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Phase Field Methods for Binary Recovery

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

Abstract

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.

Keywords

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

Notes

Acknowledgements

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

References

  1. [1]
    J.W. Barrett, C.M. Elliott, Finite element approximation of a free boundary problem arising in the theory of liquid drops and plasma physics. Math. Modell. Numer. Anal. 25, 213–252 (1991)MathSciNetzbMATHGoogle Scholar
  2. [2]
    P. Bastian, M. Blatt, A. Dedner, C. Engwer, R. Klöfkorn, M. Ohlberger, O. Sander, A generic grid interface for parallel and adaptive scientific computing. Part I: Abstract framework. Computing 82, 103–119 (2008)zbMATHGoogle Scholar
  3. [3]
    P. Bastian, M. Blatt, A. Dedner, C. Engwer, R. Klöfkorn, R. Kornhuber, M. Ohlberger, O. Sander, A generic grid interface for parallel and adaptive scientific computing. Part II: Implementation and tests in DUNE. Computing 82, 121–138 (2008)zbMATHGoogle Scholar
  4. [4]
    P. Bastian, M. Blatt, A. Dedner, C. Engwer, J. Fahlke, C. Gräser, R. Klöfkorn, M. Nolte, M. Ohlberger, O. Sander, DUNE web page, 2011. http://www.dune-project.org
  5. [5]
    M. Blatt, P. Bastian, The iterative solver template library. In: Applied Parallel Computing. State of the Art in Scientific Computing, ed. by B. Kagström, E. Elmroth, J. Dongarra, J. Wasniewski, pp. 666–675, Vol. 4699 of Lecture Notes in Computer Science (Springer, Berlin-Heidelberg-New York, 2007)Google Scholar
  6. [6]
    L. Blank, M. Butz, H. Garcke, Solving the Cahn-Hilliard variational inequality with a semi-smooth Newton method. ESAIM Contr Optim. Calculus Variat. 17, 931–954 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    L. Blank, H. Garcke, L. Sarbu, V. Styles, Primal-dual active set methods for Allen-Cahn variational inequalities with nonlocal constraints. Numer. Meth. Part. Differ. Equat. 29(3), 999–1030 (2013) http://onlinelibrary.wiley.com/doi/10.1002/num.v29.3/issuetoc
  8. [8]
    J.F. Blowey, C.M. Elliott, The Cahn-Hilliard gradient theory for phase separation with non-smooth free energy. Part I: Mathematical analysis. Eur. J. Appl. Math. 2, 233–279 (1991)MathSciNetzbMATHGoogle Scholar
  9. [9]
    J.F. Blowey, C.M. Elliott, The Cahn-Hilliard gradient theory for phase separation with non-smooth free energy. Part II: Numerical analysis. Eur. J. Appl. Math. 3, 147–179 (1992)MathSciNetzbMATHGoogle Scholar
  10. [10]
    J.E. Blowey, C.M. Elliott, Curvature dependent phase boundary motion and parabolic double obstacle problems. In: Degenerate Diffusions, ed. by W.M. Ni, L.A. Peletier, L. Vazquez (Springer, Berlin-Heidelberg-New York, 1993), pp. 19–60CrossRefGoogle Scholar
  11. [11]
    A. Chambolle, P. Lions, Image recovery via total variation minimization and related problems. Numer. Math. 76, 167–188 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    A. Chambolle, G. Del Maso, Discrete approximation of the Mumford-Shah functional in dimension two. ESAIM Math. Modell. Numer. Anal. 33, 651–672 (1999)CrossRefzbMATHGoogle Scholar
  13. [13]
    T.F. Chan, S. Esedoglu, Aspects of total variation regularized L1 function approximation. SIAM J. Appl. Math. 65, 1817–1837 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    T. Chan, S. Esedoglu, M. Nikolova, Algorithms for finding global minimizers of image segmentation and denoising models. SIAM J. Appl. Math. 66, 1632–1648 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    X. Chen, C.M. Elliott, Asymptotics for a parabolic double obstacle problem. Proc. R. Soc. Lond. A. 444(1922), 429–445 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    R. Choksi, Y. Van Gennip, Deblurring of one dimensional bar codes via total variation energy minimisation. SIAM J. Imag. Sci. 3, 735–764 (2010)CrossRefzbMATHGoogle Scholar
