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Journal of Mathematical Imaging and Vision

, Volume 59, Issue 3, pp 515–533 | Cite as

Optimal Selection of the Regularization Function in a Weighted Total Variation Model. Part II: Algorithm, Its Analysis and Numerical Tests

  • Michael Hintermüller
  • Carlos N. Rautenberg
  • Tao Wu
  • Andreas Langer
Article

Abstract

Based on the weighted total variation model and its analysis pursued in Hintermüller and Rautenberg 2016, in this paper a continuous, i.e., infinite dimensional, projected gradient algorithm and its convergence analysis are presented. The method computes a stationary point of a regularized bilevel optimization problem for simultaneously recovering the image as well as determining a spatially distributed regularization weight. Further, its numerical realization is discussed and results obtained for image denoising and deblurring as well as Fourier and wavelet inpainting are reported on.

Keywords

Image restoration Weighted total variation regularization Spatially distributed regularization weight Fenchel predual Bilevel optimization Variance corridor Projected gradient method Convergence analysis 

Mathematics Subject Classification

94A08 68U10 49K20 49K30 49K40 49M37 65K15 

References

  1. 1.
    Adams, R.A., Fournier, J.J.F.: Sobolev Spaces. Volume 140 of Pure and Applied Mathematics. Elsevier/Academic Press, Amsterdam (2003)Google Scholar
  2. 2.
    Almansa, A., Ballester, C., Caselles, V., Haro, G.: A TV based restoration model with local constraints. J. Sci. Comput. 34(3), 209–236 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Athavale, P., Jerrard, R., Novaga, M., Orlandi, G.: Weighted TV minimization and applications to vortex density models. Technical report, University of Pisa, Department of Mathematics, (2015)Google Scholar
  4. 4.
    Attouch, H., Buttazzo, G., Michaille, G.: Variational analysis in Sobolev and BV spaces. MPS-SIAM, (2006)Google Scholar
  5. 5.
    Barbu, V.: Optimal control of variational inequalities. Res, vol. 100. Notes Math. Pitman, London, United Kingdom (1984)Google Scholar
  6. 6.
    Bertalmio, M., Caselles, V., Rougé, B., Solé, A.: TV based image restoration with local constraints. J. Sci. Comput. 19, 95–122 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Bertsekas, D.P.: On the Goldstein-Levitin-Polyak gradient projection method. IEEE Trans. Autom. Control AC–21(2), 174–184 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Bertsekas, D.P.: Projected Newton methods for optimization problems with simple constraints. SIAM J. Control Optim. 20(2), 221–246 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Bertsekas, D.P. Gafni, E.M.: Convergence of a gradient projection method. Report P-121, Laboratory for Information and Decision Systems Report, Laboratory for Information and Decision Systems, Massachusetts Institute of Technology, Cambridge, MA, (1982)Google Scholar
  10. 10.
    Brézis, H.: Problèmes Unilatéraux. PhD thesis, Sc. math. Paris VI. 1971., (1972)Google Scholar
  11. 11.
    Bui-Thanh, T., Willcox, K., Ghattas, O.: Model reduction for large-scale systems with high-dimensional parametric input space. SIAM J. Sci. Comput. 30(6), 3270–3288 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Cao, V.C., De los Reyes, J. C., Schoenlieb, C.B.: Learning optimal spatially-dependent regularization parameters in total variation image restoration. ArXiv e-prints, Mar. (2016)Google Scholar
  13. 13.
    Chan, R.H., Yang, J., Yuan, X.: Alternating direction method for image inpainting in wavelet domain. SIAM J. Imaging Sci. 4, 807–826 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Chan, T.F., Shen, J., Zhou, H.-M.: Total variation wavelet inpainting. J. Math. Imaging Vis. 25, 107–125 (2006)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Chen, K., Dong, Y., Hintermüller, M.: A nonlinear multigrid solver with line Gauss-Seidel-semismooth-Newton smoother for the Fenchel pre-dual in total variation based image restoration. Inverse Probl. Imaging 5(2), 323–339 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Chipot, M.: Variational Inequalities and Flow in Porous Media. Springer, New York (1984)CrossRefzbMATHGoogle Scholar
