Reprint of Inexact Bregman iteration for deconvolution of superimposed extended and point sources

Abstract In this paper we consider the deconvolution of high contrast images consisting of very bright stars (point component) and smooth structures underlying the stars (diffuse component). A typical case is a weak diffuse jet line emission superimposed to a strong stellar continuum. In order to reconstruct the diffuse component, the original object can be regarded as the sum of these two components. When the position of the point sources is known, a regularization term can be introduced for the second component. An approximation of the original object can be obtained by solving a reduced variational problem whose unknowns are the intensities of the stars and the diffuse component. We analyze this problem when the detected image is corrupted by Poisson noise and Tikhonov-like regularization is used, giving conditions for the existence and the uniqueness of the solution. Furthermore, since only an overestimation of the regularization parameter is available, we propose to solve the variational problem by inexact Bregman iteration combined with a Scaled Gradient Projection method (SGP). Numerical simulations show that the images obtained with this approach enable us to reconstruct the original intensity distribution around the point source with satisfactory accuracy.

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

[2]  L. Zanni,et al.  New adaptive stepsize selections in gradient methods , 2008 .

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

[4]  Alessandra Staglianò,et al.  Analysis of an approximate model for Poisson data reconstruction and a related discrepancy principle , 2011 .

[5]  Valeria Ruggiero,et al.  On the Uniqueness of the Solution of Image Reconstruction Problems with Poisson Data , 2010 .

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

[7]  Michel Barlaud,et al.  Deterministic edge-preserving regularization in computed imaging , 1997, IEEE Trans. Image Process..

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

[9]  L. Bregman.  The relaxation method of finding the common point of convex sets and its application to the solution of problems in convex programming , 1967 .

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

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

[12]  R. Chan,et al.  Minimization and parameter estimation for seminorm regularization models with I-divergence constraints , 2013 .

[13]  Valeria Ruggiero,et al.  Iterative regularization algorithms for constrained image deblurring on graphics processors , 2010, J. Glob. Optim..

[14]  Thomas Rodet,et al.  A variational Bayesian approach for unsupervised super-resolution using mixture models of point and smooth sources applied to astrophysical map-making , 2012 .

[15]  Valeria Ruggiero,et al.  An alternating extragradient method for total variation-based image restoration from Poisson data , 2011 .

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

[17]  L. Zanni,et al.  A scaled gradient projection method for constrained image deblurring , 2008 .

[18]  Jon A. Morse,et al.  Hubble Space Telescope Images of the HH 34 Jet and Bow Shock: Structure and Proper Motions , 2002 .

[19]  Emiliano Diolaiti,et al.  Analysis of LBT LINC-NIRVANA simulated images of galaxies and young stellar objects , 2008, Astronomical Telescopes + Instrumentation.

[20]  L. Lucy.  An iterative technique for the rectification of observed distributions , 1974 .

[21]  H. Engl,et al.  Regularization of Inverse Problems , 1996 .

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

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

[24]  Jack J. Dongarra,et al.  An extended set of FORTRAN basic linear algebra subprograms , 1988, TOMS.

[25]  Valeria Ruggiero,et al.  Inexact Bregman iteration with an application to Poisson data reconstruction , 2013 .

[26]  Mario Bertero,et al.  Image reconstruction for observations with a high dynamic range: LINC-NIRVANA simulations of a stellar jet , 2012, Other Conferences.

[27]  Donald Geman,et al.  Stochastic Relaxation, Gibbs Distributions, and the Bayesian Restoration of Images , 1984, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[28]  Bin Zhou,et al.  Gradient Methods with Adaptive Step-Sizes , 2006, Comput. Optim. Appl..

[29]  Laure Blanc-Féraud,et al.  Two constrained formulations for deblurring Poisson noisy images , 2011, 2011 18th IEEE International Conference on Image Processing.

[30]  Mario Bertero,et al.  Reconstruction of High Dynamic Range Images: Simulations of LBT Observations of a Stellar Jet, a Pathfinder Study for Future AO-Assisted Giant Telescopes , 2014 .

[31]  J. Giovannelli,et al.  Positive deconvolution for superimposed extended source and point sources , 2005, astro-ph/0507691.

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