WIAS Preprint No. 2703, (2020)

Uncertainty quantification in image segmentation using the Ambrosio--Tortorelli approximation of the Mumford--Shah energy


  • Hintermüller, Michael
    ORCID: 0000-0001-9471-2479
  • Stengl, Steven-Marian
  • Surowiec, Thomas M.
    ORCID: 0000-0003-2473-4984

2010 Mathematics Subject Classification

  • 62D99 65N75 68U10 65K10


  • Image segmentation, Mumford-Shah, Ambrosio-Tortorelli, measurable selection, Monte-Carlo, sampling




The quantification of uncertainties in image segmentation based on the Mumford-Shah model is studied. The aim is to address the error propagation of noise and other error types in the original image to the restoration result and especially the reconstructed edges (sharp image contrasts). Analytically, we rely on the Ambrosio-Tortorelli approximation and discuss the existence of measurable selections of its solutions as well as sampling-based methods and the limitations of other popular methods. Numerical examples illustrate the theoretical findings.

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