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International Workshop on Energy Minimization Methods in Computer Vision and Pattern Recognition

EMMCVPR 2013: Energy Minimization Methods in Computer Vision and Pattern Recognition pp 151–164Cite as

An Optimal Control Approach to Find Sparse Data for Laplace Interpolation

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Part of the Lecture Notes in Computer Science book series (LNIP,volume 8081)

Abstract

Finding optimal data for inpainting is a key problem in the context of partial differential equation-based image compression. We present a new model for optimising the data used for the reconstruction by the underlying homogeneous diffusion process. Our approach is based on an optimal control framework with a strictly convex cost functional containing an L 1 term to enforce sparsity of the data and non-convex constraints. We propose a numerical approach that solves a series of convex optimisation problems with linear constraints. Our numerical examples show that it outperforms existing methods with respect to quality and computation time.

Keywords

  • Laplace Interpolation
  • Optimal Control
  • Inpainting
  • Non-convex Optimisation

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References

  1. Belhachmi, Z., Bucur, D., Burgeth, B., Weickert, J.: How to choose interpolation data in images. SIAM Journal on Applied Mathematics 70(1), 333–352 (2009)

    MathSciNet  MATH  CrossRef  Google Scholar 

  2. Bertalmío, M., Sapiro, G., Caselles, V., Ballester, C.: Image inpainting. In: Proc. 27th Annual Conference on Computer Graphics and Interactive Techniques, pp. 417–424. ACM Press/Addison-Wesley Publishing Company (2000)

    Google Scholar 

  3. Bonnans, J.F., Shapiro, A.: Perturbation Analysis of Optimization Problems. Springer Series in Operations Research. Springer (2000)

    Google Scholar 

  4. Buckheit, J., Chen, S.S., Donoho, D., Huo, X., Johnstone, I., Levi, O., Scargle, J., Yu, T.: WAVELAB 850 toolbox for matlab (2012), http://www-stat.stanford.edu/~wavelab/Wavelab_850/download.html

  5. Chambolle, A., Pock, T.: A first order primal-dual algorithm for convex problems with applications to imaging. Journal of Mathematical Imaging and Vision 40(1), 120–145 (2011)

    MathSciNet  MATH  CrossRef  Google Scholar 

  6. Clarke, F.H.: Optimization and Nonsmooth Analysis. SIAM (1990)

    Google Scholar 

  7. Clason, C., Kunisch, K.: A duality-based approach to elliptic control problems in non-reflexive banach spaces. ESAIM: Control, Optimisation and Calculus of Variations 17(1), 243–266 (2011)

    MathSciNet  MATH  CrossRef  Google Scholar 

  8. Demaret, L., Dyn, N., Iske, A.: Image compression by linear splines over adaptive triangulations. Signal Processing 86(7), 1604–1616 (2006)

    MATH  CrossRef  Google Scholar 

  9. Demaret, L., Iske, A.: Advances in digital image compression by adaptive thinning. Annals of the MCFA 3, 105–109 (2004)

    Google Scholar 

  10. Esser, E., Zhang, X., Chan, T.F.: A general framework for a class of first order primal-dual algorithms for convex optimization in imaging science. SIAM Journal on Imaging Sciences 3(4), 1015–1046 (2010)

    MathSciNet  MATH  CrossRef  Google Scholar 

  11. Friedlander, M.P., Saunders, M.A.: A globally convergent linearly constrained lagrangian method for nonlinear optimization. SIAM Journal on Optimization 15(3), 863–897 (2005)

    MathSciNet  MATH  CrossRef  Google Scholar 

  12. Galić, I., Weickert, J., Welk, M., Bruhn, A., Belyaev, A., Seidel, H.P.: Image compression with anisotropic diffusion. Journal of Mathematical Imaging and Vision 31(2-3), 255–269 (2008)

    MathSciNet  CrossRef  Google Scholar 

  13. Griffith, R.E., Stewart, R.A.: A nonlinear programming technique for the optimization of continuous processing systems. Management Science 7(4), 379–392 (1961)

    MathSciNet  MATH  CrossRef  Google Scholar 

  14. Haslinger, J., Mäkinen, R.A.E.: Introduction to Shape Optimization: Theory, Approximation, and Computation. SIAM (1987)

    Google Scholar 

  15. Lin, C.J.: Projected gradient methods for nonnegative matrix factorization. Neural Computation 19(10), 2756–2779 (2007)

    MathSciNet  MATH  CrossRef  Google Scholar 

  16. Mainberger, M., Bruhn, A., Weickert, J., Forchhammer, S.: Edge-based image compression of cartoon-like images with homogeneous diffusion. Pattern Recognition 44(9), 1859–1873 (2011)

