Variational Bayesian Blind Image Deconvolution: A review <s> 1) Sparse General Representation: <s> SUMMARY This paper presents necessary and sufficient conditions under which a random variable X may be generated as the ratio ZI V where Z and V are independent and Z has a standard normal distribution. This representation is useful in Monte Carlo calculations. It is established that when 7 V2 is exponential, X is double exponential; and that when WV has the asymptotic distribution of the Kolmogorov distance statistic, X is logistic. <s> BIB001 </s> Variational Bayesian Blind Image Deconvolution: A review <s> 1) Sparse General Representation: <s> Abstract We present some Bayesian algorithms for the detection and estimation of Bernoulli-Gaussian processes, when just a filtered and noise-corrupted version of the original sequence is available. In a Bayesian framework, reconstruction is obtained from the posterior distribution of this sequence. Then, we conventionally define the likelihood of a Bernoulli-Gaussian process and show that this prior definition allows one to control the errors of detection, as well as the maximum intensity value that can be detected. Some of the traditional algorithms, generally used in image restoration, as MAP, MPM or ICM, are shown to be very efficient for the deconvolution of Bernoulli-Gaussian processes. Simulations and applications to real data are presented. <s> BIB002 </s> Variational Bayesian Blind Image Deconvolution: A review <s> 1) Sparse General Representation: <s> In this chapter, ensemble learning is applied to the problem of blind source separation and deconvolution of images. It is assumed that the observed images were constructed by mixing a set of images (consisting of independent, identically distributed pixels), convolving the mixtures with unknown blurring filters and then adding Gaussian noise. <s> BIB003 </s> Variational Bayesian Blind Image Deconvolution: A review <s> 1) Sparse General Representation: <s> Bayesian sparse factor analysis has many applications; for example, it has been applied to the problem of inferring a sparse regulatory network from gene expression data. We describe a number of inference algorithms for Bayesian sparse factor analysis using a slab and spike mixture prior. These include well-established Markov chain Monte Carlo (MCMC) and variational Bayes (VB) algorithms as well as a novel hybrid of VB and Expectation Propagation (EP). For the case of a single latent factor we derive a theory for learning performance using the replica method. We compare the MCMC and VB/EP algorithm results with simulated data to the theoretical prediction. The results for MCMC agree closely with the theory as expected. Results for VB/EP are slightly sub-optimal but show that the new algorithm is effective for sparse inference. In large-scale problems MCMC is infeasible due to computational limitations and the VB/EP algorithm then provides a very useful computationally efficient alternative. <s> BIB004 </s> Variational Bayesian Blind Image Deconvolution: A review <s> 1) Sparse General Representation: <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> BIB005
A probability distribution is considered to be sparse when it is Super Gaussian (SG)  , i.e., compared to the Gaussian distribution, it has heavier tails, it is more peaked, and has a positive excess kurtosis. These distributions are referred to as sparse since most of the distribution mass is located around zero (hence strongly favoring zero values), but the probability of occurrence of large signal values is higher compared to the Gaussian distribution. Babacan et al. BIB005 propose the use of the following general framework to define the prior model either in the image or the filter space. First they consider L high-pass filters {f γ } L γ=1 (such as derivatives, wavelets, curvelets, etc.) and define Using these filters on the real underlying image the following prior model in the image space can be defined where ρ(·) is an energy function symmetric around zero with ρ( √ s) increasing and concave for s ∈ (0, ∞)  and α γ a scale parameter. Alternatively, the filtered original images can be assumed to be independent. The following set of independent priors is then considered In this case, a set of blurred and noisy observations can be defined, associated with the filtered original images where n γ is assumed to be Gaussian independent noise with precision β. It is important to note that the observations y γ , γ = 1, . . . , L, are assumed to be independent and they provide information on the blur but not exactly on x but on its filtered versions. Notice that the most popular recent prior models, such as TV, p , or MOG are Super Gaussian distributions (see Fig. 2 for some examples), and therefore can be represented using Eq. BIB003 . Notice also in Fig. 2 , that log enforces sparsity very strongly due to its infinite peak at zero and heavy tails. A sub-class of Super Gaussian distributions is the so called Scale Mixture of Gaussians (SMG), proposed by Andrews and Mallows BIB001 and used as a general framework for BID in Babacan et al. BIB005 . Here, associated with each filter γ and each pixel i we have where p(x γ (i)|ξ γ (i)) is a Gaussian distribution with precision ξ γ (i). This model can also benefit from the introduction of a global scale parameter α γ in Eq. BIB003 . SMG requires complete monotonicity of p( √ s), i.e., (−1) n p( √ s) n ≥ 0 must be satisfied for all n = 0, 1, 2, . . .. As can be seen in  , this representation is more strict, in the sense that fewer classes of sparse priors can be represented with it than using Eq. (5). Finding p(ξ γ (i)) is in general a difficult task; however, as we will see in Section III its full knowledge is not needed for our purposes. One example of SMG is the Student-t prior proposed by Mohammad-Djafari  . It is clear that an MOG is an SMG model and that spike and slab distributions on z ∈ R, p(z) = λδ(z) are the limit of MOG models with two components, one of them with very small variance. Inference with these models is complicated due to the image size; however variational inference can still be carried out, see BIB004 . Notice that sparse promoting spike and slab and Bernoulli-Gaussian  , BIB002 priors will May 7, 2015 DRAFT very likely receive more attention by the BID community especially when estimating the blur in the filter space.