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Operator splittings, Bregman methods and frame shrinkage in image processing. (English) Zbl 1235.68314
Summary: We examine the underlying structure of popular algorithms for variational methods used in image processing. We focus here on operator splittings and Bregman methods based on a unified approach via fixed point iterations and averaged operators. In particular, the recently proposed alternating split Bregman method can be interpreted from different points of view-as a Bregman, as an augmented Lagrangian and as a Douglas-Rachford splitting algorithm which is a classical operator splitting method. We also study similarities between this method and the forward-backward splitting method when applied to two frequently used models for image denoising which employ a Besov-norm and a total variation regularization term, respectively. In the first setting, we show that for a discretization based on Parseval frames the gradient descent reprojection and the alternating split Bregman algorithm are equivalent and turn out to be a frame shrinkage method. For the total variation regularizer, we also present a numerical comparison with multistep methods.

MSC:
68U10 Computing methodologies for image processing
94A08 Image processing (compression, reconstruction, etc.) in information and communication theory
Software:
RecPF
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[1] Aubin, J. P., & Frankowska, H. (2009). Set-valued analysis. Boston: Birkhäuser. · Zbl 1168.49014
[2] Aujol, J. F. (2009). Some first-order algorithms for total variation based image restoration. Journal of Mathematical Imaging and Vision, 34(3), 307–327. · Zbl 1287.94012 · doi:10.1007/s10851-009-0149-y
[3] Barzilai, J., & Borwein, J. M. (1988). Two-point step size gradient methods. IMA Journal of Numerical Analysis, 8, 141–148. · Zbl 0638.65055 · doi:10.1093/imanum/8.1.141
[4] Bauschke, H. H., & Borwein, J. M. (1996). On projection algorithms for solving convex feasibility problems. SIAM Review, 38(3), 367–426. · Zbl 0865.47039 · doi:10.1137/S0036144593251710
[5] Bechler, P., DeVore, R., Kamont, A., Petrova, G., & Wojtaszczyk, P. (2006). Greedy wavelet projections are bounded on BV. Transactions of the American Mathematical Society, 359(2), 619–635. · Zbl 1134.42022 · doi:10.1090/S0002-9947-06-03903-1
[6] Beck, A., & Teboulle, M. (2009a). Fast iterative shrinkage-thresholding algorithm for linear inverse problems. SIAM Journal on Imaging Sciences, 2(1), 183–202. · Zbl 1175.94009 · doi:10.1137/080716542
[7] Beck, A., & Teboulle, M. (2009b). Fast gradient-based algorithms for constrained total variation image denoising and deblurring. IEEE Transactions on Image Processing, 18(11), 2419–2434. · Zbl 1371.94049 · doi:10.1109/TIP.2009.2028250
[8] Bioucas-Dias, J., & Figueiredo, M. (2009). Total variation restoration of speckled images using a split-Bregman algorithm. In Proceedings of the international conference on image processing, ICIP 2009, Cairo, Egypt (pp. 3717–3720). New York: IEEE.
