Abstract
Domain decomposition is one of the most efficient techniques to derive efficient methods for large-scale problems. In this chapter such decomposition methods for the minimization of the total variation are discussed. We differ between approaches which directly tackle the (primal) total variation minimization and approaches which deal with their predual formulation. Thereby we mainly concentrate on the presentation of domain decomposition methods which guarantee to converge to a solution of the global problem.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Acar, R., Vogel, C.R.: Analysis of bounded variation penalty methods for ill-posed problems. Inverse Probl. 10(6), 1217–1229 (1994)
Alliney, S.: A property of the minimum vectors of a regularizing functional defined by means of the absolute norm. IEEE Trans. Signal Process. 45(4), 913–917 (1997)
Ambrosio, L., Fusco, N., Pallara, D.: Functions of Bounded Variation and Free Discontinuity Problems. Oxford Mathematical Monographs. The Clarendon Press/Oxford University Press, New York (2000)
Attouch, H., Buttazzo, G., Michaille, G.: Variational Analysis in Sobolev and BV Spaces. MOS-SIAM Series on Optimization, 2nd edn. Society for Industrial and Applied Mathematics (SIAM)/Mathematical Optimization Society, Philadelphia (2014). Applications to PDEs and optimization
Aubert, G., Aujol, J.-F.: A variational approach to removing multiplicative noise. SIAM J. Appl. Math. 68(4), 925–946 (2008)
Beck, A., Teboulle, M.: A fast iterative shrinkage-thresholding algorithm for linear inverse problems. SIAM J. Imaging Sci. 2(1), 183–202 (2009)
Bertsekas, D.P.: Constrained Optimization and Lagrange Multiplier Methods. Academic Press, New York (2014)
Burger, M., Sawatzky, A., Steidl, G.: First order algorithms in variational image processing. In: Splitting Methods in Communication, Imaging, Science, and Engineering. Scientific Computation, pp. 345–407. Springer, Cham (2016)
Cai, J.-F., Chan, R.H., Nikolova, M.: Two-phase approach for deblurring images corrupted by impulse plus Gaussian noise. Inverse Probl. Imaging 2(2), 187–204 (2008)
Calatroni, L., De Los Reyes, J.C., Schönlieb, C.-B.: Infimal convolution of data discrepancies for mixed noise removal. SIAM J. Imaging Sci. 10(3), 1196–1233 (2017)
Carstensen, C.: Domain decomposition for a non-smooth convex minimization problem and its application to plasticity. Numer. Linear Algebra Appl. 4(3), 177–190 (1997)
Chambolle, A.: An algorithm for total variation minimization and applications. J. Math. Imaging Vis. 20(1–2), 89–97 (2004). Special issue on mathematics and image analysis
Chambolle, A., Pock, T.: A First-order Primal-dual Algorithm for Convex Problems with Applications to Imaging. J. Math. Imaging Vis. 40(1), 120–145 (2011)
Chambolle, A., Pock, T.: A remark on accelerated block coordinate descent for computing the proximity operators of a sum of convex functions. SMAI J. Comput. Math. 1, 29–54 (2015)
Chambolle, A., Pock, T.: An introduction to continuous optimization for imaging. Acta Numer. 25, 161–319 (2016)
Chambolle, A., Caselles, V., Cremers, D., Novaga, M., Pock, T.: An introduction to total variation for image analysis. Theor. Found. Numer. Methods Sparse Recovery 9, 263–340 (2010)
Chan, T.F., Mathew, T.P.: Domain decomposition algorithms. In: Acta Numerica, pp. 61–143. Cambridge University Press, Cambridge (1994)
Chan, T.F., Shen, J.J.: Image Processing and Analysis: Variational, PDE, Wavelet, and Stochastic Methods. SIAM, Philadelphia (2005)
Chan, T.F., Esedoglu, S., Nikolova, M.: Algorithms for finding global minimizers of image segmentation and denoising models. SIAM J. Appl. Math. 66(5), 1632–1648 (2006)