  17. [17]
    R. Choksi, Y. Van Gennip, A. Oberman, Anisotropic total variation regularized L1-approximation and denoising/deblurring of 2D bar codes. Preprint, arXiv:1007.1035 (2010)Google Scholar
  18. [18]
    A. Dedner, R. Klöfkorn, M. Nolte, M. Ohlberger, A generic interface for parallel and adaptive scientific computing: Abstraction principles and the DUNE-FEM module. Computing 90, 165–196 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    A. Dedner, R. Klöfkorn, M. Nolte, M. Ohlberger, DUNE-FEM web page. 2011. http://dune.mathematik.uni-freiburg.de
  20. [20]
    C.M. Elliott, A.M. Stuart, The global dynamics of discrete semilinear parabolic equations. SIAM J. Numer. Anal. 30, 1622–1663 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    S. Esedoglu, Blind deconvolution of bar code signals. Inverse Probl. 20, 121–135 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  22. [22]
    D.J. Eyre, An unconditionally stable one-step scheme for gradient systems. Unpublished article (1998)Google Scholar
  23. [23]
    C. Gräser, R. Kornhuber, Multigrid methods for obstacle problems. J. Comput. Math. 27, 1–44 (2009)MathSciNetzbMATHGoogle Scholar
  24. [24]
    R. Glowinski, Numerical Methods for Nonlinear Variational Problems (Springer, Berlin-Heidelberg-New York, 1984)CrossRefzbMATHGoogle Scholar
  25. [25]
    C. Gräser, Convex minimization and phase field models. PhD thesis, Freie Universität Berlin, 2011Google Scholar
  26. [26]
    B. Hackl, Geometry variations, level set and phase-field methods for perimeter regularized geometric inverse problems. PhD thesis, Johannes Kepler Universität Linz, 2006Google Scholar
  27. [27]
    J.K. Hale, Asymptotic Behavior of Dissipative Systems (American Mathematical Society, Providence, 1988)zbMATHGoogle Scholar
  28. [28]
    L. Modica, S. Mortola, Un esempio di \(\Gamma \)-convergenza. Bollettino dell’Unione Matematica Italiana 14-B, 285–299 (1977)MathSciNetGoogle Scholar
  29. [29]
    D.B. Mumford, J. Shah, Optimal approximations by piecewise smooth functions and associated variational problems. Comm. Pure Appl. Math. 42, 577–685 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  30. [30]
    N. Petra, G. Stadler, Model variational inverse problems governed by partial differential equations. Technical Report ADA555315, University of Texas at Austin, Institute for Computational Engineering and Sciences, 2011Google Scholar
  31. [31]
    L.I. Rudin, S. Osher, E. Fatemi, Nonlinear total variation based noise removal algorithms. Phys. D 60, 259–268 (1992)CrossRefzbMATHGoogle Scholar
  32. [32]
    L. Sarbu, Primal-dual active set methods for Allen-Cahn variational inequalities. PhD thesis, University of Sussex, 2010Google Scholar
  33. [33]
    X. Tai, T.F. Chan, A survey on multiple level set methods with applications for identifying piecewise constant functions. Int. J. Numer. Anal. Modell. 1, 25–47 (2004)MathSciNetzbMATHGoogle Scholar
  34. [34]
    X. Tai, H. Li, A piecewise constant level set method for elliptic inverse problems. Appl. Numer. Math. 57, 686–696 (2007)MathSciNetCrossRefzbMATHGoogle Scholar

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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