Quantitative analysis of finite-difference approximations of free-discontinuity problems

  • Annika Bach

    Technische Universität München, Garching, Germany
  • Andrea Braides

    Università di Roma Tor Vergata, Italy
  • Caterina Ida Zeppieri

    Universität Münster, Germany
Quantitative analysis of finite-difference approximations of free-discontinuity problems cover

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Abstract

Motivated by applications to image reconstruction, in this paper we analyse a finite-difference discretisation of the Ambrosio–Tortorelli functional. Denoted by the elliptic-approximation parameter and by the discretisation step-size, we fully describe the relative impact of and in terms of -limits for the corresponding discrete functionals, in the three possible scaling regimes. We show, in particular, that when and are of the same order, the underlying lattice structure affects the -limit which turns out to be an anisotropic free-discontinuity functional.

Cite this article

Annika Bach, Andrea Braides, Caterina Ida Zeppieri, Quantitative analysis of finite-difference approximations of free-discontinuity problems. Interfaces Free Bound. 22 (2020), no. 3, pp. 317–381

DOI 10.4171/IFB/443