Chang, H., Zhang, X., Tai, X.-C., Yang, D.: Domain decomposition methods for nonlocal total variation image restoration. J. Sci. Comput. 60(1), 79–100 (2014)
Chang, H., Tai, X.-C., Wang, L.-L., Yang, D.: Convergence rate of overlapping domain decomposition methods for the Rudin–Osher–Fatemi model based on a dual formulation. SIAM J. Imaging Sci. 8(1), 564–591 (2015)
Chen, K., Tai, X.-C.: A nonlinear multigrid method for total variation minimization from image restoration. J. Sci. Comput. 33(2), 115–138 (2007)
Combettes, P.L., Wajs, V.R.: Signal recovery by proximal forward-backward splitting. Multiscale Model. Simul. 4(4), 1168–1200 (electronic) (2005)
Daubechies, I., Defrise, M., De Mol, C.: An iterative thresholding algorithm for linear inverse problems with a sparsity constraint. Commun. Pure Appl. Math. 57(11), 1413–1457 (2004)
Daubechies, I., Teschke, G., Vese, L.: Iteratively solving linear inverse problems under general convex constraints. Inverse Probl. Imaging 1(1), 29–46 (2007)
Dolean, V., Jolivet, P., Nataf, F.: An Introduction to Domain Decomposition Methods: Algorithms, Theory, and Parallel Implementation, vol. 144. SIAM, Philadelphia (2015)
Duan, Y., Tai, X.-C.: Domain decomposition methods with graph cuts algorithms for total variation minimization. Adv. Comput. Math. 36(2), 175–199 (2012)
Duan, Y., Chang, H., Tai, X.-C.: Convergent non-overlapping domain decomposition methods for variational image segmentation. J. Sci. Comput. 69(2), 532–555 (2016)
Fornasier, M.: Domain decomposition methods for linear inverse problems with sparsity constraints. Inverse Probl. Int. J. Theory Pract. Inverse Probl. Inverse Methods Comput. Inversion Data 23(6), 2505–2526 (2007)
Fornasier, M., Schönlieb, C.-B.: Subspace correction methods for total variation and l1-minimization. SIAM J. Numer. Anal. 47(5), 3397–3428 (2009)
Fornasier, M., Langer, A., Schönlieb, C.-B.: Domain decomposition methods for compressed sensing. In: Proceedings of the International Conference of SampTA09, Marseilles, arXiv preprint arXiv:0902.0124 (2009)
Fornasier, M., Langer, A., Schönlieb, C.-B.: A convergent overlapping domain decomposition method for total variation minimization. Numerische Mathematik 116(4), 645–685 (2010)
Fornasier, M., Kim, Y., Langer, A., Schönlieb, C.: Wavelet decomposition method for L2/TV-image deblurring. SIAM J. Imaging Sci. 5(3), 857–885 (2012)
Getreuer, P., Tong, M., Vese, L.A.: A variational model for the restoration of mr images corrupted by blur and rician noise. In: International Symposium on Visual Computing, pp. 686–698. Springer (2011)
Gilboa, G., Osher, S.: Nonlocal operators with applications to image processing. Multiscale Model. Simul. 7(3), 1005–1028 (2009)
Giusti, E.: Minimal Surfaces and Functions of Bounded Variation. Monographs in Mathematics, vol. 80. Birkhäuser Verlag, Basel (1984)
Goldstein, T., Osher, S.: The split Bregman method for L1-regularized problems. SIAM J. Imaging Sci. 2(2), 323–343 (2009)
Hintermüller, M., Kunisch, K.: Total bounded variation regularization as a bilaterally constrained optimization problem. SIAM J. Appl. Math. 64(4), 1311–1333 (2004)
Hintermüller, M., Langer, A.: Subspace correction methods for a class of nonsmooth and nonadditive convex variational problems with mixed L1∕L2 data-fidelity in image processing. SIAM J. Imaging Sci. 6(4), 2134–2173 (2013)
Hintermüller, M., Langer, A.: Surrogate functional based subspace correction methods for image processing. In: Domain Decomposition Methods in Science and Engineering XXI, pp. 829–837. Springer, Cham (2014)