  17. 17.
    Combettes, P.L., Pesquet, J.-C.: Proximal splitting methods in signal processing. In: Fixed-point algorithms for inverse problems in science and engineering, volume 49 of Springer Optim. Appl., pp.185–212. Springer, New York, (2011)Google Scholar
  18. 18.
    De los Reyes, J.C., Schönlieb, C.-B., Valkonen, T.: Bilevel parameter learning for higher-order total variation regularisation models. Journal of Mathematical Imaging and Vision, pages 1–25, (2016)Google Scholar
  19. 19.
    De Los Reyes, J.C., Schönlieb, C.-B., Valkonen, T.: The structure of optimal parameters for image restoration problems. J. Math. Anal. Appl. 434(1), 464–500 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Deledalle, C.-A., Vaiter, S., Fadili, J., Peyré, G.: Stein Unbiased GrAdient estimator of the Risk (SUGAR) for multiple parameter selection. SIAM J. Imaging Sci. 7(4), 2448–2487 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Dong, Y., Hintermüller, M., Rincon-Camacho, M.: Automated regularization parameter selection in multi-scale total variation models for image restoration. J. Math. Imaging Vis. 40(1), 82–104 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Dong, Y., Hintermüller, M., Rincon-Camacho, M.: A multi-scale vectorial l\(^{\tau }\)-TV framework for color image restoration. Int. J. Comput. Vis. 92(3), 296–307 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Dong, Y., Hintermüller, M., Rincon-Camacho, M.M.: Automated regularization parameter selection in multi-scale total variation models for image restoration. J. Math. Imaging Vis. 40(1), 82–104 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Frick, K., Marnitz, P., Munk, A.: Statistical multiresolution Dantzig estimation in imaging: fundamental concepts and algorithmic framework. Electron. J. Stat. 6, 231–268 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Grisvard, P.: Elliptic Problems in Nonsmooth Domains. Volume 24 of Monographs and Studies in Mathematics. Pitman. Advanced Publishing Program, Boston, MA (1985)Google Scholar
  26. 26.
    Gumbel, E.: Les valeurs extrêmes des distributions statistiques. Ann. Inst. H. Poincaré 5(2), 115–158 (1935)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Gumbel, E.J.: Statistics of extremes. Dover Publications, Inc., Mineola, NY, 2004. Reprint of the 1958 original [Columbia University Press, New York; MR0096342]Google Scholar
  28. 28.
    Haber, E., Tenorio, L.: Learning regularization functionals—a supervised training approach. Inverse Probl. 19(3), 611–626 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Hintermüller, M., Kopacka, I.: Mathematical programs with complementarity constraints in function space: \(C\)- and strong stationarity and a path-following algorithm. SIAM J. Optim. 20(2), 868–902 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Hintermüller, M., Kunisch, K.: Path-following methods for a class of constrained minimization problems in function space. SIAM J. Optim. 17(1), 159–187 (2006). (electronic)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Hintermüller, M., Rautenberg, C.N.: Optimal selection of the regularization function in a generalized total variation model. Part I: Modelling and theory. WIAS Preprint No. 2235, (2016)Google Scholar
  32. 32.
    Hintermüller, M., Rautenberg, C.N.: On the density of classes of closed convex sets with pointwise constraints in Sobolev spaces. J. Math. Anal. Appl. 426(1), 585–593 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Hintermüller, M., Rincon-Camacho, M.: Expected absolute value estimators for a spatially adapted regularization parameter choice rule in L1-TV-based image restoration. Inverse Probl. 26(8), 085005 (2010)CrossRefzbMATHGoogle Scholar
