models of Phase Transitions in: Lecture Notes in Phys. vol. 334 1989 Springer-Verlag Berlin 207-215
6. J.M. Ball R.D. James Fine phase mixtures as minimizers of energy Arch. Rational Mech. Anal. 100 1987 13-52
7. J.M. Ball R.D. James Proposed experimental tests of a theory of fine microstructure and the two well problem Philos. Trans. Roy. Soc. London Ser. A 338 1992 389-450
8. G. Bellettini, G. Fusco, A regularized Perona - Malik functional: Some aspects of the gradient dynamics, preprint, 1993
9. I. Barnes K. Zhang Instability of the Eikonal equation and shape from shading M2AN Math. Model. Numer. Anal. 34 2000
10. B. Dacorogna Direct Methods in the Calculus of Variations 1989 Springer-Verlag Berlin
11. S. Demoulini Variational methods for Young measure solutions of nonlinear parabolic evolutions of forward - backward type and of high spatial order Appl. Anal. 63 1996 363-373
12. S. Demoulini Young measure solutions for a nonlinear parabolic equation of forward - backward type SIAM J. Math. Anal. 27 1996 376-403
13. S. Demoulini Young measure solutions for nonlinear evolutionary systems of mixed type Ann. Inst. H. Poincaré Anal. Non Linéaire 14 1997 143-162
14. L.C. Evans R.F. Gariepy Measure Theory and Fine Properties of Functions 1992 CRC Press Boca Raton, FL
15. A. Friedman Partial Differential Equations of Parabolic Type 1964 Prentice Hall New York
16. S. Guillermo Geometric Partial Differential Equations and Image Analysis 2001 Cambridge Univ. Press Cambridge
17. K. Höllig Existence of infinitely many solutions for a forward backward heat equation Trans Amer. Math. Soc. 26 1983 299-316
18. D. Horstmann, K.J. Painter, H.G. Othmer, Aggregation under local reinforcement: From lattice to continuum, European J. Appl. Math. (2004), in press
19. S. Kichenassamy The Perona - Malik paradox SIAM J. Appl. Math. 57 1997 1328-1342
20. B. Kawohl N. Kutev Maximum and comparison principle for one-dimensional anisotropic diffusion Math. Ann. 311 1998 107-123
21. B. Kirchheim S. Müller V. Sverák Studying nonlinear PDE by geometry in matrix space in: S. Hildebrandt Stefan et al. (Eds.), Geometric Analysis and Nonlinear Partial Differential Equations 2003 Springer-Verlag Berlin 347-395
22. D. Kinderlehrer P. Pedregal Characterizations of Young measures generated by gradients Arch. Rational Mech. Anal. 115 1991 329-365
23. D. Kinderlehrer P. Pedregal Weak convergence of integrands and the Young measure representation SIAM J. Math. Anal. 23 1992 1-19
24. G.M. Lieberman Second Order Parabolic Differential Equations 1996 World Scientific Singapore
25. O.A. Ladyzhenskaya V.A. Solonnikov N.N. Uralceva Linear and Quasilinear Equations of Parabolic type 1967 Nauka Moscow
26. C.B. Morrey Jr. Multiple Integrals in the Calculus of Variations 1966 Springer-Verlag Berlin
27. S. Müller Variational models for microstructure and phase transitions in: Calculus of Variations and Geometric Evolution Problems Cetraro, 1996 in: Lecture Notes in Math. vol. 1713 1999 Springer-Verlag Berlin 85-210
28. S. Müller V. Šverák Attainment results for the two-well problem by convex integration in: Geometric Analysis and the Calculus of Variations 1996 Internat. Press Cambridge, MA 239-251
29. S. Müller V. Šverák Convex integration for Lipschitz mappings and counterexamples to regularity Ann. of Math. 157 2003 715-742
30. S. Müller, M.O. Rieger, V. Sverák, Parabolic systems with nowhere smooth solutions, Arch. Rational Mech. Anal. (2002)
31. P. Pedregal Parametrized measures and variational principles Bull. Amer. Math. Soc. 30 1997
32. P. Perona J. Malik Scale space and edge detection using anisotropic diffusion IEEE Trans. Pattern Anal. Mach. Intell. 12 1990 629-639
33. V. Scheffer, Regularity and irregularity of solutions to nonlinear second order elliptic systems of partial differential equations and inequalities, Dissertation, Princeton University, 1974
34. V. Šverák Rank one convexity does not imply quasiconvexity Proc. Roy. Soc. Edinburgh 120 1992 185-189
35. V. Šverák Lower-semicontinuity of variational integrals and compensated compactness in: Proceedings ICM 94 1995 Birkhäuser Zürich
36. V. Šverák On the Problem of Two Wells, in Microstructure and Phase Transition IMA Vol. Math. Appl. vol. 54 1993 Springer-Verlag New York pp. 183-189
37. L. Szekelyhidi Jr. The regularity of critical points of polyconvex functionals Arch. Rational Mech. Anal. 172 2004 133-152
38. L. Tartar Compensated compactness and applications to partial differential equations, in nonlinear analysis and mechanics in: R.J. Knops (Eds.), Heriot-Watt Symp. IV 1979 Pitman London
39. Q. Tang, K. Zhang, An evolutionary double-well problem, Ann. Inst. H. Poincaré Anal. Non Linéaire
40. J. Weickert Anisotropic diffusion in image processing in: ECMI Ser. 1998 Teubner Stuttgart
41. J. Ying C. Wang Young measure solutions for a class of forward-backward diffusion equations J. Math. Anal. Appl. 279 2003 659-683
42. K.-W. Zhang Quasiconvex functions, SO(n) and two elastic wells Ann. Inst. H. Poincaré Anal. Non Linéaire 14 1997 759-785
43. K.-W. Zhang On various semiconvex hulls in the calculus of variations Calc. Var. Partial Differential Equation 6 1998 143-160
44. K.-W. Zhang Rank-one connections at infinity and quasiconvex hulls J. Convex Anal. 7 2000 19-45