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Total bounded variation-based Poissonian images recovery by split Bregman iteration. (English) Zbl 1252.94014

Summary: This paper presents a new total bounded variation regularization-based Poissonian images deconvolution scheme. Computationally, an extended split Bregman iteration is described to obtain the optimal solution recursively. Moreover, the rigorous convergence analysis of the proposed algorithm is also expatiated here. Compared with the computational speed and the recovered results of the total variation-based method, numerical simulations definitely demonstrate the competitive performance of the proposed strategy in Poissonian images restoration.

MSC:

94A08 Image processing (compression, reconstruction, etc.) in information and communication theory
65K10 Numerical optimization and variational techniques
65M32 Numerical methods for inverse problems for initial value and initial-boundary value problems involving PDEs

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[1] Rudin, Nonlinear total variation based noise removal algorithms, Physica D 60 pp 259– (1992) · Zbl 0780.49028
[2] Acar, Analysis of bounded variation penalty methods for ill-posed problems, Inverse Problem 10 pp 1217– (1994) · Zbl 0809.35151
[3] Chambolle, Image recovery via total variational minimization and related problems, Numerische Mathematik 76 pp 167– (1997) · Zbl 0874.68299
[4] Chan, On the convergence of the lagged diffusivity fixed point method in total variation image restoration, SIAM Journal on Mathematical Analysis 36 (2) pp 354– (1999) · Zbl 0923.65037
[5] Vogel, Iteration methods for total variation denoising, SIAM Journal on Scientific Computing 17 pp 227– (1996) · Zbl 0847.65083
[6] Chambolle, An algorithm for total variation minimization and application, Journal of Mathematical Imaging and Vision 20 pp 89– (2004) · Zbl 1366.94048
[7] Chan, A nonlinear primal-dual method for total variation-based image restoration, SIAM Journal on Scientific Computing 20 pp 1964– (1999) · Zbl 0929.68118
[8] Osher, An iterative regularization method for total variation-based image restoration, SIAM Multiscale Modeling and Simulation 4 pp 460– (2005) · Zbl 1090.94003
[9] Wang, A new alternating minimization algorithm for total variation image reconstruction, SIAM Journal on Imaging Sciences 1 pp 248– (2008) · Zbl 1187.68665
[10] Yin, Bregman iterative algorithms for L1-minimization with applications to compressend sensing, SIAM Journal on Imaging Sciences 1 pp 143– (2008) · Zbl 1203.90153
[11] Burger, Nonlinear inverse scale space methods, Communications in Mathematical Sciences 4 pp 179– (2006) · Zbl 1106.68117
[12] Lie, Inverse scale spaces for nonlinear regularization, Journal of Mathematical Imaging and Vision 27 pp 41– (2007) · Zbl 1478.94059
[13] Lysaker, Noise removel using smoothed normals and surface fitting, IEEE Transactions on Image Processing 13 pp 1345– (2004) · Zbl 1286.94022
[14] Cai JF Osher S Shen Z Split Bregman methods and frame based image restoration CAM Report 09-28 2009
[15] Goldstein, The Split Bregman algorithm for L1 regularized problems, SIAM Journal on Imaging Sciences 2 pp 323– (2009) · Zbl 1177.65088
[16] Wang Y Yin W Zhang Y A fast algorithm for image deblurring with total variation regularization CAAM Technical Report 2007
[17] Esser E Applications of Lagrangian-based alternating direction methods and connections to split Bregman CAM Report 09-31 2009
[18] Wen Z Goldfarb D Yin W Alternating direction augmented Lagrangian methods for semidefinite programming Rice CAAM Report, TR09-42 2009
[19] Wu C Tai X-C Augmented Lagrangian method, dual methods, and split Bregman iteration for ROF, vectorial TV, and high order models CAM Report 09-76 2009
[20] Wu C Zhang J Tai X-C Augmented Lagrangian method for total variation restoration with non-quadratic fidelity CAM Report 09-82 2009
[21] Chavent, Regularization of linear least squares problems by total bounded variation, ESAIM: Control, Optimisation and Calculus of Variations 2 pp 359– (1997) · Zbl 0890.49010
[22] Hintermüller, Total bounded variation regularization as a bilaterally constrained optimization problem, SIAM Journal on Applied Mathematics 64 pp 1311– (2004) · Zbl 1055.94504
[23] Liu, Split Bregman iteration algorithm for total bounded variation regularization based image deblurring, Journal of Mathematical Analysis and Applications 372 pp 486– (2010) · Zbl 1202.94062
[24] Bratsolis, A spatial regularization method preserving local photometry for Richardson-Lucy restoration, Astronomy Astrophysics 375 (3) pp 1120– (2001)
[25] Day, Richardson-Lucy algorithm with total variation regularization for 3D confocal microscope deconvolution, Microscopy Research and Technique 69 pp 260– (2006)
[26] Kryvanos, Nonlinear image restoration methods for marker extraction in 3D fluorescent microscopy, Proceedings of SPIE, San Jose, CA, USA. Computational Imaging III 5674 pp 432– (2005)
[27] Le, A variational approach to constructing images corrupted by Poisson noise, Journal of Mathematical Imaging and Vision 27 pp 257– (2007)
[28] Sawatzky A Brune C Wübbeling F Kösters T Schäfers K Burger M Accurate EM-TV algorithm in PET with low SNR IEEE Nuclear Science Symposium Conference Record 2008 5133 5137
[29] Brune C Sawatzky A Kösters T Wübbeling F Burger M An analytical view on EM-TV based methods for inverse problems with Poisson noise http://wwwmath.uni-muenster.de/num/publications/2009/BSKWB09/emtv-paper.pdf
[30] Setzer, Deblurring Poissonian images by split Bregman techniques, Journal of Visual Communication and Image Representation 21 pp 193– (2010) · Zbl 05742901
[31] Giusti, Minimal surfaces and functions of bounded variations 80 (1984) · Zbl 0545.49018
[32] Bertsekas, Convex analysis and optimization (2006)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.