Abstract
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.
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References
Agarwal, R., O’Regan, D.: Fixed-point theory for weakly sequentially upper-semicontinuous maps with applications to differential inclusions. Nonlinear Oscillations 5(3) (2002)
Alt, H.W.: Linear functional analysis. An application oriented introduction. Springer (2006)
Berkels, B., Burger, M., Droske, M., Nemitz, O., Rumpf, M.: Cartoon extraction based on anisotropic image classification. In: VMV (2006)
Bredies, K., Kunisch, K., Pock, T.: Total Generalized Variation. SIAM J. Imaging Sciences 3(3), 492–526 (2010)
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)
Dabov, K., Foi, A., Katkovnik, V., Egiazarian, K., et al.: BM3D image denoising with shape-adaptive principal component analysis. In: SPARS (2009)
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)
Estellers, V., Soato, S., Bresson, X.: Adaptive regularization with the structure tensor. Technical report, UCLA VisionLab (2014)
Evans, L.C., Gariepy, R.F.: Measure theory and fine properties of functions, vol. 5. CRC Press (1992)
Garcia, D.: Robust smoothing of gridded data in one and higher dimensions with missing values. Computational statistics & data analysis 54(4), 1167–1178 (2010)
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)
Grasmair, M., Lenzen, F.: Anisotropic total variation filtering. Applied Mathematics & Optimization 62, 323–339 (2010)
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)
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)
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)
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)
Rudin, L., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Physica D 60, 259–268 (1992)
Scherzer, O., Grasmair, M., Grossauer, H., Haltmeier, M., Lenzen, F.: Variational methods in imaging, vol. 167 of Applied Mathematical Sciences. Springer (2009)
Schmidt, U., Schelten, K., Roth, S.: Bayesian deblurring with integrated noise estimation. In: CVPR (2011)
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)
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)
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Lenzen, F., Berger, J. (2015). Solution-Driven Adaptive Total Variation Regularization. In: Aujol, JF., Nikolova, M., Papadakis, N. (eds) Scale Space and Variational Methods in Computer Vision. SSVM 2015. Lecture Notes in Computer Science(), vol 9087. Springer, Cham. https://doi.org/10.1007/978-3-319-18461-6_17
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DOI: https://doi.org/10.1007/978-3-319-18461-6_17
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