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Acta Applicandae Mathematicae

, Volume 156, Issue 1, pp 79–107 | Cite as

Static, Quasistatic and Dynamic Analysis for Scaled Perona-Malik Functionals

  • Andrea Braides
  • Valerio Vallocchia
Article

Abstract

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 Morini and Negri. 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.

Keywords

Perona-Malik functional Image processing Fracture mechanics Minimizing movements Variational evolution Local minima \(\varGamma \)-convergence 

Mathematics Subject Classification

49J45 74S20 49M25 94A08 

Notes

Acknowledgements

The authors acknowledge the many suggestions of the anonymous referee, which greatly improved the paper.

Conflict of Interest

The authors declare that they have no conflict of interest.

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© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Dipartimento di MatematicaUniversità di Roma “Tor Vergata”RomeItaly

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