An Optimal Control Approach to Find Sparse Data for Laplace Interpolation

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)

   Article  MathSciNet  MATH  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)

   Article  MathSciNet  MATH  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)

   Article  MathSciNet  MATH  Google Scholar 

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

   Article  MATH  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)

    Article  MathSciNet  MATH  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)

    Article  MathSciNet  MATH  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)

    Article  MathSciNet  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)

    Article  MathSciNet  MATH  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)

    Article  MathSciNet  MATH  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)

    Article  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)

    Chapter  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)

    Article  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)

    Chapter  Google Scholar 

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

    Article  MathSciNet  MATH  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)

    Chapter  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)

    Article  MathSciNet  MATH  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)

    Article  MathSciNet  MATH  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)

    Article  MathSciNet  MATH  Google Scholar 

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