Bregmanized Domain Decomposition for Image Restoration
Computational problems of large-scale data are gaining attention recently due to better hardware and hence, higher dimensionality of images and data sets acquired in applications. In the last couple of years non-smooth minimization problems such as total variation minimization became increasingly important for the solution of these tasks. While being favorable due to the improved enhancement of images compared to smooth imaging approaches, non-smooth minimization problems typically scale badly with the dimension of the data. Hence, for large imaging problems solved by total variation minimization domain decomposition algorithms have been proposed, aiming to split one large problem into N>1 smaller problems which can be solved on parallel CPUs. The N subproblems constitute constrained minimization problems, where the constraint enforces the support of the minimizer to be the respective subdomain.
In this paper we discuss a fast computational algorithm to solve domain decomposition for total variation minimization. In particular, we accelerate the computation of the subproblems by nested Bregman iterations. We propose a Bregmanized Operator Splitting–Split Bregman (BOS-SB) algorithm, which enforces the restriction onto the respective subdomain by a Bregman iteration that is subsequently solved by a Split Bregman strategy. The computational performance of this new approach is discussed for its application to image inpainting and image deblurring. It turns out that the proposed new solution technique is up to three times faster than the iterative algorithm currently used in domain decomposition methods for total variation minimization.
KeywordsDomain decomposition Bregman distance Iterative Bregman algorithms Image restoration Total variation
Andreas Langer acknowledges the financial support provided by the FWF project Y 432-N15 START-Preis Sparse Approximation and Optimization in High Dimensions and the project WWTF Five senses-Call 2006, Mathematical Methods for Image Analysis and Processing in the Visual Arts. Stanley Osher acknowledges the NSF grants DMS0835863 and DMS0914561, the ONR grant N000140910360, and ARO Muri subs from Rice University and the University of South Carolina. Carola-Bibiane Schönlieb acknowledges the financial support provided by the Cambridge Centre for Analysis (CCA), the DFG Graduiertenkolleg 1023 Identification in Mathematical Models: Synergy of Stochastic and Numerical Methods and the Royal Society International Exchanges Award IE110314. Further, this publication is based on work supported by Award No. KUK-I1-007-43, made by King Abdullah University of Science and Technology (KAUST).
- 2.Aubert, G., Kornprobst, P.: Mathematical Problems in Image Processing. Partial Differential Equations and the Calculus of Variation. Springer, Berlin (2002) Google Scholar
- 11.Darbon, J., Sigelle, M.: A fast and exact algorithm for total variation minimization. In: IbPRIA 2005. Lecture Notes in Computer Science, vol. 3522, pp. 717–765 (2005) Google Scholar
- 13.Engl, HW., Hanke, M., Neubauer, A.: Regularization of inverse problems. In: Mathematics and Its Applications (Dordrecht), vol. 375. Kluwer Academic Publishers, Dordrecht (1996) Google Scholar
- 22.Langer, A.: Convergence analysis of a non-overlapping domain decomposition algorithm for total variation minimization. J. Comput. Appl. Math. (2009, submitted) Google Scholar
- 23.Langer, A.: Subspace correction and domain decomposition methods for total variation minimization. PhD Thesis, Johannes Kepler University Linz (2011) Google Scholar
- 28.Schönlieb, C.-B.: Total variation minimization with an H −1 constraint. In: Singularities in Nonlinear Evolution Phenomena and Applications Proceedings, Scuola Normale Superiore Pisa. CRM Series, vol. 9, pp. 201–232 (2009) Google Scholar
- 29.Setzer, S.: Split Bregman algorithm, Douglas–Rachford splitting and frame shrinkage. In: Tai, X.-C., et al. (eds.) Proceedings of the Second International Conference on Scale Space Methods and Proceedings of the 2nd International Conference on Scale Space and Variational Methods in Computer Vision. LNCS, vol. 5567, pp. 464–476. Springer, Berlin (2009) CrossRefGoogle Scholar