[9] Bonnans, J. F., & Shapiro, A. (2000). Springer series in operations research : Vol. 7. Perturbation analysis of optimization problems. Berlin: Springer. · Zbl 0966.49001
[10] Borwein, J. M., & Zhu, Q. J. (2005). Techniques of variational analysis. New York: Springer. · Zbl 1076.49001
[11] Borwein, J. M., & Zhu, Q. J. (2006). Variational methods in convex analysis. Journal of Global Optimization, 35(2), 197–213. · Zbl 1103.49005 · doi:10.1007/s10898-005-3835-3
[12] Bregman, L. M. (1967). The relaxation method of finding the common point of convex sets and its application to the solution of problems in convex programming. USSR Computational Mathematics and Mathematical Physics, 7(3), 200–217. · Zbl 0186.23807 · doi:10.1016/0041-5553(67)90040-7
[13] Browder, F. E., & Petryshyn, W. V. (1966). The solution by iteration of nonlinear functional equations in Banach spaces. Bulletin of the American Mathematical Society, 72, 571–575. · Zbl 0138.08202 · doi:10.1090/S0002-9904-1966-11544-6
[14] Buades, A., Coll, B., & Morel, J. M. (2008). Nonlocal image and movie denoising. International Journal of Computer Vision, 76(2), 123–139. · Zbl 05322199 · doi:10.1007/s11263-007-0052-1
[15] Byrne, C. (2004). A unified treatment of some iterative algorithms in signal processing and image reconstruction. Inverse Problems, 20, 103–120. · Zbl 1051.65067 · doi:10.1088/0266-5611/20/1/006
[16] Cai, J. F., Chan, R. H., & Shen, Z. (2008). A framelet-based image inpainting algorithm. Applied and Computational Harmonic Analysis, 24, 131–149. · Zbl 1135.68056 · doi:10.1016/j.acha.2007.10.002
[17] Cai, J. F., Osher, S., & Shen, Z. (2009). Split Bregman methods and frame based image restoration (Technical report). UCLA Computational and Applied Mathematics. · Zbl 1189.94014
[18] Censor, Y., & Lent, A. (1981). An iterative row-action method for interval convex programming. Journal of Optimization Theory and Applications, 34(3), 321–353. · Zbl 0431.49042 · doi:10.1007/BF00934676
[19] Censor, Y., & Zenios, S. A. (1992). Proximal minimization algorithm with D-functions. Journal of Optimization Theory and Applications, 73(3), 451–464. · Zbl 0794.90058 · doi:10.1007/BF00940051
[20] Censor, Y., & Zenios, S. A. (1997). Parallel optimization: Theory, algorithms, and applications. New York: Oxford University Press. · Zbl 0945.90064
[21] Chambolle, A. (2004). An algorithm for total variation minimization and applications. Journal of Mathematical Imaging and Vision, 20(1–2), 89–97. · Zbl 1366.94056 · doi:10.1023/B:JMIV.0000011320.81911.38
[22] Chambolle, A. (2005). Total variation minimization and a class of binary MRF models. In A. Rangarajan, B. C. Vemuri, & A. L. Yuille (Eds.), Lecture notes in computer science : Vol. 3757. Energy minimization methods in computer vision and pattern recognition, EMMCVPR (pp. 136–152). Berlin: Springer.
[23] Chan, R. H., Setzer, S., & Steidl, G. (2008). Inpainting by flexible Haar-wavelet shrinkage. SIAM Journal on Imaging Science, 1, 273–293. · Zbl 1187.68649 · doi:10.1137/070711499
[24] Cohen, A., DeVore, R., Petrushev, P., & Xu, H. (1999). Nonlinear approximation and the space BV(\(\mathbb{R}\)2). American Journal of Mathematics, 121, 587–628. · Zbl 0931.41019 · doi:10.1353/ajm.1999.0016
[25] Combettes, P. L. (2004). Solving monotone inclusions via compositions of nonexpansive averaged operators. Optimization, 53(5–6), 475–504. · Zbl 1153.47305 · doi:10.1080/02331930412331327157
[26] Combettes, P. L., & Pesquet, J. C. (2007). A Douglas-Rachford splitting approach to nonsmooth convex variational signal recovery. IEEE Journal of Selected Topics in Signal Processing, 1(4), 564–574. · doi:10.1109/JSTSP.2007.910264
[27] Combettes, P. L., & Wajs, V. R. (2005). Signal recovery by proximal forward-backward splitting. Multiscale Modelling & Simulation, 4, 1168–1200. · Zbl 1179.94031 · doi:10.1137/050626090
[28] Daubechies, I., Han, B., Ron, A., & Shen, Z. (2003). Framelets: MRA-based construction of wavelet frames. Applied and Computational Harmonic Analysis, 14, 1–46. · Zbl 1035.42031 · doi:10.1016/S1063-5203(02)00511-0
[29] DeVore, R. A., & Lucier B. J. (1992). Fast wavelet techniques for near-optimal image processing. In IEEE MILCOM ’92 conf. rec. (Vol. 3, pp. 1129–1135). San Diego: IEEE Press.
