Primal-dual algorithms for total variation based image restoration under Poisson noise

详细信息    查看全文

• 作者：YouWei Wen ; Raymond Honfu Chan ; TieYong Zeng
• 关键词：image restoration ; Poisson noise ; total variation (TV) ; alternating direction method of multipliers (ADMM) ; primal ; dual ; minimax problem ; 65K10 ; 68U10
• 刊名：SCIENCE CHINA Mathematics
• 出版年：2016
• 出版时间：January 2016
• 年：2016
• 卷：59
• 期：1
• 页码：141-160
• 全文大小：885 KB
• 参考文献：1.Andrew H, Hunt B. Digital Image Restoration. Englewood Cliffs: Prentice-Hall, 1977
  2.Bardsley J, Goldes J. Regularization parameter selection methods for ill-posed Poisson maximum likelihood estimation. Inverse Probl, 2009, 25: 095005MathSciNet CrossRef
  3.Bardsley J, Goldes J. An iterative method for edge-preserving MAP estimation when data-noise is Poisson. SIAM J Sci Comput, 2010, 32: 171–185MATH MathSciNet CrossRef
  4.Bertero M, Boccacci P, Desiderà G, et al. Image deblurring with poisson data: From cells to galaxies. Inverse Probl, 2009, 25: 123006CrossRef
  5.Bertsekas D. Convex Optimization Theory. Belmont: Athena Scientific, 2009
  6.Blomgren P, Chan T. Color TV: Total variation methods for restoration of vector-valued images. IEEE Trans Image Process, 1998, 7: 304–309CrossRef
  7.Bovik A. Handbook of image and video processing. New York: Academic Press, 2010
  8.Boyd S, Parikh N, Chu E, et al. Distributed optimization and statistical learning via the alternating direction method of multipliers. Found Trends Mach Learn, 2011, 3: 1–122CrossRef
  9.Brune C, Sawatzky A, Burger M. Bregman-EM-TV methods with application to optical nanoscopy. In: Scale Space and Variational Methods in Computer Vision. New York: Springer, 2009, 235–246CrossRef
  10.Brune C, Sawatzky A, Burger M. Primal and dual Bregman methods with application to optical nanoscopy. Internat J Comput Vision, 2011, 92: 211–229MATH MathSciNet CrossRef
  11.Chambolle A. An algorithm for total variation minimization and applications. J Math Imaging Vision, 2004, 20: 89–97MathSciNet CrossRef
  12.Chambolle A, Pock T. A first-order primal-dual algorithm for convex problems with applications to imaging. J Math Imaging Vision, 2011, 40: 120–145MATH MathSciNet CrossRef
  13.Chan R, Yang H, Zeng T. A two-stage image segmentation method for blurry images with poisson or multiplicative gamma noise. SIAM J Imaging Sci, 2014, 7: 98–127MATH MathSciNet CrossRef
  14.Chan R, Chen K. Multilevel algorithm for a Poisson noise removal model with total-variation regularization. Int J Comput Math, 2007, 84: 1183–1198MATH MathSciNet CrossRef
  15.Chen G, Teboulle M. A proximal-based decomposition method for convex minimization problems. Math Program Ser A, 1994, 64: 81–101MATH MathSciNet CrossRef
  16.Combettes P, Pesquet J. A proximal decomposition method for solving convex variational inverse problems. Inverse Problems, 2008, 24: 065014MathSciNet CrossRef
  17.Dupe F, Fadili J, Starck J. A proximal iteration for deconvolving poisson noisy images using sparse representations. IEEE Trans Image Process, 2009, 18: 310–321MathSciNet CrossRef
  18.Eckstein J, Bertsekas D. On the Douglas-Rachford splitting method and the proximal point algorithm for maximal monotone operators. Math Program Ser A, 1992, 55: 293–318MATH MathSciNet CrossRef
  19.Fessler J, Booth S. Conjugate-gradient preconditioning methods for shift-variant pet image reconstruction. Image Process IEEE Trans, 1999, 8: 688–699MATH MathSciNet CrossRef
  20.Fessler J, Lee S, Olafsson V, et al. Toeplitz-based iterative image reconstruction for mri with correction for magnetic field inhomogeneity. Signal Process IEEE Trans, 2005, 53: 3393–3402MathSciNet CrossRef
  21.Figueiredo M, Bioucas-Dias J. Restoration of Poissonian images using alternating direction optimization. IEEE Trans Image Process, 2010, 19: 3133–3145MathSciNet CrossRef
  22.Gabay D, Mercier B. A dual algorithm for the solution of nonlinear variational problems via finite element approximation. Comput Math Appl, 1976, 2: 17–40MATH CrossRef
  23.Goldstein T, Osher S. The split Bregman method for L1 regularized problems. SIAM J Imaging Sci, 2009, 2: 323–343MATH MathSciNet CrossRef
  24.Hansen P. Rank-Deficient and Discrete Ill-Posed Problems: Numerical Aspects of Linear Inversion. Philadelphia: SIAM, 1998
  25.He B, Yuan X. On the o(1/n) convergence rate of the Douglas-Rachford alternating direction method. SIAM J Numer Anal, 2012, 50: 700–709MATH MathSciNet CrossRef
  26.Le T, Chartrand R, Asaki T J. A variational approach to reconstructing images corrupted by Poisson noise. J Math Imaging Vision, 2007, 27: 257–263MathSciNet CrossRef
  27.Lucy L. An iterative technique for the rectification of observed distributions. Astron J, 1974, 79: 745–754CrossRef
  28.Murtagh F, Starck J. Astronomical Image and Data Analysis. New York: Springer, 2006
  29.Ng M, Chan R, Tang W. A fast algorithm for deblurring models with Neumann boundary conditions. SIAM J Sci Comput, 1999, 21: 851–866MATH MathSciNet CrossRef
  30.Puetter R, Gosnell T, Yahil A. Digital image reconstruction: Deblurring and denoising. Annu Rev Astron Astrophys, 2005, 43: 139–194CrossRef
  31.Richarson W. Bayesian-based iterative method of image restoration. J Opt Soc Amer A, 1972, 62: 55–59CrossRef
  32.Rockafellar R. Augmented Lagrangians and applications of the proximal point algorithm in convex programming. Math Oper Res, 1976, 1: 97–116MATH MathSciNet CrossRef
  33.Rockafellar R. Monotone operators and the proximal point algorithm. SIAM J Control Optim, 1976, 14: 877–898MATH MathSciNet CrossRef
  34.Rudin L, Osher S, Fatemi E. Nonlinear total variation based noise removal algorithms. Physica D, 1992, 60: 259–268MATH CrossRef
  35.Setzer S, Steidl G, Teuber T. Deblurring Poissonian images by split Bregman techniques. J Vis Comm Image Represent, 2010, 21: 193–199CrossRef
  36.Setzer S, Steidl G, Teuber T. Infimal convolution regularizations with discrete l1-type functionals. Comm Math Sci, 2011, 9: 797–872MATH MathSciNet CrossRef
  37.Snyder D, Hammoud A, White R. Image recovery from data acquired with a charge-coupled-device camera. JOSA A, 1993, 10: 1014–1023CrossRef
  38.Tai X, Wu C. Augmented Lagrangian method, dual methods and split Bregman iteration for ROF model. In: Scale Space and Variational Methods in Computer Vision. Lecture Notes in Computer Science, vol. 5567. New York: Springer, 2009, 502–513CrossRef
  39.Timmermann K, Nowak R. Multiscale modeling and estimation of Poisson processes with application to photon-limited imaging. IEEE Trans Inform Theory, 1999, 45: 846–862MATH MathSciNet CrossRef
  40.Tseng P. Applications of a splitting algorithm to decomposition in convex programming and variational inequalities. SIAM J Control Optim, 1991, 29: 119–138MATH MathSciNet CrossRef
  41.Wernick M, Aarsvold J. Emission Tomography: The Fundamentals of PET and SPECT. New York: Academic Press, 2004
  42.Willett R, Nowak R. Platelets: A multiscale approach for recovering edges and surfaces in photon-limited medical imaging. IEEE Trans Medical Imaging, 2003, 22: 332–350CrossRef
  43.Yin W, Osher S, Goldfarb D, et al. Bregman iterative algorithms for ℓ_{1-minimization with applications to compressed sensing. SIAM J Imaging Sci, 2008, 1: 143–168MATH MathSciNet CrossRef
  44.Zanella R, Boccacci P, Zanni L, et al. Efficient gradient projection methods for edge-preserving removal of Poisson noise. Inverse Problems, 2009, 25: 045010MathSciNet CrossRef
  45.Zhou W, Bovik A, Sheikh H, et al. Image quality assessment: From error visibility to structural similarity. IEEE Trans Image Process, 2004, 13: 600–612CrossRef
  46.Zhu M. Fast Numerical Algorithms for Total Variation Based Image Restoration. PhD thesis. Los Angeles: University of California, 2008
  47.Zhu M, Chan T. An efficient primal-dual hybrid gradient algorithm for total variation image restoration. UCLA CAM Report, 2007, 08–34}
  
