2010

Journal article
Open Access

Kayabol K., Kuruoglu E. E., Sanz J. L., Sankur B., Salerno E., Herranz D.

Langevin stochastic equation Instrumentation and Methods for Astrophysics (astro-ph.IM) Bayesian source separation T-distribution FOS: Physical sciences Computer Graphics and Computer-Aided Design Markov Random fields Student's t-distribution Image analysis Software Cosmology and Nongalactic Astrophysics (astro-ph.CO) Astrophysics - Instrumentation and Methods for Astrophysics Astrophysics - Cosmology and Nongalactic Astrophysics

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.

**Source: **IEEE transactions on image processing 19 (2010): 2357–2368. doi:10.1109/TIP.2010.2048613

**Publisher: **Institute of Electrical and Electronics Engineers,, New York, NY , Stati Uniti d'America

[1] K. Kayabol, E. E. Kuruoglu and B. Sankur, “Bayesian separation of images modelled with MRFs using MCMC,” IEEE Trans. Image Process., vol.18, no.5, pp. 982-994, May 2009.

[2] K. Kayabol, and E. E. Kuruoglu, and B. Sankur, “Image Source Separation using Color Channel Dependencies,” in ICA 2009, LNCS, vol. 5441, pp. 499-506, Paraty, Brasil, Springer-Verlag, 2009.

[3] J.-F. Cardoso, “The three easy routes to independent component analysis; Contrasts and Geometry,” in Int. Conf. on Indepen. Comp. Anal. ICA'01, San Diego, Dec. 2001.

[4] Student (W. S. Gosset), “The probable error of a mean,” Biometrika, vol.6, no.1, pp. 1-25, 1908.

[5] D. Higdon, Spatial Applications of Markov Chain Monte Carlo for Bayesian Inference. PhD Thesis, University of Washingthon, 1994.

[6] I. Prudyus, S. Voloshynovskiy and A. Synyavskyy, “Wavelet-based MAP image denoising using provably better class of stochastic i.i.d. images models,” in Int. Conf. on Telecomm. in Modern Satell., Cable and Broadcas. TELSIKS'01, pp. 583-586, Sep. 2001.

[7] G. Chantas, N. Galatsanos, A. Likas and M. Saunders, “Variational Bayesian image restoration based on a product of t-distributions image priors,” IEEE Trans. Image Process., vol.17, no.10, pp. 1795-1805, Oct. 2008.

[8] D. Tzikas, A. Likas and N. Galatsanos, “Variational Bayesian sparse kernel-based blind image deconvolution with Student's-t priors,” IEEE Trans. Image Process., vol.18, no.4, pp. 753-764, Apr. 2009.

[9] C. Fevotte and S. J. Godsill “A Bayesian approach for blind separation of sparse sources,” IEEE Trans. Audio, Speech Language Process., vol. 14, no. 6, pp. 2174-2188, Nov. 2006.

[10] T. Hebert and R. Leahy, “A generalized EM algorithm for 3-D Bayesian reconstruction from Poisson data using Gibbs priors,” IEEE Trans. Medical Imaging, vol.8, no.2, pp. 194-202, June 1989.

[11] D. B. Rowe, “A Bayesian approach to blind source separation,” Journal of Interdisciplinary Mathematics, vol.5, no.1, 2002.

[12] K. H. Knuth, “A Bayesian approach to source separation, ” in Int. Conf. on Indepen. Comp. Anal. ICA'99, pp. 283-288, Jan. 1999.

[13] A. Mohammad-Djafari, “A Bayesian approach to source separation,” in Int. Workshop on Maximum Entropy and Bayesian Methods, MaxEnt'99, July, 1999.

[14] A. Tonazzini, L. Bedini, and E. Salerno “A Markov model for blind image separation by a mean-field EM algorithm,” IEEE Trans. Image Process., vol. 15, pp. 473-482, Feb. 2006.

[15] E. E. Kuruoglu, A. Tonazzini, and L. Bianchi, “Source separation in noisy astrophysical images modelled by Markov random fields,” in Int. Conf. on Image Proc. ICIP'04, pp. 24-27, Oct., 2004.

[16] P. Langevin, “Sur la theorie du mouvement brownien,” (On the theory of Brownian motion), C.R. Acad. Sci., (Paris), vol. 146, pp. 530-533, 1908.

[17] R.J. Rossky, J.D. Doll, and H.L. Friedman, “Brownian dynamics as a smart Monte Carlo simulation,” J. Chem. Phys., vol. 69, pp. 4628-4633, 1978.