Hintermüller, M., Langer, A.: Non-overlapping domain decomposition methods for dual total variation based image denoising. J. Sci. Comput. 62(2), 456–481 (2015)
Hintermüller, M., Rautenberg, C.: On the density of classes of closed convex sets with pointwise constraints in sobolev spaces. J. Math. Anal. Appl. 426(1), 585–593 (2015)
Hintermüller, M., Rautenberg, C.N.: Optimal selection of the regularization function in a weighted total variation model. Part I: Modelling and theory. J. Math. Imaging Vis. 59(3), 498–514 (2017)
Ito, K., Kunisch, K.: Lagrange Multiplier Approach to Variational Problems and Applications, vol. 15. SIAM, Philadelphia (2008)
Langer, A.: Automated parameter selection for total variation minimization in image restoration. J. Math. Imaging Vis. 57(2), 239–268 (2017a)
Langer, A.: Automated parameter selection in the L1-L2-TV model for removing Gaussian plus impulse noise. Inverse Probl. 33(7), 74002 (2017b)
Langer, A.: Locally adaptive total variation for removing mixed Gaussian–impulse noise. Int. J. Comput. Math. 96(2), 298–316 (2019)
Langer, A., Gaspoz, F.: Overlapping domain decomposition methods for total variation denoising. SIAM J. Numer. Anal. 57(3), 1411–1444 (2019)
Langer, A., Osher, S., Schönlieb, C.-B.: Bregmanized domain decomposition for image restoration. J. Sci. Comput. 54(2–3), 549–576 (2013)
Le, T., Chartrand, R., Asaki, T.J.: A variational approach to reconstructing images corrupted by poisson noise. J. Math. Imaging Vis. 27(3), 257–263 (2007)
Lee, C.-O., Nam, C.: Primal domain decomposition methods for the total variation minimization, based on dual decomposition. SIAM J. Sci. Comput. 39(2), B403–B423 (2017)
Lee, C.-O., Park, J.: Fast nonoverlapping block Jacobi method for the dual Rudin–Osher–Fatemi model. SIAM J. Imaging Sci. 12(4), 2009–2034 (2019a)
Lee, C.-O., Park, J.: A finite element nonoverlapping domain decomposition method with lagrange multipliers for the dual total variation minimizations. J. Sci. Comput. 81(3), 2331–2355 (2019b)
Lee, C.-O., Lee, J.H., Woo, H., Yun, S.: Block decomposition methods for total variation by primal–dual stitching. J. Sci. Comput. 68(1), 273–302 (2016)
Lee, C.-O., Nam, C., Park, J.: Domain decomposition methods using dual conversion for the total variation minimization with L1 fidelity term. J. Sci. Comput. 78(2), 951–970 (2019a)
Lee, C.-O., Park, E.-H., Park, J.: A finite element approach for the dual Rudin–Osher–Fatemi model and its nonoverlapping domain decomposition methods. SIAM J. Sci. Comput. 41(2), B205–B228 (2019b)
Lions, J.L.: Optimal Control of Systems Governed by Partial Differential Equations. Die Grundlehren der mathematischen Wissenschaften, vol. 170. Springer (1971)
Lions, P.-L.: On the Schwarz alternating method. I. In: First International Symposium on Domain Decomposition Methods for Partial Differential Equations, Paris, pp. 1–42 (1988)
Marini, L.D., Quarteroni, A.: A relaxation procedure for domain decomposition methods using finite elements. Numerische Mathematik 55(5), 575–598 (1989)
Mathew, T.: Domain Decomposition Methods for the Numerical Solution of Partial Differential Equations, vol. 61. Springer Science & Business Media, Berlin (2008)
Nikolova, M.: Minimizers of cost-functions involving nonsmooth data-fidelity terms. Application to the processing of outliers. SIAM J. Numer. Anal. 40(3), 965–994 (electronic) (2002)
Nikolova, M.: A variational approach to remove outliers and impulse noise. J. Math. Imaging Vis. 20(1–2), 99–120 (2004)
