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Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.78.

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A posteriori error control for the binary Mumford-Shah model
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by Benjamin Berkels, Alexander Effland and Martin Rumpf PDF
Math. Comp. 86 (2017), 1769-1791 Request permission


The binary Mumford-Shah model is a widespread tool for image segmentation and can be considered as a basic model in shape optimization with a broad range of applications in computer vision, ranging from basic segmentation and labeling to object reconstruction. This paper presents robust a posteriori error estimates for a natural error quantity, namely the area of the non-properly segmented region. To this end, a suitable uniformly convex and non-constrained relaxation of the originally non-convex functional is investigated and Repin’s functional approach for a posteriori error estimation is used to control the numerical error for the relaxed problem in the $L^2$-norm. In combination with a suitable cut out argument, fully practical estimates for the area mismatch are derived. This estimate is incorporated in an adaptive mesh refinement strategy. Two different adaptive primal-dual finite element schemes, a dual gradient descent scheme, and the most frequently used finite difference discretization are investigated and compared. Numerical experiments show qualitative and quantitative properties of the estimates and demonstrate their usefulness in practical applications.
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Additional Information
  • Benjamin Berkels
  • Affiliation: AICES Graduate School, RWTH Aachen University, 52062 Aachen, Germany
  • Email:
  • Alexander Effland
  • Affiliation: Institute for Numerical Simulation, University of Bonn, 53115 Bonn, Germany
  • Email:
  • Martin Rumpf
  • Affiliation: Institute for Numerical Simulation, University of Bonn, 53115 Bonn, Germany
  • MR Author ID: 604100
  • Email:
  • Received by editor(s): May 19, 2015
  • Received by editor(s) in revised form: December 8, 2015, and December 17, 2015
  • Published electronically: October 20, 2016
  • © Copyright 2016 American Mathematical Society
  • Journal: Math. Comp. 86 (2017), 1769-1791
  • MSC (2010): Primary 49M25, 53C22, 65D18, 65L20
  • DOI:
  • MathSciNet review: 3626536