[18] R.M. Neal, “Probabilistic inference using Markov chain Monte Carlo methods,” Tech. Rep. CRG-TR-93-1, Dept. Comp. Scien., University of Toronto”, Sep. 1993.

[19] D.M. Higdon, J.E. Bowsher, V.E. Johnson, T.G. Turkington, D.R. Gilland, and R. J. Jaszczak”, “Fully Bayesian estimation of Gibbs hyperparameters for emission computed tomography data,” IEEE Trans. Medical Imaging, vol. 16, no. 5, pp. 516-516, Oct. 1997.

[20] X. Descombes, R.D. Morris, J. Zerubia, and M. Berthod, “Estimation of Markov random field prior parameters using Markov chain Monte Carlo maximum likelihood,” IEEE Trans. Image Process.”, vol. 8, no. 7, pp. 954-963, July 1999.

[21] R. Molina, A. K. Katsaggelos and J. Mateos, “Bayesian and regularization methods for hyperparameter estimation in image restoration,” IEEE Trans. Image Process.”, vol. 8, no. 2, pp. 231-246, Feb. 1999.

[22] K. Kayabol, E. E. Kuruoglu, B. Sankur, E. Salerno and L. Bedini, “Fast MCMC separation for MRF Modelled astrophysical components,” in Int. Conf. on Image Proc. ICIP'09, pp. 2769-2772, Nov. 2009.

[23] Planck Science Team, “PLANCK: The scientific programme,” European Space Agency (ESA), 2005. [Online]. Available: http://www.esa.int/SPECIALS/Planck/index.html

[24] W. Hu and S. Dodelson, Cosmic Microwave Background Anisotropies, Annual Review of Astronomy and Astrophysics, vol. 40, pp. 171-216, 2002.

[25] C. Baccigalupi, L. Bedini, C. Burigana, G. De Zotti, A. Farusi, D. Maino, M. Maris, F. Perrotta, E. Salerno, L. Toffolatti, A. Tonazzini “Neural networks and the separation of cosmic microwave background and astrophysical signals in sky maps,” Mon. Not. Royal Astronom. Soc., vol.318, pp. 769-780, 2000.

[26] C. A. Bonaldi, L. Bedini, E. Salerno, C. Baccigalupi, and G. De Zotti, “Estimating the spectral indices of correlated astrophysical foregrounds by a second-order statistical approach,” Mon. Not. Royal Astronom. Soc., vol.373, 271-279, 2006.

[27] G. E. Hinton,“Products of experts,” in Int. Conf. on Artificial Neural Net. ICANN'99, vol. 1, pp. 1-6, 1999.

[28] U. Grenander and M. I. Miller, “Representations of knowledge in complex systems (with discussion), ” J. R. Statist. Soc. B”, vol. 56, pp. 549-603, 1994.

[29] P. Dostert, Y. Efendiev, T. Y. Hou and W. Luo, “Coarse-gradient Langevin algorithms for dynamic data integration and uncertainty quantification, ” J. Comput. Phys.”, vol. 217, pp. 123-142, 2006.

[30] C. Liu and D. B. Rubin, “ML estimation of the t distribution using EM and its extensions, ECM and ECME,” Statistica Sinica”, vol. 5, pp. 19-39, 1995.

[31] L. Bedini, and E. Salerno, “Extracting astrophysical source from channel-dependent convolutional mixtures by correlated component analysis in the frequency domain,” in Lecture Notes in Artificial Intelligence, vol. 4694, pp. 9-16, Springer-Verlag, 2007.

[32] National Aeronautics and Space Administration, “Cosmic Background Explorer,” NASA. [Online]. Available: http://lambda.gsfc.nasa.gov/product/cobe/

[33] National Aeronautics and Space Administration, “Wilkinson Microwave Anisotropy Probe,” NASA. [Online]. Available: http://map.gsfc.nasa.gov/

[34] W. K. Hastings, “Monte Carlo sampling methods using Markov chains and their applications,” Biometrika”, vol. 57, no. 1, pp. 97-109, Apr. 1970.

[35] S. Becker and Y. Le Cun, “Improving the convergence of backpropagation learning with second-order methods,” in Proc. of the 1988 Connectionist Models Summer School, pp. 29-37, 1989.

[36] L. Bedini, D. Herranz, E. Salerno, C. Baccigalupi, E. Kuruoglu, A. Tonazzini, ”Separation of correlated astrophysical sources using multiplelag data covariance matrices”, Eurasip J. on Appl. Sig. Proc., vol. 2005, no. 15, pp. 2400-2412, Aug. 2005.