[30] Dong, B., & Shen, Z. (2007). Pseudo-splines, wavelets and framelets. Applied and Computational Harmonic Analysis, 22, 78–104. · Zbl 1282.42035 · doi:10.1016/j.acha.2006.04.008
[31] Douglas, J., & Rachford, H. H. (1956). On the numerical solution of heat conduction problems in two and three space variables. Transactions of the American Mathematical Society, 82(2), 421–439. · Zbl 0070.35401 · doi:10.1090/S0002-9947-1956-0084194-4
[32] Eckstein, J. (1989). Splitting methods for monotone operators with applications to parallel optimization. PhD thesis, Massachusetts Institute of Technology.
[33] Eckstein, J. (1993). Nonlinear proximal point algorithms using Bregman functions, with applications to convex programming. Mathematics of Operations Research, 18(1), 202–226. · Zbl 0807.47036 · doi:10.1287/moor.18.1.202
[34] Eckstein, J., & Bertsekas, D. P. (1992). On the Douglas–Rachford splitting method and the proximal point algorithm for maximal monotone operators. Mathematical Programming, 55, 293–318. · Zbl 0765.90073 · doi:10.1007/BF01581204
[35] Ekeland, I., & Temam, R. (1976). Convex analysis and variational problems. Amsterdam: North-Holland. · Zbl 0322.90046
[36] Esser, E. (2009). Applications of Lagrangian-based alternating direction methods and connections to split Bregman (Technical report). UCLA Computational and Applied Mathematics.
[37] Esser, E., Zhang, X., & Chan, T. F. (2009). A general framework for a class of first order primal-dual algorithms for TV minimization (Technical report). UCLA Computational and Applied Mathematics.
[38] Figueiredo, M., & Bioucas-Dias, J. (2009). Deconvolution of Poissonian images using variable splitting and augmented Lagrangian optimization. In IEEE workshop on statistical signal processing. Cardiff.
[39] Frick, K. (2008). The augmented Lagrangian method and associated evolution equations. PhD thesis.
[40] Gabay, D. (1983). Applications of the method of multipliers to variational inequalities. In M. Fortin & R. Glowinski (Eds.), Studies in mathematics and its applications : Vol. 15. Augmented Lagrangian methods: Applications to the numerical solution of boundary-value problems (pp. 299–331). Amsterdam: North-Holland. Chap. 9.
[41] Gabay, D., & Mercier, B. (1976). A dual algorithm for the solution of nonlinear variational problems via finite element approximation. Computers & Mathematics with Applications, 2, 17–40. · Zbl 0352.65034 · doi:10.1016/0898-1221(76)90003-1
[42] Gilboa, G., & Osher, S. (2008). Nonlocal operators with applications to image processing. Multiscale Modelling & Simulation, 7(3), 1005–1028. · Zbl 1181.35006 · doi:10.1137/070698592
[43] Gilboa, G., Darbon, J., Osher, S., & Chan, T. F. (2006). Nonlocal convex functionals for image regularization (UCLA CAM Report 06-57).
[44] Glowinski, R., & Marroco, A. (1975). Sur l’approximation par éléments finis d’ordre un, et la résolution, par pénalisation-dualité d’une classe de problèmes de Dirichlet non linéaires. Revue Française d’Automatique, Informatique, Recherche Opérationnelle Analyse Numérique, 9(2), 41–76. · Zbl 0368.65053
[45] Goldstein, T., & Osher, S. (2009). The split Bregman method for L1-regularized problems. SIAM Journal on Imaging Sciences, 2(2), 323–343. · Zbl 1177.65088 · doi:10.1137/080725891
[46] Goldstein, T., Bresson, X., & Osher, S. (2009). Geometric applications of the split Bregman method: Segmentation and surface reconstruction (Technical report). UCLA Computational and Applied Mathematics. · Zbl 1203.65044
[47] Hestenes, M. R. (1969). Multiplier and gradient methods. Journal of Optimization Theory and Applications, 4, 303–320. · Zbl 0174.20705 · doi:10.1007/BF00927673
[48] Iusem, A. N. (1999). Augmented Lagrangian methods and proximal point methods for convex optimization. Investigación Operativa, 8, 11–49.
