A Convex Variational Model for Restoring Blurred Images with Multiplicative Noise

In this paper, a new variational model for restoring blurred images with multiplicative noise is proposed. Based on the statistical property of the noise, a quadratic penalty function technique is utilized in order to obtain a strictly convex model under a mild condition, which guarantees the uniqueness of the solution and the stabilization of the algorithm. For solving the new convex variational model, a primal-dual algorithm is proposed, and its convergence is studied. The paper ends with a report on numerical tests for the simultaneous deblurring and denoising of images subject to multiplicative noise. A comparison with other methods is provided as well.

[1]  Michael K. Ng,et al.  On the Total Variation Dictionary Model , 2010, IEEE Transactions on Image Processing.

[2]  Jian-Feng Cai,et al.  A framelet-based image inpainting algorithm , 2008 .

[3]  L. Rudin,et al.  Nonlinear total variation based noise removal algorithms , 1992 .

[4]  OsherStanley,et al.  A Nonlinear Inverse Scale Space Method for a Convex Multiplicative Noise Model , 2008 .

[5]  Rachid Deriche,et al.  Image Sequence Analysis via Partial Differential Equations , 1999, Journal of Mathematical Imaging and Vision.

[6]  Florence Tupin,et al.  Iterative Weighted Maximum Likelihood Denoising With Probabilistic Patch-Based Weights , 2009, IEEE Transactions on Image Processing.

[7]  Tony F. Chan,et al.  Image processing and analysis - variational, PDE, wavelet, and stochastic methods , 2005 .

[8]  Leon M. Hall,et al.  Special Functions , 1998 .

[9]  S. Osher,et al.  Geometric Level Set Methods in Imaging, Vision, and Graphics , 2011, Springer New York.

[10]  Guy Gilboa,et al.  Nonlocal Operators with Applications to Image Processing , 2008, Multiscale Model. Simul..

[11]  J. Hadamard Sur les problemes aux derive espartielles et leur signification physique , 1902 .

[12]  L. Ambrosio,et al.  Functions of Bounded Variation and Free Discontinuity Problems , 2000 .

[13]  Tom Goldstein,et al.  The Split Bregman Method for L1-Regularized Problems , 2009, SIAM J. Imaging Sci..

[14]  Jerry D. Gibson,et al.  Handbook of Image and Video Processing , 2000 .

[15]  E. Giusti Minimal surfaces and functions of bounded variation , 1977 .

[16]  P. Bickel,et al.  Mathematical Statistics: Basic Ideas and Selected Topics , 1977 .

[17]  Barry G. Quinn,et al.  Normal approximations to discrete unimodal distributions , 1986 .

[18]  Milton Abramowitz,et al.  Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables , 1964 .

[19]  Khodabina Morteza,et al.  SOME PROPERTIES OF GENERALIZED GAMMA DISTRIBUTION , 2010 .

[20]  DongYiqiu,et al.  An Efficient Primal-Dual Method for $L^1$TV Image Restoration , 2009 .

[21]  Nelly Pustelnik,et al.  Nested Iterative Algorithms for Convex Constrained Image Recovery Problems , 2008, SIAM J. Imaging Sci..

[22]  J. Einmahl Poisson and Gaussian approximation of weighted local empirical processes , 1997 .

[23]  Gabriele Steidl,et al.  Removing Multiplicative Noise by Douglas-Rachford Splitting Methods , 2010, Journal of Mathematical Imaging and Vision.

[24]  Gjlles Aubert,et al.  Mathematical problems in image processing , 2001 .

[25]  Carolyn Pillers Dobler,et al.  Mathematical Statistics , 2002 .

[26]  Kjell A. Doksum,et al.  Mathematical Statistics: Basic Ideas and Selected Topics, Volume I, Second Edition , 2015 .

[27]  Jian Yu,et al.  Restoration of images corrupted by mixed Gaussian-impulse noise via l1-l0 minimization , 2011, Pattern Recognit..

[28]  Tanja Teuber,et al.  Nonlocal Filters for Removing Multiplicative Noise , 2011, SSVM.

[29]  Stephen P. Boyd,et al.  Distributed Optimization and Statistical Learning via the Alternating Direction Method of Multipliers , 2011, Found. Trends Mach. Learn..

[30]  Michael K. Ng,et al.  A New Total Variation Method for Multiplicative Noise Removal , 2009, SIAM J. Imaging Sci..

[31]  Jianing Shi,et al.  A Nonlinear Inverse Scale Space Method for a Convex Multiplicative Noise Model , 2008, SIAM J. Imaging Sci..

[32]  J.-C. Pesquet,et al.  A Douglas–Rachford Splitting Approach to Nonsmooth Convex Variational Signal Recovery , 2007, IEEE Journal of Selected Topics in Signal Processing.

[33]  José M. Bioucas-Dias,et al.  Multiplicative Noise Removal Using Variable Splitting and Constrained Optimization , 2009, IEEE Transactions on Image Processing.

[34]  Daniel Cremers,et al.  An algorithm for minimizing the Mumford-Shah functional , 2009, 2009 IEEE 12th International Conference on Computer Vision.

[35]  José M. Bioucas-Dias,et al.  Restoration of Poissonian Images Using Alternating Direction Optimization , 2010, IEEE Transactions on Image Processing.

[36]  Yiqiu Dong,et al.  An Efficient Primal-Dual Method for L1TV Image Restoration , 2009, SIAM J. Imaging Sci..

[37]  P. Lions,et al.  Image recovery via total variation minimization and related problems , 1997 .

[38]  Gilles Aubert,et al.  A Variational Approach to Removing Multiplicative Noise , 2008, SIAM J. Appl. Math..

[39]  Michael K. Ng,et al.  Multiplicative Noise Removal with Spatially Varying Regularization Parameters , 2010, SIAM J. Imaging Sci..

[40]  Stanley Osher,et al.  Multiplicative Denoising and Deblurring: Theory and Algorithms , 2003 .

[41]  Tony F. Chan,et al.  A General Framework for a Class of First Order Primal-Dual Algorithms for Convex Optimization in Imaging Science , 2010, SIAM J. Imaging Sci..

[42]  M. Nikolova An Algorithm for Total Variation Minimization and Applications , 2004 .

[43]  Michael K. Ng,et al.  Multiplicative Noise Removal via a Learned Dictionary , 2012, IEEE Transactions on Image Processing.

[44]  Antonin Chambolle,et al.  A First-Order Primal-Dual Algorithm for Convex Problems with Applications to Imaging , 2011, Journal of Mathematical Imaging and Vision.

[45]  José M. Bioucas-Dias,et al.  Total variation restoration of speckled images using a split-bregman algorithm , 2009, 2009 16th IEEE International Conference on Image Processing (ICIP).

[46]  Thomas J. Asaki,et al.  A Variational Approach to Reconstructing Images Corrupted by Poisson Noise , 2007, Journal of Mathematical Imaging and Vision.

[47]  Tony F. Chan,et al.  Image processing and analysis , 2005 .

[48]  Pierre Kornprobst,et al.  Mathematical problems in image processing - partial differential equations and the calculus of variations , 2010, Applied mathematical sciences.

[49]  G. Aubert,et al.  A VARIATIONAL APPROACH TO REMOVE MULTIPLICATIVE NOISE , 2006 .

[50]  Mohamed-Jalal Fadili,et al.  Multiplicative Noise Removal Using L1 Fidelity on Frame Coefficients , 2008, Journal of Mathematical Imaging and Vision.

[51]  D. Owen Handbook of Mathematical Functions with Formulas , 1965 .