    CrossRef  Google Scholar 

  17. Mainberger, M., Hoffmann, S., Weickert, J., Tang, C.H., Johannsen, D., Neumann, F., Doerr, B.: Optimising spatial and tonal data for homogeneous diffusion inpainting. In: Bruckstein, A.M., ter Haar Romeny, B.M., Bronstein, A.M., Bronstein, M.M. (eds.) SSVM 2011. LNCS, vol. 6667, pp. 26–37. Springer, Heidelberg (2012)

    CrossRef  Google Scholar 

  18. Masnou, S., Morel, J.M.: Level lines based disocclusion. In: Proc. of the International Conference on Image Processing, vol. 3, pp. 259–263. IEEE (1998)

    Google Scholar 

  19. Murthagh, B.A., Saunders, M.A.: A projected lagrangian algorithm and its implementation for sparse nonlinear constraints. Mathematical Programming Study 16, 84–117 (1982)

    CrossRef  Google Scholar 

  20. Pock, T., Schoenemann, T., Graber, G., Bischof, H., Cremers, D.: A convex formulation of continuous multi-label problems. In: Forsyth, D., Torr, P., Zisserman, A. (eds.) ECCV 2008, Part III. LNCS, vol. 5304, pp. 792–805. Springer, Heidelberg (2008)

    CrossRef  Google Scholar 

  21. Robinson, S.M.: A quadratically-convergent algorithm for general nonlinear programming problems. Mathematical Programming 3, 145–156 (1972)

    MathSciNet  MATH  CrossRef  Google Scholar 

  22. Schmaltz, C., Weickert, J., Bruhn, A.: Beating the quality of JPEG 2000 with anisotropic diffusion. In: Denzler, J., Notni, G., Süße, H. (eds.) DAGM 2009. LNCS, vol. 5748, pp. 452–461. Springer, Heidelberg (2009)

    CrossRef  Google Scholar 

  23. Sokolowski, J., Zolesio, J.P.: Introduction to Shape Optimization. Springer (1992)

    Google Scholar 

  24. Stadler, G.: Elliptic optimal control problems with L 1-control cost and applications for the placement of control devices. Computational Optimization and Applications 44(2), 159–181 (2009)

    MathSciNet  MATH  CrossRef  Google Scholar 

  25. Tröltzsch, F.: Optimale Steuerung Partieller Differentialgleichungen: Theorie, Verfahren und Anwendungen, 2nd edn. Vieweg+Teubner (2009)

    Google Scholar 

  26. Wachsmuth, G., Wachsmuth, D.: Convergence and regularization results for optimal control problems with sparsity functional. ESAIM: Control, Optimisation and Calculus of Variations 17(3), 858–886 (2011)

    MathSciNet  MATH  CrossRef  Google Scholar 

  27. Xu, Y., Yin, W.: A block coordinate descent method for multi-convex optimization with applications to nonnegative tensor factorization and completion. Rice CAAM Technical Report TR12-15, Rice University (2012)

    Google Scholar 

  28. Xu, Y., Yin, W., Wen, Z., Zhang, Y.: An alternating direction algorithm for matrix completion with nonnegative factors. Frontiers of Mathematics in China 7(2), 365–384 (2012)

    MathSciNet  MATH  CrossRef  Google Scholar 

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Author information

Authors and Affiliations

  1. Mathematical Image Analysis Group, Faculty of Mathematics and Computer Science, Saarland University, Campus E1.7, 66041, Saarbrücken, Germany

    Laurent Hoeltgen, Simon Setzer & Joachim Weickert

Authors
  1. Laurent Hoeltgen
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  2. Simon Setzer
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  3. Joachim Weickert
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Editor information

Editors and Affiliations

  1. Centre for Mathematical Sciences, Lund University, Sweden

    Anders Heyden, Fredrik Kahl, Carl Olsson & Magnus Oskarsson, ,  & 

  2. Dept. of Mathematics, University of Bergen, Johaness Brunsgate 12, 5007, Bergen, Norway

    Xue-Cheng Tai

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Hoeltgen, L., Setzer, S., Weickert, J. (2013). An Optimal Control Approach to Find Sparse Data for Laplace Interpolation. In: Heyden, A., Kahl, F., Olsson, C., Oskarsson, M., Tai, XC. (eds) Energy Minimization Methods in Computer Vision and Pattern Recognition. EMMCVPR 2013. Lecture Notes in Computer Science, vol 8081. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-40395-8_12

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  • DOI: https://doi.org/10.1007/978-3-642-40395-8_12

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-40394-1

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