Phase Field Methods for Binary Recovery

  • Charles BrettEmail author
  • Charles M. Elliott
  • Andreas S. Dedner
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 101)


We consider the inverse problem of recovering a binary function from blurred and noisy data. Such problems arise in many applications, for example image processing and optimal control of PDEs. Our formulation is based on the Mumford-Shah model, but with a phase field approximation to the perimeter regularisation. We use a double obstacle potential as well as a smooth double well potential. We introduce an iterative method for solving the problem, develop a suitable discretisation of this iterative method, and prove some convergence results. Numerical simulations are presented which illustrate the usefulness of the approach and the relative merits of the phase field models.


Binary recovery Image processing Mumford-Shah model Optimal control Phase field models 



We are grateful to Carsten Gräser for sharing his Dune-Solvers code for the TNNMG method.


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Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Charles Brett
    • 1
    Email author
  • Charles M. Elliott
    • 1
  • Andreas S. Dedner
    • 1
  1. 1.Mathematics InstituteUniversity of WarwickCoventryUK

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