Γ-Convergence of an Ambrosio-Tortorelli approximation scheme for image segmentation

Irene Fonseca, Lisa Maria Kreusser, Carola-Bibiane Schönlieb, Matthew Thorpe

Research output: Contribution to journalArticlepeer-review

Abstract

Given an image u0, the aim of minimising the Mumford-Shah functional is to find a decomposition of the image domain into sub-domains and a piecewise smooth approximation u of u0 such that u varies smoothly within each sub-domain. Since the Mumford-Shah functional is highly non-smooth, regularizations such as the Ambrosio-Tortorelli approximation can be considered which is one of the most computationally efficient approximations of the MumfordShah functional for image segmentation. Our main result is the Γ-convergence of the AmbrosioTortorelli approximation of the Mumford-Shah functional for piecewise smooth approximations. This requires the introduction of an appropriate function space. As a consequence of our Γconvergence result, we can infer the convergence of minimizers of the respective functionals.
Original languageEnglish
JournalIndiana University Mathematics Journal
Volume73
Issue number1
Publication statusPublished - 2024

Keywords

  • math.OC
  • cs.NA
  • math.AP
  • math.NA

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