Spatially adapted regularization parameter selection based on the local discrepancy function for Poissonian image deblurring

The total variation (TV) model with a fidelity term of the generalized Kullback–Leibler (KL) divergence is a classical method for Poissonian image deblurring. In this paper, we propose a new TV-KL model with a spatially dependent regularization parameter. This model is able to preserve small details of images while homogeneous regions still remain sufficiently smooth. The automated selection of the regularization parameter is based on the local discrepancy function. The corresponding minimization problem with a spatially adapted regularization parameter can be solved efficiently by the split Bregman method. Numerical experiments demonstrate that the proposed algorithm has the potential to enhance regions of images containing detail and remove Poisson noise simultaneously, which leads to an improvement in the signal-to-noise ratio and the mean absolute error for deblurring results.

[1]  C. Stein.  Estimation of the Mean of a Multivariate Normal Distribution , 1981 .

[2]  Hui Zhang,et al.  A Fast Fixed Point Algorithm for Total Variation Deblurring and Segmentation , 2012, Journal of Mathematical Imaging and Vision.

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

[4]  M. Bertero,et al.  Image deblurring with Poisson data: from cells to galaxies , 2009 .

[5]  Stanley Osher,et al.  Total variation based image restoration with free local constraints , 1994, Proceedings of 1st International Conference on Image Processing.

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

[7]  A. D. Korshunov,et al.  Computational complexity of Boolean functions , 2012 .

[8]  Gabriele Steidl,et al.  Deblurring Poissonian images by split Bregman techniques , 2010, J. Vis. Commun. Image Represent..

[9]  V. Morozov.  On the solution of functional equations by the method of regularization , 1966 .

[10]  I. Csiszár.  Why least squares and maximum entropy? An axiomatic approach to inference for linear inverse problems , 1991 .

[11]  Mohamed-Jalal Fadili,et al.  A Proximal Iteration for Deconvolving Poisson Noisy Images Using Sparse Representations , 2008, IEEE Transactions on Image Processing.

[12]  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..

[13]  Bernard Rougé,et al.  TV Based Image Restoration with Local Constraints , 2003, J. Sci. Comput..

[14]  Josiane Zerubia,et al.  Richardson–Lucy algorithm with total variation regularization for 3D confocal microscope deconvolution , 2006, Microscopy research and technique.

[15]  Johnathan M. Bardsley,et al.  Regularization parameter selection methods for ill-posed Poisson maximum likelihood estimation , 2009 .

[16]  Gene H. Golub,et al.  Generalized cross-validation as a method for choosing a good ridge parameter , 1979, Milestones in Matrix Computation.

[17]  M. Burger,et al.  Accurate EM-TV algorithm in PET with low SNR , 2008, 2008 IEEE Nuclear Science Symposium Conference Record.

[18]  Yiqiu Dong,et al.  Automated Regularization Parameter Selection in Multi-Scale Total Variation Models for Image Restoration , 2011, Journal of Mathematical Imaging and Vision.

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

[20]  Tony F. Chan,et al.  Scale Recognition, Regularization Parameter Selection, and Meyer's G Norm in Total Variation Regularization , 2006, Multiscale Model. Simul..

[21]  C. Vogel,et al.  Analysis of bounded variation penalty methods for ill-posed problems , 1994 .

[22]  Michael Hintermüller,et al.  Expected absolute value estimators for a spatially adapted regularization parameter choice rule in L1-TV-based image restoration , 2010 .

[23]  Eitan Tadmor,et al.  A Multiscale Image Representation Using Hierarchical (BV, L2 ) Decompositions , 2004, Multiscale Model. Simul..

[24]  L. Vese,et al.  MULTISCALE HIERARCHICAL DECOMPOSITION OF IMAGES WITH APPLICATIONS TO DEBLURRING, DENOISING AND SEGMENTATION ∗ , 2008 .

[25]  Luca Zanni,et al.  A discrepancy principle for Poisson data , 2010 .

[26]  Yiqiu Dong,et al.  A Multi-Scale Vectorial Lτ-TV Framework for Color Image Restoration , 2011, International Journal of Computer Vision.

[27]  Giorgio C. Buttazzo,et al.  Variational Analysis in Sobolev and BV Spaces - Applications to PDEs and Optimization, Second Edition , 2014, MPS-SIAM series on optimization.

[28]  Fang Su,et al.  A New TV-Stokes Model with Augmented Lagrangian Method for Image Denoising and Deconvolution , 2012, J. Sci. Comput..

[29]  Dimitri P. Bertsekas,et al.  On the Douglas—Rachford splitting method and the proximal point algorithm for maximal monotone operators , 1992, Math. Program..

[30]  Michel L. Lapidus,et al.  Riemann zeros and phase transitions via the spectral operator on fractal strings , 2012, 1203.4828.

[31]  Grant T. Gullberg,et al.  Total variation regulated EM algorithm , 1998, 1998 IEEE Nuclear Science Symposium Conference Record. 1998 IEEE Nuclear Science Symposium and Medical Imaging Conference (Cat. No.98CH36255).

[32]  Christoph Brune,et al.  EM-TV Methods for Inverse Problems with Poisson Noise , 2013 .

[33]  M. Bertero,et al.  Efficient gradient projection methods for edge-preserving removal of Poisson noise , 2009 .

[34]  Dianne P. O'Leary,et al.  The Use of the L-Curve in the Regularization of Discrete Ill-Posed Problems , 1993, SIAM J. Sci. Comput..

[35]  Wotao Yin,et al.  An Iterative Regularization Method for Total Variation-Based Image Restoration , 2005, Multiscale Model. Simul..

[36]  E. Bratsolis,et al.  A spatial regularization method preserving local photometry for Richardson-Lucy restoration , 2001 .

[37]  Martin Burger,et al.  Primal and Dual Bregman Methods with Application to Optical Nanoscopy , 2011, International Journal of Computer Vision.

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

[39]  Gengsheng L. Zeng,et al.  Total variation regulated EM algorithm [SPECT reconstruction] , 1999 .

[40]  Andrés Almansa,et al.  A TV Based Restoration Model with Local Constraints , 2008, J. Sci. Comput..