[37] K. M. Go´rski, E. Hivon, A. J. Banday, B. D. Wandelt, F. K. Hansen, M. Reinecke, M. Bartelmann, ”HEALPix: A framework for high-resolution discretization and fast analysis of data distributed on the sphere”, The Astrophysical Journal, vol 622, Issue 2, pp. 759-771, 2005.

[2] K. Kayabol, and E. E. Kuruoglu, and B. Sankur, “Image Source Separation using Color Channel Dependencies,” in ICA 2009, LNCS, vol. 5441, pp. 499-506, Paraty, Brasil, Springer-Verlag, 2009.

[3] J.-F. Cardoso, “The three easy routes to independent component analysis; Contrasts and Geometry,” in Int. Conf. on Indepen. Comp. Anal. ICA'01, San Diego, Dec. 2001.

[4] Student (W. S. Gosset), “The probable error of a mean,” Biometrika, vol.6, no.1, pp. 1-25, 1908.

[5] D. Higdon, Spatial Applications of Markov Chain Monte Carlo for Bayesian Inference. PhD Thesis, University of Washingthon, 1994.

[6] I. Prudyus, S. Voloshynovskiy and A. Synyavskyy, “Wavelet-based MAP image denoising using provably better class of stochastic i.i.d. images models,” in Int. Conf. on Telecomm. in Modern Satell., Cable and Broadcas. TELSIKS'01, pp. 583-586, Sep. 2001.

[7] G. Chantas, N. Galatsanos, A. Likas and M. Saunders, “Variational Bayesian image restoration based on a product of t-distributions image priors,” IEEE Trans. Image Process., vol.17, no.10, pp. 1795-1805, Oct. 2008.

[8] D. Tzikas, A. Likas and N. Galatsanos, “Variational Bayesian sparse kernel-based blind image deconvolution with Student's-t priors,” IEEE Trans. Image Process., vol.18, no.4, pp. 753-764, Apr. 2009.

[9] C. Fevotte and S. J. Godsill “A Bayesian approach for blind separation of sparse sources,” IEEE Trans. Audio, Speech Language Process., vol. 14, no. 6, pp. 2174-2188, Nov. 2006.

[10] T. Hebert and R. Leahy, “A generalized EM algorithm for 3-D Bayesian reconstruction from Poisson data using Gibbs priors,” IEEE Trans. Medical Imaging, vol.8, no.2, pp. 194-202, June 1989.

[11] D. B. Rowe, “A Bayesian approach to blind source separation,” Journal of Interdisciplinary Mathematics, vol.5, no.1, 2002.

[12] K. H. Knuth, “A Bayesian approach to source separation, ” in Int. Conf. on Indepen. Comp. Anal. ICA'99, pp. 283-288, Jan. 1999.

[13] A. Mohammad-Djafari, “A Bayesian approach to source separation,” in Int. Workshop on Maximum Entropy and Bayesian Methods, MaxEnt'99, July, 1999.

[14] A. Tonazzini, L. Bedini, and E. Salerno “A Markov model for blind image separation by a mean-field EM algorithm,” IEEE Trans. Image Process., vol. 15, pp. 473-482, Feb. 2006.

[15] E. E. Kuruoglu, A. Tonazzini, and L. Bianchi, “Source separation in noisy astrophysical images modelled by Markov random fields,” in Int. Conf. on Image Proc. ICIP'04, pp. 24-27, Oct., 2004.

[16] P. Langevin, “Sur la theorie du mouvement brownien,” (On the theory of Brownian motion), C.R. Acad. Sci., (Paris), vol. 146, pp. 530-533, 1908.

[17] R.J. Rossky, J.D. Doll, and H.L. Friedman, “Brownian dynamics as a smart Monte Carlo simulation,” J. Chem. Phys., vol. 69, pp. 4628-4633, 1978.

[18] R.M. Neal, “Probabilistic inference using Markov chain Monte Carlo methods,” Tech. Rep. CRG-TR-93-1, Dept. Comp. Scien., University of Toronto”, Sep. 1993.

[19] D.M. Higdon, J.E. Bowsher, V.E. Johnson, T.G. Turkington, D.R. Gilland, and R. J. Jaszczak”, “Fully Bayesian estimation of Gibbs hyperparameters for emission computed tomography data,” IEEE Trans. Medical Imaging, vol. 16, no. 5, pp. 516-516, Oct. 1997.

[20] X. Descombes, R.D. Morris, J. Zerubia, and M. Berthod, “Estimation of Markov random field prior parameters using Markov chain Monte Carlo maximum likelihood,” IEEE Trans. Image Process.”, vol. 8, no. 7, pp. 954-963, July 1999.