• 作者单位：YouWei Wen (1)
  Raymond Honfu Chan (2)
  TieYong Zeng (3)
  
  1. Faculty of Science, Kunming University of Science and Technology, Kunming, 650093, China
  2. Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong, China
  3. Department of Mathematics, Hong Kong Baptist University, Kowloon, Hong Kong, China
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Chinese Library of Science
  Applications of Mathematics
  
• 出版者：Science China Press, co-published with Springer
• ISSN：1869-1862

文摘

We consider the problem of restoring images corrupted by Poisson noise. Under the framework of maximum a posteriori estimator, the problem can be converted into a minimization problem where the objective function is composed of a Kullback-Leibler (KL)-divergence term for the Poisson noise and a total variation (TV) regularization term. Due to the logarithm function in the KL-divergence term, the non-differentiability of TV term and the positivity constraint on the images, it is not easy to design stable and efficiency algorithm for the problem. Recently, many researchers proposed to solve the problem by alternating direction method of multipliers (ADMM). Since the approach introduces some auxiliary variables and requires the solution of some linear systems, the iterative procedure can be complicated. Here we formulate the problem as two new constrained minimax problems and solve them by Chambolle-Pock’s first order primal-dual approach. The convergence of our approach is guaranteed by their theory. Comparing with ADMM approaches, our approach requires about half of the auxiliary variables and is matrix-inversion free. Numerical results show that our proposed algorithms are efficient and outperform the ADMM approach. Keywords image restoration Poisson noise total variation (TV) alternating direction method of multipliers (ADMM) primal-dual minimax problem