Peyré, G., Bougleux, S., Cohen, L.: Non-local regularization of inverse problems. In: European Conference on Computer Vision, pp. 57–68. Springer (2008)
Pock, T., Unger, M., Cremers, D., Bischof, H.: Fast and exact solution of total variation models on the gpu. In: 2008 IEEE Computer Society Conference on Computer Vision and Pattern Recognition Workshops, pp. 1–8. IEEE (2008)
Quarteroni, A., Valli, A.: Domain Decomposition Methods for Partial Differential Equations. Oxford University Press, New York (1999)
Raviart, P.-A., Thomas, J.-M.: A mixed finite element method for 2-nd order elliptic problems. In: Mathematical Aspects of Finite Element Methods, pp. 292–315. Springer, Berlin (1977)
Rudin, L.I., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Phys. D: Nonlinear Phenom. 60(1), 259–268 (1992)
Schönlieb, C.-B.: Total variation minimization with an H−1 constraint. CRM Ser. 9, 201–232 (2009)
Schwarz, H.A.: Über einige Abbildungsaufgaben. Journal für die reine und angewandte Mathematik 1869(70), 105–120 (1869)
Smith, B., Bjorstad, P., Gropp, W.: Domain Decomposition: Parallel Multilevel Methods for Elliptic Partial Differential Equations. Cambridge University Press, Dordrecht (2004)
Tai, X.-C.: Rate of convergence for some constraint decomposition methods for nonlinear variational inequalities. Numerische Mathematik 93(4), 755–786 (2003)
Tai, X.-C., Tseng, P.: Convergence rate analysis of an asynchronous space decomposition method for convex minimization. Math. Comput. 71(239), 1105–1135 (2002)
Tai, X.-C., Xu, J.: Global and uniform convergence of subspace correction methods for some convex optimization problems. Math. Comput. 71(237), 105–124 (2002)
Toselli, A., Widlund, O.: Domain Decomposition Methods: Algorithms and Theory, vol. 34. Springer Science & Business Media, Dordrecht (2006)
Tseng, P.: Convergence of a block coordinate descent method for nondifferentiable minimization. J. Optim. Theory Appl. 109(3), 475–494 (2001)
Tseng, P., Yun, S.: A coordinate gradient descent method for nonsmooth separable minimization. Math. Prog. 117(1–2), 387–423 (2009)
Vonesch, C., Unser, M.: A fast multilevel algorithm for wavelet-regularized image restoration. IEEE Trans. Image Process. 18(3), 509–523 (2009)
Warga, J.: Minimizing Certain Concex Functions. J. Soc. Indust. Appl. Math. 11, 588–593 (1963)
Wright, S.J.: Coordinate descent algorithms. Math. Prog. 151(1), 3–34 (2015)
Wu, C., Tai, X.-C.: Augmented lagrangian method, dual methods, and split bregman iteration for rof, vectorial tv, and high order models. SIAM J. Imaging Sci. 3(3), 300–339 (2010)
Xu, J., Tai, X.-C., Wang, L.-L.: A two-level domain decomposition method for image restoration. Inverse Probl. Imaging 4(3), 523–545 (2010)
Xu, J., Chang, H.B., Qin, J.: Domain decomposition method for image deblurring. J. Comput. Appl. Math. 271, 401–414 (2014)
Zhang, X., Burger, M., Bresson, X., Osher, S.: Bregmanized nonlocal regularization for deconvolution and sparse reconstruction. SIAM J. Imaging Sci. 3(3), 253–276 (2010)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2023 Springer Nature Switzerland AG
About this entry
Cite this entry
Langer, A. (2023). Domain Decomposition for Non-smooth (in Particular TV) Minimization. In: Chen, K., Schönlieb, CB., Tai, XC., Younes, L. (eds) Handbook of Mathematical Models and Algorithms in Computer Vision and Imaging. Springer, Cham. https://doi.org/10.1007/978-3-030-98661-2_104
Download citation
DOI: https://doi.org/10.1007/978-3-030-98661-2_104
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-98660-5
Online ISBN: 978-3-030-98661-2
eBook Packages: Mathematics and StatisticsReference Module Computer Science and Engineering