Adaptive Langevin Sampler for Separation of $t$-Distribution Modelled Astrophysical Maps

We propose to model the image differentials of astrophysical source maps by Student's t-distribution and to use them in the Bayesian source separation method as priors. We introduce an efficient Markov Chain Monte Carlo (MCMC) sampling scheme to unmix the astrophysical sources and describe the derivation details. In this scheme, we use the Langevin stochastic equation for transitions, which enables parallel drawing of random samples from the posterior, and reduces the computation time significantly (by two orders of magnitude). In addition, Student's t-distribution parameters are updated throughout the iterations. The results on astrophysical source separation are assessed with two performance criteria defined in the pixel and the frequency domains.

[1]  Josiane Zerubia,et al.  Estimation of Markov random field prior parameters using Markov chain Monte Carlo maximum likelihood , 1999, IEEE Trans. Image Process..

[2]  K. Gorski,et al.  HEALPix: A Framework for High-Resolution Discretization and Fast Analysis of Data Distributed on the Sphere , 2004, astro-ph/0409513.

[3]  Emanuele Salerno,et al.  Extracting Astrophysical Sources from Channel-Dependent Convolutional Mixtures by Correlated Component Analysis in the Frequency Domain , 2007, KES.

[4]  C. Baccigalupi,et al.  Neural networks and the separation of cosmic microwave background and astrophysical signals in sky maps , 2000, astro-ph/0002257.

[5]  Anna Tonazzini,et al.  Separation of Correlated Astrophysical Sources Using Multiple-Lag Data Covariance Matrices , 2005, EURASIP J. Adv. Signal Process..

[6]  P. Mazur.  On the theory of brownian motion , 1959 .

[7]  W. K. Hastings,et al.  Monte Carlo Sampling Methods Using Markov Chains and Their Applications , 1970 .

[8]  D. Rubin,et al.  ML ESTIMATION OF THE t DISTRIBUTION USING EM AND ITS EXTENSIONS, ECM AND ECME , 1999 .

[9]  Nikolas P. Galatsanos,et al.  Variational Bayesian Sparse Kernel-Based Blind Image Deconvolution With Student's-t Priors , 2009, IEEE Transactions on Image Processing.

[10]  Nikolas P. Galatsanos,et al.  Variational Bayesian Image Restoration Based on a Product of $t$-Distributions Image Prior , 2008, IEEE Transactions on Image Processing.

[11]  J. D. Doll,et al.  Brownian dynamics as smart Monte Carlo simulation , 1978 .

[12]  S. Dodelson,et al.  Cosmic Microwave Background Anisotropies , 2001, astro-ph/0110414.

[13]  Yann LeCun,et al.  Improving the convergence of back-propagation learning with second-order methods , 1989 .

[14]  Bülent Sankur,et al.  Bayesian Separation of Images Modeled With MRFs Using MCMC , 2009, IEEE Transactions on Image Processing.

[15]  Anna Tonazzini,et al.  A Markov model for blind image separation by a mean-field EM algorithm , 2006, IEEE Transactions on Image Processing.

[16]  Simon J. Godsill,et al.  A Bayesian Approach for Blind Separation of Sparse Sources , 2006, IEEE Transactions on Audio, Speech, and Language Processing.

[17]  Gustav Zeuner,et al.  Markov chain Monte Carlo for Bayesian inference , 2007 .

[18]  Bülent Sankur,et al.  Fast MCMC separation for MRF modelled astrophysical components , 2009, 2009 16th IEEE International Conference on Image Processing (ICIP).

[19]  Max Welling Donald,et al.  Products of Experts , 2007 .

[20]  Michael I. Miller,et al.  REPRESENTATIONS OF KNOWLEDGE IN COMPLEX SYSTEMS , 1994 .

[21]  Anna Tonazzini,et al.  Source separation in noisy astrophysical images modelled by Markov random fields , 2004, 2004 International Conference on Image Processing, 2004. ICIP '04..

[22]  Bülent Sankur,et al.  Image Source Separation Using Color Channel Dependencies , 2009, ICA.

[23]  Jean-Francois Cardoso,et al.  THE THREE EASY ROUTES TO INDEPENDENT COMPONENT ANALYSIS; CONTRASTS AND GEOMETRY , 2001 .

[24]  Kevin H. Knuth.  A Bayesian approach to source separation , 1999 .

[25]  Yalchin Efendiev,et al.  Coarse-gradient Langevin algorithms for dynamic data integration and uncertainty quantification , 2006, J. Comput. Phys..

[26]  Radford M. Neal.  Probabilistic Inference Using Markov Chain Monte Carlo Methods , 2011 .

[27]  Charles L. Bennett,et al.  Wilkinson microwave anisotropy probe , 2007, Scholarpedia.

[28]  I. Prudyus,et al.  Wavelet-based MAP image denoising using provably better class of stochastic i.i.d. image models , 2001, 5th International Conference on Telecommunications in Modern Satellite, Cable and Broadcasting Service. TELSIKS 2001. Proceedings of Papers (Cat. No.01EX517).

[29]  Student,et al.  THE PROBABLE ERROR OF A MEAN , 1908 .

[30]  T. Hebert,et al.  A generalized EM algorithm for 3-D Bayesian reconstruction from Poisson data using Gibbs priors. , 1989, IEEE transactions on medical imaging.

[31]  D. Rowe.  A Bayesian approach to blind source separation , 2002 .

[32]  Aggelos K. Katsaggelos,et al.  Bayesian and regularization methods for hyperparameter estimation in image restoration , 1999, IEEE Trans. Image Process..

[33]  Gary F. Hinshaw,et al.  Cosmic background explorer , 2008, Scholarpedia.

[34]  C. Baccigalupi,et al.  Estimating the spectral indices of correlated astrophysical foregrounds by a second-order statistical approach , 2006, astro-ph/0609701.

[35]  Ronald J. Jaszczak,et al.  Fully Bayesian estimation of Gibbs hyperparameters for emission computed tomography data , 1997, IEEE Transactions on Medical Imaging.