Solution-Driven Adaptive Total Variation Regularization

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9087)


We consider solution-driven adaptive variants of Total Variation, in which the adaptivity is introduced as a fixed point problem. We provide existence theory for such fixed points in the continuous domain. For the applications of image denoising, deblurring and inpainting, we provide experiments which demonstrate that our approach in most cases outperforms state-of-the-art regularization approaches.


Regularization Inverse problems Adaptive total variation Solution-driven adaptivity Fixed point problems Image restoration 


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  1. 1.
    Agarwal, R., O’Regan, D.: Fixed-point theory for weakly sequentially upper-semicontinuous maps with applications to differential inclusions. Nonlinear Oscillations 5(3) (2002)Google Scholar
  2. 2.
    Alt, H.W.: Linear functional analysis. An application oriented introduction. Springer (2006)Google Scholar
  3. 3.
    Berkels, B., Burger, M., Droske, M., Nemitz, O., Rumpf, M.: Cartoon extraction based on anisotropic image classification. In: VMV (2006)Google Scholar
  4. 4.
    Bredies, K., Kunisch, K., Pock, T.: Total Generalized Variation. SIAM J. Imaging Sciences 3(3), 492–526 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 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)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Dabov, K., Foi, A., Katkovnik, V., Egiazarian, K., et al.: BM3D image denoising with shape-adaptive principal component analysis. In: SPARS (2009)Google Scholar
  7. 7.
    Dong, Y., Hintermüller, M., Rincon-Camacho, M.M.: Automated regularization parameter selection in multi-scale total variation models for image restoration. Journal of Mathematical Imaging and Vision 40(1), 82–104 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Estellers, V., Soato, S., Bresson, X.: Adaptive regularization with the structure tensor. Technical report, UCLA VisionLab (2014)Google Scholar
  9. 9.
    Evans, L.C., Gariepy, R.F.: Measure theory and fine properties of functions, vol. 5. CRC Press (1992)Google Scholar
  10. 10.
    Garcia, D.: Robust smoothing of gridded data in one and higher dimensions with missing values. Computational statistics & data analysis 54(4), 1167–1178 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Grasmair, M.: Locally adaptive total variation regularization. In: Tai, X.-C., Mørken, K., Lysaker, M., Lie, K.-A. (eds.) SVM 2009. LNCS 5567, vol. 5567, pp. 331–342. Springer, Heidelberg (2009) CrossRefGoogle Scholar
  12. 12.
    Grasmair, M., Lenzen, F.: Anisotropic total variation filtering. Applied Mathematics & Optimization 62, 323–339 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Lefkimmiatis, S., Roussos, A., Unser, M., Maragos, P.: Convex generalizations of total variation based on the structure tensor with applications to inverse problems. In: Pack, T. (ed.) SSVM 2013. LNCS, vol. 7893, pp. 48–60. Springer, Heidelberg (2013) CrossRefGoogle Scholar
  14. 14.
    Lenzen, F., Becker, F., Lellmann, J., Petra, S., Schnörr, C.: Variational Image Denoising with Adaptive Constraint Sets. In: Bruckstein, A.M., ter Haar Romeny, B.M., Bronstein, A.M., Bronstein, M.M. (eds.) SSVM 2011. LNCS, vol. 6667, pp. 206–217. Springer, Heidelberg (2012) CrossRefGoogle Scholar
  15. 15.
    Lenzen, F., Becker, F., Lellmann, J., Petra, S., Schnörr, C.: A class of Quasi-Variational Inequalities for adaptive image denoising and decomposition. Computational Optimization and Applications 54(2), 371–398 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Lenzen, F., Lellmann, J., Becker, F., Schnörr, C.: Solving Quasi-Variational Inequalities for image restoration with adaptive constraint sets. SIAM Journal on Imaging Sciences (SIIMS) 7, 2139–2174 (2014)CrossRefzbMATHGoogle Scholar
  17. 17.
    Rudin, L., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Physica D 60, 259–268 (1992)CrossRefzbMATHGoogle Scholar
  18. 18.
    Scherzer, O., Grasmair, M., Grossauer, H., Haltmeier, M., Lenzen, F.: Variational methods in imaging, vol. 167 of Applied Mathematical Sciences. Springer (2009)Google Scholar
  19. 19.
    Schmidt, U., Schelten, K., Roth, S.: Bayesian deblurring with integrated noise estimation. In: CVPR (2011)Google Scholar
  20. 20.
    Steidl, G., Teuber, T.: Anisotropic smoothing using double orientations. In: Tai, X.-C., Mørken, K., Lysaker, M., Lie, K.-A. (eds.) SVM 2009. LNCS 5567, vol. 5567, pp. 477–489. Springer, Heidelberg (2009) CrossRefGoogle Scholar
  21. 21.
    Wang, Z., Bovik, A., Sheikh, H., Simoncelli, E.: Image quality assessment: from error visibility to structural similarity. IEEE Transactions on Image Processing 13(4), 600–612 (2004)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.HCI & IPA, University of HeidelbergHeidelbergGermany

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