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Unformatted text preview: IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 18, NO. 9, SEPTEMBER 2009 2059 Hierarchical Bayesian Sparse Image Reconstruction With Application to MRFM Nicolas Dobigeon , Member, IEEE , Alfred O. Hero , Fellow, IEEE , and Jean-Yves Tourneret , Senior Member, IEEE Abstract— This paper presents a hierarchical Bayesian model to reconstruct sparse images when the observations are obtained from linear transformations and corrupted by an additive white Gaussian noise. Our hierarchical Bayes model is well suited to such naturally sparse image applications as it seamlessly accounts for properties such as sparsity and positivity of the image via appropriate Bayes priors. We propose a prior that is based on a weighted mixture of a positive exponential distribution and a mass at zero. The prior has hyperparameters that are tuned automatically by marginalization over the hierarchical Bayesian model. To overcome the complexity of the posterior distribution, a Gibbs sampling strategy is proposed. The Gibbs samples can be used to estimate the image to be recovered, e.g., by maximizing the estimated posterior distribution. In our fully Bayesian ap- proach, the posteriors of all the parameters are available. Thus, our algorithm provides more information than other previously proposed sparse reconstruction methods that only give a point estimate. The performance of the proposed hierarchical Bayesian sparse reconstruction method is illustrated on synthetic data and real data collected from a tobacco virus sample using a prototype MRFM instrument. Index Terms— Bayesian inference, deconvolution, Markov chain Monte Carlo (MCMC) methods, magnetic resonance force mi- croscopy (MRFM) imaging, sparse representation. I. INTRODUCTION F OR several decades, image deconvolution has been of increasing interest [2], [47]. Image deconvolution is a method for reconstructing images from observations provided by optical or other devices and may include denoising, deblur- ring or restoration. The applications are numerous including astronomy [49], medical imagery [48], remote sensing [41] and photography [55]. More recently, a new imaging tech- nology, called magnetic resonance force microscopy (MRFM), has been developed (see [38] and [29] for reviews). This nondestructive method allows one to improve the detection Manuscript received September 19, 2008; revised May 11, 2009. First pub- lished May 29, 2009; current version published August 14, 2009. This work was supported in part by ARO MURI Grant W911NF-05-1-0403. The associate ed- itor coordinating the review of this manuscript and approving it for publication was Dr. Erik H. W. Meijering. N. Dobigeon was with the Department of Electrical Engineering and Com- puter Science, University of Michigan, Ann Arbor, MI 48109-2122 USA. He is now with the University of Toulouse, IRIT/INP-ENSEEIHT, 31071 Toulouse cedex 7, France (e-mail:
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This note was uploaded on 05/28/2010 for the course EE EE564 taught by Professor Runyiyu during the Spring '10 term at Eastern Mediterranean University.

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