Hierarchical Bayesian Sparse Image Reconstruction With Application to MRFM

Abstract

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 approach, 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 microscopy (MRFM) imaging, sparse representation.

Myopic sparse image reconstruction with application to MRFM

“…12 Here an nominal PSF κ 0 was assumed such that it corresponds to the mathematical MRFM point response model proposed by Mamin et al 13 This nominal PSF is used in AM algorithm and the parameter values of AM algorithm were set empirically according to the procedure in Herrity et al 18 …”

“…In this paper, we propose a hierarchical Bayesian approach to myopic image deconvolution that uses prior information on the PSF model. We propose a myopic modification of the Bayesian MRFM reconstruction approach in Dobigeon et al, 12 whereby one performs a simple additional step in the initial Gibbs sampler, producing a Bayesian estimate of the PSF and a Bayesian reconstruction of the image. Our approach can be related to the recent paper of Orieux et al 19 who introduced a Metropolis-within-Gibbs algorithm to estimate the parameters that tune the device response.…”

“…This image prior is similar to the LAZE distribution (weighted average of a Laplacian pdf and an atom at zero) used, for example, in Ting et al 11,20 As motivated by Dobigeon et al, 12 the image prior in (6) has the interesting property of enforcing the pixel value to be zero, reflecting the natural sparsity of the MRFM images. Furthermore, the proposed prior in (6) ensures positivity of the pixel values (spin density) to be estimated.…”

“…where x −i stands for the vector x whose ith component has been removed and µ i and η 2 i are given in equation (36) in Dobigeon et al 12 In (22), φ + ·, m, s 2 stands for the pdf of the truncated Gaussian distribution defined on R * + with hidden mean m and hidden variance s 2 . Therefore, from (22), x i |w, a, λ , σ 2 , x −i , y is a Bernoulli-truncated Gaussian variable with parameter w i , µ i , η 2 i .…”

“…We performed simulations of MRFM measurements for PSF and image models similar to those described in Dobigeon et al 12 The signal-to-noise ratio was set to SNR = 10dB. Several 32 × 32 synthetic sparse images, one of which is depicted in Fig.…”