[49] Kindermann, S., Osher, S., & Jones, P. W. (2005). Deblurring and denoising of images by nonlocal functionals. Multiscale Modelling & Simulation, 4(4), 1091–1115. · Zbl 1161.68827 · doi:10.1137/050622249
[50] Kiwiel, K. C. (1997). Proximal minimization methods with generalized Bregman functions. SIAM Journal on Control and Optimization, 35(4), 1142–1168. · Zbl 0890.65061 · doi:10.1137/S0363012995281742
[51] Krasnoselskii, M. A. (1955). Two observations about the method of successive approximations. Uspekhi Matematicheskikh Nauk, 10, 123–127 (in Russian).
[52] Levenberg, K. (1944). A method for the solution of certain problems in least squares. Quarterly of Applied Mathematics, 2, 164–168. · Zbl 0063.03501
[53] Lions, P. L., & Mercier, B. (1979). Splitting algorithms for the sum of two nonlinear operators. SIAM Journal on Numerical Analysis, 16(6), 964–979. · Zbl 0426.65050 · doi:10.1137/0716071
[54] Mann, W. R. (1953). Mean value methods in iteration. Proceedings of the American Mathematical Society, 16(4), 506–510. · Zbl 0050.11603 · doi:10.1090/S0002-9939-1953-0054846-3
[55] Marquardt, D. (1963). An algorithm for least-squares estimation of nonlinear parameters. SIAM Journal of Applied Mathematics, 11, 431–441. · Zbl 0112.10505 · doi:10.1137/0111030
[56] Moreau, J. J. (1965). Proximité et dualité dans un espace hilbertien. Bulletin de la Societé Mathématique de France, 93, 273–299. · Zbl 0136.12101
[57] Mrázek, P., & Weickert, J. (2003). Rotationally invariant wavelet shrinkage. In B. Michaelis & G. Krell (Eds.), Pattern recognition (Vol. 2781, pp. 156–163). Berlin: Springer. · Zbl 1067.68758
[58] Nesterov, Y. E. (1983). A method of solving a convex programming problem with convergence rate O(1/k 2). Soviet Mathematics Doklady, 27(2), 372–376. · Zbl 0535.90071
[59] Nesterov, Y. E. (2005). Smooth minimization of non-smooth functions. Mathematical Programming, 103, 127–152. · Zbl 1079.90102 · doi:10.1007/s10107-004-0552-5
[60] Opial, Z. (1967). Weak convergence of a sequence of successive approximations for nonexpansive mappings. Bulletin of the American Mathematical Society, 73, 591–597. · Zbl 0179.19902 · doi:10.1090/S0002-9904-1967-11761-0
[61] Osher, S., Burger, M., Goldfarb, D., Xu, J., & Yin, W. (2005). An iterative regularization method for the total variation based image restoration. Multiscale Modeling & Simulation, 4, 460–489. · Zbl 1090.94003 · doi:10.1137/040605412
[62] Passty, G. B. (1979). Ergodic convergence to a zero of the sum of monotone operators in Hilbert space. Journal of Mathematical Analysis and Applications, 72, 383–390. · Zbl 0428.47039 · doi:10.1016/0022-247X(79)90234-8
[63] Powell, M. J. D. (1969). A method for nonlinear constraints in minimization problems. London: Academic Press. · Zbl 0194.47701
[64] Rockafellar, R. T. (1970). Convex analysis. Princeton: Princeton University Press. · Zbl 0193.18401
[65] Rockafellar, R. T. (1976). Augmented Lagrangians and applications of the proximal point algorithm in convex programming. Mathematics of Operations Research, 1(2), 97–116. · Zbl 0402.90076 · doi:10.1287/moor.1.2.97
[66] Ron, A., & Shen, Z. (1997). Affine systems in L 2(\(\mathbb{R}\) d ): The analysis of the analysis operator. Journal of Functional Analysis, 148, 408–447. · Zbl 0891.42018 · doi:10.1006/jfan.1996.3079
[67] Rudin, L. I., Osher, S., & Fatemi, E. (1992). Nonlinear total variation based noise removal algorithms. Physica D, 60, 259–268. · Zbl 0780.49028 · doi:10.1016/0167-2789(92)90242-F
[68] Schäfer, H. (1957). Über die Methode sukzessiver Approximationen. Jahresbericht der Deutschen Mathematiker-Vereinigung, 59, 131–140. · Zbl 0077.11002
[69] Setzer, S. (2009a). Split Bregman algorithm, Douglas-Rachford splitting and frame shrinkage. In A. Lie, M. Lysaker, K. Morken, & X. C. Tai (Eds.), Lecture notes in computer science : Vol. 5567. Second international conference on scale space methods and variational methods in computer vision, SSVM 2009, Proceedings (pp. 464–476). Voss, Norway, June 1–5, 2009. Berlin: Springer.