  34. 34.
    Hintermüller, M., Surowiec, T.M., Mordukhovich, B.S.: Several approaches for the derivation of stationarity conditions for elliptic MPECs with upper-level control constraints. Math. Program. 146(1–2), 555–582 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Hintermüller, M., Wu, T.: Bilevel optimization for calibrating point spread functions in blind deconvolution. Inverse Probl. Imaging 9(4), 1139–1169 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Hinze, M., Pinnau, R., Ulbrich, M., Ulbrich, S.: Optimization with PDE Constraints. Mathematical Modelling: Theory and Applications, volume 23. Springer, New York (2009)zbMATHGoogle Scholar
  37. 37.
    Hotz, T., Marnitz, P., Stichtenroth, R., Davies, L., Kabluchko, Z., Munk, A.: Locally adaptive image denoising by a statistical multiresolution criterion. Comput. Stat. Data Anal. 56(3), 543–558 (2012)MathSciNetzbMATHGoogle Scholar
  38. 38.
    Jalalzai, K.: Regularization of inverse problems in image processing. Ph.D. thesis, Ecole Polytechnique (2012)Google Scholar
  39. 39.
    Kinderlehrer, D., Stampacchia, G.: An introduction to Variational Inequalities and Their Applications. SIAM, Philadelphia (2000)CrossRefzbMATHGoogle Scholar
  40. 40.
    Kunisch, K., Pock, T.: A bilevel optimization approach for parameter learning in variational models. SIAM J. Imaging Sci. 6, 938–983 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Luo, T., Pang, J.-S., Ralph, D.: Mathematical Programs with Equilibrum Constraints. Cambridge University Press, Cambridge (1996)CrossRefGoogle Scholar
  42. 42.
    Nittka, R.: Elliptic and parabolic problems with Robin boundary conditions on Lipschitz domains. Ph.D. thesis, Universität Ulm (2010)Google Scholar
  43. 43.
    Nittka, R.: Quasilinear elliptic and parabolic Robin problems on Lipschitz domains. NoDEA Nonlinear Differ. Equ. Appl. 20(3), 1125–1155 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Outrata, J., Kocvara, M., Zowe, J.: Nonsmooth Approach to Optimization Problems with Equilibrium Constraints. Nonconvex Optimization and its Applications, vol. 28. Kluwer Academic, Dordrecht (1998)CrossRefzbMATHGoogle Scholar
  45. 45.
    Pesquet, J.-C., Benazza-Benyahia, A., Chaux, C.: A SURE approach for digital signal/image deconvolution problems. IEEE Trans. Signal Process. 57(12), 4616–4632 (2009)MathSciNetCrossRefGoogle Scholar
  46. 46.
    Rodrigues, J.F.: Obstacle Problems in Mathematical Physics. North-Holland, Amsterdam (1987)zbMATHGoogle Scholar
  47. 47.
    Schönlieb, C., De Los Reyes, J.C.: Image denoising: learning noise distribution via PDE-constrained optimisation. Inverse Probl. Imaging 7(4), 1183–1214 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    Serrin, J.: Local behavior of solutions of quasi-linear equations. Acta Math. 111, 247–302 (1964)MathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    Showalter, R.E.: Hilbert Space Methods for Partial Differential Equations. (Monographs and Studies in Mathematics.). Pitman, London (1977)zbMATHGoogle Scholar
  50. 50.
    Showalter, R.E.: Monotone Operators in Banach Space and Nonlinear Partial Differential Equations. American Mathematical Society, Providence (1997)zbMATHGoogle Scholar
  51. 51.
    Tröltzsch, F.: Optimal Control of Partial Differential Equations: Theory, Methods and Applications. Volume 112 of Graduate Studies in Mathematics. American Mathematical Society, Providence (2010). Translated from the 2005 German original by Jürgen SprekelsCrossRefGoogle Scholar
  52. 52.
    Zowe, J., Kurcyusz, S.: Regularity and stability for the mathematical programming problem in Banach spaces. Appl. Math. Optim. 5(1), 49–62 (1979)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Department of MathematicsHumboldt-University of BerlinBerlinGermany
  2. 2.Department of MathematicsUniversity of StuttgartStuttgartGermany

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