Bayesian wavelet-based image deconvolution: a GEM algorithm exploiting a class of heavy-tailed priors

Abstract

Abstract-Image deconvolution is formulated in the wavelet domain under the Bayesian framework. The well-known sparsity of the wavelet coefficients of real-world images is modeled by heavy-tailed priors belonging to the Gaussian scale mixture (GSM) class; i.e., priors given by a linear (finite of infinite) combination of Gaussian densities. This class includes, among others, the generalized Gaussian, the Jeffreys, and the Gaussian mixture priors. Necessary and sufficient conditions are stated under which the prior induced by a thresholding/shrinking denoising rule is a GSM. This result is then used to show that the prior induced by the "nonnegative garrote" thresholding/shrinking rule, herein termed the garrote prior, is a GSM. To compute the maximum a posteriori estimate, we propose a new generalized expectation maximization (GEM) algorithm, where the missing variables are the scale factors of the GSM densities. The maximization step of the underlying expectation maximization algorithm is replaced with a linear stationary second-order iterative method. The result is a GEM algorithm of ( log ) computational complexity. In a series of benchmark tests, the proposed approach outperforms or performs similarly to state-of-the art methods, demanding comparable (in some cases, much less) computational complexity.

Dual-Augmented Lagrangian Method for Efficient Sparse Reconstruction

“…Iteratively reweighted shrinkage (IRS) is a popular approach for solving the problem (1) (see [3], [4], [5], [6], [7]). The main idea of the IRS approach is to replace a non-differentiable (or nonconvex) optimization problem by a series of differentiable convex ones; typically the regularizer (e.g., · 1 in Eq.…”

Total Variation-Based Image Deconvolution: a Majorization-Minimization Approach

“…To assess the relative merit of the proposed methodology, TV estimation results are compared with wavelet-based state-of-theart methods [6,13,14,25,26].…”

“…In spite of the good results next presented, we are aware that there is room to improve the performance of the introduced algorithm, by developing better ways of selecting the regularization parameter. Table 1 shows improvements of SNR (ISNR ≡ y − x 2 / b x−x 2 ) of the proposed approach and of the methods described in [6,13,14,25,26], for the three experiments. The last line of Table 1 shows the ISNR obtained using l 1 regularization.…”

“…Thus, for convolution kernels, the com- Table 1. ISNR of the proposed algorithm (Algoritm 1) and of the methods [13], [14], [25], [26], [6]. plexity of the proposed algorithm is O(N ) and O(N log N ) for finite and non-finite support convolution kernels, respectively.…”

Blind Estimation of Motion Blur Parameters for Image Deconvolution

“…Among the state of the art methods, we find those based on total variation (TV) regularization (see [3,4,8] and further references therein) which favors images of bounded variation, without penalizing possible discontinuities, as well as wavelet-based methods [2,9,10], which also provide regularization without overly penalizing image discontinuities.…”