[70] Setzer, S. (2009b). Splitting methods in image processing. PhD thesis, University of Mannheim.
[71] Setzer, S., Steidl, G., & Teuber, T. (2010). Deblurring Poissonian images by split Bregman methods. Journal of Visual Communication and Image Representation, 21(3), 193–199. · Zbl 05742901 · doi:10.1016/j.jvcir.2009.10.006
[72] Steidl, G., & Teuber, T. (2010). Removing multiplicative noise by Douglas-Rachford splitting methods. International Journal of Mathematical Imaging and Vision, 36(2), 168–184. · Zbl 1287.94016 · doi:10.1007/s10851-009-0179-5
[73] Tai, X. C., & Wu, C. (2009). Augmented Lagrangian method, dual methods and split Bregman iteration for ROF model. In A. Lie, M. Lysaker, K. Morken, & X. C. Tai (Eds.), Lecture notes in computer science : Vol. 5567. Second international conference on scale space methods and variational methods in computer vision, SSVM 2009, Proceedings (pp. 502–513). Voss, Norway, June 1–5, 2009. Berlin: Springer. · Zbl 1371.94362
[74] Tseng, P. (1991). Applications of a splitting algorithm to decomposition in convex programming and variational inequalities. SIAM Journal on Control and Optimization, 29, 119–138. · Zbl 0737.90048 · doi:10.1137/0329006
[75] Wang, Y., Yang, J., Yin, W., & Zhang, Y. (2008). A new alternating minimization algorithm for total variation image reconstruction. SIAM Journal on Imaging Sciences, 1(3), 248–272. · Zbl 1187.68665 · doi:10.1137/080724265
[76] Weiss, P., Blanc-Féraud, L., & Aubert, G. (2009). Efficient schemes for total variation minimization under constraints in image processing. SIAM Journal on Scientific Computing, 31(3), 2047–2080. · Zbl 1191.94029 · doi:10.1137/070696143
[77] Welk, M., Steidl, G., & Weickert, J. (2008). Locally analytic schemes: A link between diffusion filtering and wavelet shrinkage. Applied and Computational Harmonic Analysis, 24, 195–224. · Zbl 1161.68831 · doi:10.1016/j.acha.2007.05.004
[78] Yin, W., Osher, S., Goldfarb, D., & Darbon, J. (2008). Bregman iterative algorithms for 1-minimization with applications to compressed sensing. SIAM Journal on Imaging Sciences, 1(1), 143–168. · Zbl 1203.90153 · doi:10.1137/070703983
[79] Zhu, M. (2008). Fast numerical algorithms for total variation based image restoration. PhD thesis, University of California, Los Angeles, USA.
[80] Zhu, M., & Chan, T. F. (2008). An efficient primal-dual hybrid gradient algorithm for total variation image restauration (Technical report). UCLA Computational and Applied Mathematics.
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