[21] R. Molina, A. K. Katsaggelos and J. Mateos, “Bayesian and regularization methods for hyperparameter estimation in image restoration,” IEEE Trans. Image Process.”, vol. 8, no. 2, pp. 231-246, Feb. 1999.

[22] K. Kayabol, E. E. Kuruoglu, B. Sankur, E. Salerno and L. Bedini, “Fast MCMC separation for MRF Modelled astrophysical components,” in Int. Conf. on Image Proc. ICIP'09, pp. 2769-2772, Nov. 2009.

[23] Planck Science Team, “PLANCK: The scientific programme,” European Space Agency (ESA), 2005. [Online]. Available: http://www.esa.int/SPECIALS/Planck/index.html

[24] W. Hu and S. Dodelson, Cosmic Microwave Background Anisotropies, Annual Review of Astronomy and Astrophysics, vol. 40, pp. 171-216, 2002.

[25] C. Baccigalupi, L. Bedini, C. Burigana, G. De Zotti, A. Farusi, D. Maino, M. Maris, F. Perrotta, E. Salerno, L. Toffolatti, A. Tonazzini “Neural networks and the separation of cosmic microwave background and astrophysical signals in sky maps,” Mon. Not. Royal Astronom. Soc., vol.318, pp. 769-780, 2000.

[26] C. A. Bonaldi, L. Bedini, E. Salerno, C. Baccigalupi, and G. De Zotti, “Estimating the spectral indices of correlated astrophysical foregrounds by a second-order statistical approach,” Mon. Not. Royal Astronom. Soc., vol.373, 271-279, 2006.

[27] G. E. Hinton,“Products of experts,” in Int. Conf. on Artificial Neural Net. ICANN'99, vol. 1, pp. 1-6, 1999.

[28] U. Grenander and M. I. Miller, “Representations of knowledge in complex systems (with discussion), ” J. R. Statist. Soc. B”, vol. 56, pp. 549-603, 1994.

[29] P. Dostert, Y. Efendiev, T. Y. Hou and W. Luo, “Coarse-gradient Langevin algorithms for dynamic data integration and uncertainty quantification, ” J. Comput. Phys.”, vol. 217, pp. 123-142, 2006.

[30] C. Liu and D. B. Rubin, “ML estimation of the t distribution using EM and its extensions, ECM and ECME,” Statistica Sinica”, vol. 5, pp. 19-39, 1995.

[31] L. Bedini, and E. Salerno, “Extracting astrophysical source from channel-dependent convolutional mixtures by correlated component analysis in the frequency domain,” in Lecture Notes in Artificial Intelligence, vol. 4694, pp. 9-16, Springer-Verlag, 2007.

[32] National Aeronautics and Space Administration, “Cosmic Background Explorer,” NASA. [Online]. Available: http://lambda.gsfc.nasa.gov/product/cobe/

[33] National Aeronautics and Space Administration, “Wilkinson Microwave Anisotropy Probe,” NASA. [Online]. Available: http://map.gsfc.nasa.gov/

[34] W. K. Hastings, “Monte Carlo sampling methods using Markov chains and their applications,” Biometrika”, vol. 57, no. 1, pp. 97-109, Apr. 1970.

[35] S. Becker and Y. Le Cun, “Improving the convergence of backpropagation learning with second-order methods,” in Proc. of the 1988 Connectionist Models Summer School, pp. 29-37, 1989.

[36] L. Bedini, D. Herranz, E. Salerno, C. Baccigalupi, E. Kuruoglu, A. Tonazzini, ”Separation of correlated astrophysical sources using multiplelag data covariance matrices”, Eurasip J. on Appl. Sig. Proc., vol. 2005, no. 15, pp. 2400-2412, Aug. 2005.

[37] K. M. Go´rski, E. Hivon, A. J. Banday, B. D. Wandelt, F. K. Hansen, M. Reinecke, M. Bartelmann, ”HEALPix: A framework for high-resolution discretization and fast analysis of data distributed on the sphere”, The Astrophysical Journal, vol 622, Issue 2, pp. 759-771, 2005.

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@article{oai:it.cnr:prodotti:44401, title = {Adaptive langevin sampler for separation of t-distribution modelled astrophysical maps}, author = {Kayabol K. and Kuruoglu E. E. and Sanz J. L. and Sankur B. and Salerno E. and Herranz D.}, publisher = {Institute of Electrical and Electronics Engineers,, New York, NY , Stati Uniti d'America}, doi = {10.1109/tip.2010.2048613 and 10.48550/arxiv.1101.1396}, journal = {IEEE transactions on image processing}, volume = {19}, pages = {2357–2368}, year = {2010} }