Young measure solutions of the two-dimensional Perona-malik equation in image processing

Communications on Pure and Applied Analysis

Volumn 5, Issue 3, 2006, Pages 617-637

  Chen, Yan ^a   Zhang, Kewei ^b  

^a CHINA AGRICULTURAL UNIVERSITY   (China)

^b UNIVERSITY OF SUSSEX   (United Kingdom)

Author keywords

Approximate sequence of solutions; Edge enhancement; Image processing; Partial differential inclusion; Perona Malik model; Young measure solutions

Indexed keywords

EID: 33749390514     PISSN: 15340392     EISSN: 15535258     Source Type: Journal    
DOI: 10.3934/cpaa.2006.5.617     Document Type: Article

References (36)

1. R.A. Adams, Sobolev Spaces, Academic Press (1975).
2. L. Alvarez, F. Guichard, P.L. Lions and S.M. Morel,Axioms and fundamental equations of image processing Arch. Rational Mech. Anal., 123 (1993), 199-257.
3. J.M. Ball, A version of the fundamental theorem of Young measures, Partial Differential Equations and Continuum Models of Phase Transitions, (edited by M. Rascle, D. Serre and M. Slemrod), Springer-Verlag (1989), 207-215.
4. G. Bellettini and G. Fusco, A regularized Perona-Malik functional: some aspects of the gradient dynamics, Preprint (2003).
5. J.M. Ball and R.D. James, Fine phase mixtures as minimizers of energy, Arch. Rational Mech. Anal., 100 (1987), 13 - 52.
6. [6j J.M. Ball and R.D. James, Proposed experimental tests of a theory of fine microstructures and the two-well problem, Phil. Royal Soc. Lon., 338A (1992), 389 - 450.
7. I. Barnes and K. Zhang Instability of the eikonal equation and shape from shading, MA2N Math. Model. Numer. Anal., 34 (2000), 127-138.
8. V. Caselles and J. Morel (eds), Special Issue On Partial Differential Equations And Geometry-driven Diffusion In Image Processing And Analysis, IEEE Trans. Image Processing, (3) 7 (1998).
9. B. Dacorogna, Weak continuity and weak lower semicontinuity of nonlinear functionals, Lecture Notes in Math. 922, Springer, Berlin (1982).
10. S. Demoulini, Young measure solutions for a nonlinear parabolic equation of forward-backward type, SIAM J. Math. Anal., 27 (1996), 376-403.
11. E. DiBenedetto, Degenerate Parabolic Equations, Springer-Verlag (1993).
12. B. Dacorogna, I. Fonseca A-B quasiconvexity and implicit partial differential equations, Calc. Var. PDEs, 14 (2002), 115-149.
13. C.M. Elliott, The Stefan problem with a non-monotone constitutive relation, IMA J. Appl. Math., 35 (1985), 257-264.
14. L.C. Evans and M. Portilheiro, Irreversibility and hysteresis for a forward-backward diffusion equation, Math. Models Methods Appl. Sci., 14 (2004), 1599-1620.
15. S. Esedoglu, An analysis of the Perona-Malik scheme Comm. Pure Appl. Math., 54 (2001), 1442-1487.
16. A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, (1964).
17. V. Girault and P.-A. Raviart, Finite element methods for Navier-Stokes equations. Theory and algorithms, Springer-Verlag, Berlin, (1986).
18. K. Höllig, Existence of infinitely many solutions for a forward backward heat equation, Trans. Amer. Math. Soc., 278 (1983), 299-316.
19. F. John, Partial differential equations 4th ed., Springer, (1982).
20. S. Kichenassamy, The Pernoa.-Malik paradox, SIAM J. Appl. Math., 57 (1997), 1328-1342.
21. B. Kawohl and N. Kutev, Maximum and comparision principle for one-dimensional anisotropic diffusion, Math. Ann., 311 (1998), 107-123.
22. D. Kinderlehrer and P. Pedregal, Characterizations of Young measures generated by gradients, Arch. Rational Mech. Anal., 115 (1991), 329-365.
23. G.M. Lieberman, Second order parabolic differential equations, Singapore, London: World Scientific, (1996).
24. O.A. Ladyzenskaya, V.A. Solonnikov and N.N. Uralceva, Linear and quasilinear equations of parabolic type, Nauka, Moscow, (1967).
25. C.B. Morrey Jr, Multiple integrals in the calculus of variations, Springer, (1966).
26. P.I. Plotnikov, Passing to the limit with respect to the viscosity in an equation with variable parabolicity direction, Dif. Eqns., 30 (1994), 614-622.
27. P. Perona and J. Malik, Scale space and edge detection using anisotropic diffusion, IEEE Trans. Pattern Anal. Mach. Intell., 12 (1990), 629-639.
28. M.O. Rieger, Young Measure Solutions for Nonconvex Elastodynamics, SIAM J. Math. Anal., 34 (2003), 1380-1398.
29. G. Sapiro, Geometric partial differential equations and image analysis, Cambridge: Cambridge University Press, (2001).
30. V. Šverák, On the problem of two wells, Microstructure and phase transitions, IMA Vol. Appl. Math. 54 (D. Kinderlehrer R. D. James, M. Luskin and J. Ericksen, eds), Springer, (1993), 183-189.
31. L. Tartar, Compensated compactness and applications to partial differential equations, Non-linear Analysis and Mechanics: Heriot-Watt Symposium (edited by R. J. Knops)IV (1979), 136-212.
32. S. Taheri, Q. Tang and K. Zhang, Young measure solutions and instability of the one-dimensional Perona-Malik equation, J. Math. Anal. Appl., 308 (2005), 467-490.
33. J. Weickert, Anisotropic diffusion in image processing, ECMI Series, Teubner, Stuttgart, (1998).
34. J. Yin and C. Wang, Young measure solutions for a class of forward-backward diffusion equations, J. Math. Anal. Appl., 279 (2003), 659-683.
35. K. Zhang, On the structure of quasiconvex hulls, Ann. Inst. H. Poincaré Anal. Non Linéaire, 15 (1998), 663-686.
36. K. Zhang, On existence of weak solutions for one-dimensional forward-backward diffusion equations, J. Diff. Eqns, 220 (2006) 322-353.