Abstract
In this work we consider the regularization of vectorial data such as color images. Based on the observation that edge alignment across image channels is a desirable prior for multichannel image restoration, we propose a novel scheme of minimizing the rank of the image Jacobian and extend this idea to second derivatives in the framework of total generalized variation. We compare the proposed convex and nonconvex relaxations of the rank function based on the Schatten-q norm to previous color image regularizers and show in our numerical experiments that they have several desirable properties. In particular, the nonconvex relaxations lead to better preservation of discontinuities. The efficient minimization of energies involving nonconvex and nonsmooth regularizers is still an important open question. We extend a recently proposed primal-dual splitting approach for nonconvex optimization and show that it can be effectively used to minimize such energies. Furthermore, we propose a novel algorithm for efficiently evaluating the proximal mapping of the ℓq norm appearing during optimization. We experimentally verify convergence of the proposed optimization method and show that it performs comparably to sequential convex programming.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Rudin, L.I., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Physica D 60, 259–268 (1992)
Chambolle, A., Caselles, V., Cremers, D., Novaga, M., Pock, T.: An introduction to total variation for image analysis. In: Theoretical Foundations and Numerical Methods for Sparse Recovery. De Gruyter (2010)
Attouch, H., Buttazzo, G., Michaille, G.: Variational Analysis in Sobolev and BV Spaces: Applications to PDEs and Optimization. Mps-Siam Series on Optimization, vol. 6. SIAM (2005)
Blomgren, P., Chan, T.F.: Color TV: Total variation methods for restoration of vector valued images. IEEE Trans. Image Processing 7, 304–309 (1998)
Sapiro, G., Ringach, D.: Anisotropic diffusion of multivalued images with applications to color filtering. IEEE Trans. Img. Proc. 5(11), 1582–1586 (1996)
Bresson, X., Chan, T.F.: Fast dual minimization of the vectorial total variation norm and applications to color image processing. Inverse Problems and Imaging 2(4), 255–284 (2008)
Condat, C.: Joint demosaicking and denoising by total variation minimization. In: IEEE Conference on Image Processing, pp. 2781–2784 (2012)
Miyata, T., Sakai, Y.: Vectorized total variation defined by weighted l infinity norm for utilizing inter channel dependency. In: 2012 19th IEEE International Conference on Image Processing (ICIP), pp. 3057–3060 (September 2012)
Goldluecke, B., Cremers, D.: An approach to vectorial total variation based on geometric measure theory. In: IEEE Conference on Computer Vision and Pattern Recognition (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)
Ehrhardt, M.J., Arridge, S.: Vector-valued image processing by parallel level sets. IEEE Trans. on Image Processing 23, 9–18 (2014)
Moeller, M., Brinkmann, E., Burger, M., Seybold, T.: Color bregman tv Preprint. On ArXiv, http://arxiv.org/abs/1310.3146
Huang, J., Mumford, D.: Statistics of natural images and models. In: Int. Conf. on Computer Vision and Pattern Recognition (CVPR) (1999)
Krishnan, D., Fergus, R.: Fast Image Deconvolution using Hyper-Laplacian Priors. In: Proc. Neural Information Processing Systems, pp. 1033–1041 (2009)
Bredies, K., Kunisch, K., Pock, T.: Total generalized variation. SIAM J. Img. Sci. 3(3), 492–526 (2010)
Ochs, P., Dosovitskiy, A., Pock, T., Brox, T.: An iterated L1 Algorithm for Non-smooth Non-convex Optimization in Computer Vision. In: IEEE Conference on Computer Vision and Pattern Recognition (CVPR) (2013)
Bredies, K.: Recovering piecewise smooth multichannel images by minimization of convex functionals with total generalized variation penalty. In: Bruhn, A., Pock, T., Tai, X.-C. (eds.) Global Optimization Methods. LNCS, vol. 8293, pp. 44–77. Springer, Heidelberg (2014)
Pock, T., Cremers, D., Bischof, H., Chambolle, A.: An algorithm for minimizing the piecewise smooth Mumford-Shah functional. In: IEEE Int. Conf. on Comp. Vis. (ICCV) (2009)
Esser, E., Zhang, X., Chan, T.: A general framework for a class of first order primal-dual algorithms for convex optimization in imaging science. SIAM J. Img. Sci. 3(4), 1015–1046 (2010)
Chambolle, A., Pock, T.: A first-order primal-dual algorithm for convex problems with applications to imaging. J. Math. Imaging Vis. 40, 120–145 (2011)
Strekalovskiy, E., Cremers, D.: Real-Time Minimization of the Piecewise Smooth Mumford-Shah Functional. In: Proceedings of the European Conference on Computer Vision (2014)
Storath, M., Weinmann, A., Demaret, L.: Jump-sparse and sparse recovery using potts functionals. CoRR, pp. 1–1 (2013)
Möllenhoff, T., Strekalovskiy, E., Möller, M., Cremers, D.: The Primal-Dual Hybrid Gradient Method for Semiconvex Splittings (preprint, 2014), http://arxiv.org/abs/1407.1723
Ochs, P., Chen, Y., Brox, T., Pock, T.: iPiano: Inertial Proximal Algorithm for Non-convex Optimization. SIAM Journal on Imaging Sciences (SIIMS) (Preprint, 2014)
Bouaziz, S., Tagliasacchi, A., Pauly, M.: Sparse Iterative Closest Point. Computer Graphics Forum (Symposium on Geometry Processing) 32(5), 1–11 (2013)
McKelvey, J.P.: Simple transcendental expressions for the roots of cubic equations. Amer. J. Phys. 52(3), 269–270 (1984)
Mirsky, L.: Symmetric gauge functions and unitarily invariant norms. Quart. J. Math. Oxford Ser. (2), 50–59 (1960)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this paper
Cite this paper
Möllenhoff, T., Strekalovskiy, E., Moeller, M., Cremers, D. (2015). Low Rank Priors for Color Image Regularization. In: Tai, XC., Bae, E., Chan, T.F., Lysaker, M. (eds) Energy Minimization Methods in Computer Vision and Pattern Recognition. EMMCVPR 2015. Lecture Notes in Computer Science, vol 8932. Springer, Cham. https://doi.org/10.1007/978-3-319-14612-6_10
Download citation
DOI: https://doi.org/10.1007/978-3-319-14612-6_10
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-14611-9
Online ISBN: 978-3-319-14612-6
eBook Packages: Computer ScienceComputer Science (R0)