Static, quasistatic and dynamic analysis for scaled Perona-Malik functionals. (English) Zbl 1397.49018
Summary: We present an asymptotic description of local minimization problems, and of quasistatic and dynamic evolutions of discrete one-dimensional scaled Perona-Malik functionals. The scaling is chosen in such a way that these energies \(\varGamma \)-converge to the Mumford-Shah functional by a result by M. Morini and M. Negri [Math. Models Methods Appl. Sci. 13, No. 6, 785–805 (2003; Zbl 1045.49022)]. This continuum approximation still provides a good description of quasistatic and gradient-flow type evolutions, while it must be suitably corrected to maintain the pattern of local minima and to account for long-time evolution.
MSC:
| 49J45 | Methods involving semicontinuity and convergence; relaxation |
| 74S20 | Finite difference methods applied to problems in solid mechanics |
| 49M25 | Discrete approximations in optimal control |
| 94A08 | Image processing (compression, reconstruction, etc.) in information and communication theory |
Keywords:
Perona-Malik functional; image processing; fracture mechanics; minimizing movements; variational evolution; local minima; \(\varGamma \)-convergenceCitations:
Zbl 1045.49022References:
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