Variational Bayesian Blind Image Deconvolution: A review <s> B. Estimation of blur, image, and variational parameters <s> In blind deconvolution one aims to estimate from an input blurred image y a sharp image x and an unknown blur kernel k. Recent research shows that a key to success is to consider the overall shape of the posterior distribution p(x, k\y) and not only its mode. This leads to a distinction between MAP x, k strategies which estimate the mode pair x, k and often lead to undesired results, and MAP k strategies which select the best k while marginalizing over all possible x images. The MAP k principle is significantly more robust than the MAP x, k one, yet, it involves a challenging marginalization over latent images. As a result, MAP k techniques are considered complicated, and have not been widely exploited. This paper derives a simple approximated MAP k algorithm which involves only a modest modification of common MAP x, k algorithms. We show that MAP k can, in fact, be optimized easily, with no additional computational complexity. <s> BIB001 </s> Variational Bayesian Blind Image Deconvolution: A review <s> B. Estimation of blur, image, and variational parameters <s> We present a general method for blind image deconvolution using Bayesian inference with super-Gaussian sparse image priors. We consider a large family of priors suitable for modeling natural images, and develop the general procedure for estimating the unknown image and the blur. Our formulation includes a number of existing modeling and inference methods as special cases while providing additional flexibility in image modeling and algorithm design. We also present an analysis of the proposed inference compared to other methods and discuss its advantages. Theoretical and experimental results demonstrate that the proposed formulation is very effective, efficient, and flexible. <s> BIB002 </s> Variational Bayesian Blind Image Deconvolution: A review <s> B. Estimation of blur, image, and variational parameters <s> Super Gaussian (SG) distributions have proven to be very powerful prior models to induce sparsity in Bayesian Blind Deconvolution (BD) problems. Their conjugate based representations make them specially attractive when Variational Bayes (VB) inference is used since their variational parameters can be calculated in closed form with the sole knowledge of the energy function of the prior model. In this work we show how the introduction in the SG distribution of a global strength (not necessary scale) parameter can be used to improve the quality of the obtained restorations as well as to introduce additional information on the global weight of the prior. A model to estimate the new unknown parameter within the Bayesian framework is provided. Experimental results, on both synthetic and real images, demonstrate the effectiveness of the proposed approach. <s> BIB003
For the latent image, we obtain from (21), which is a multivariate Gaussian with precision matrix where F γ is an N × N convolution matrix formed by the filter f γ , andĤ is an N × N convolution matrix obtained fromĥ. The mean valuex is used as the estimate for x, which is obtained by solving the following system of linear equations DRAFT For the variational parameterξ, we obtain from   where ν γ (i) = E[x 2 γ (i)], 1 ≤ i ≤ N , with the expected value calculated using the distributionq(x). To estimate the blur we haveĥ subject to h(i) ≥ 0, where D x is a K × K matrix given by To estimate the variational parameters in Eq. (26) and the blur in Eq. (27), the matrix C x is required. This means that the N × N matrix C −1 x has to be inverted, a very time and memory demanding task. Following BIB001 and BIB002 , we approximate C x as the inverse of the diagonal of C −1 x . This inverse approximation is commented on in the open issues section IV-C. The prior image model can also be made dependent on a global parameter, which models the general scale behavior of the prior model. Its estimation is a very hard problem which can be approached under some assumptions on the prior model, see BIB003 .