Variational Bayesian Blind Image Deconvolution: A review <s> B. Bottom-up approach <s> We propose a new measure, the method noise, to evaluate and compare the performance of digital image denoising methods. We first compute and analyze this method noise for a wide class of denoising algorithms, namely the local smoothing filters. Second, we propose a new algorithm, the nonlocal means (NL-means), based on a nonlocal averaging of all pixels in the image. Finally, we present some experiments comparing the NL-means algorithm and the local smoothing filters. <s> BIB001 </s> Variational Bayesian Blind Image Deconvolution: A review <s> B. Bottom-up approach <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. Bottom-up approach <s> Blind deconvolution involves the estimation of a sharp signal or image given only a blurry observation. Because this problem is fundamentally ill-posed, strong priors on both the sharp image and blur kernel are required to regularize the solution space. While this naturally leads to a standard MAP estimation framework, performance is compromised by unknown trade-off parameter settings, optimization heuristics, and convergence issues stemming from non-convexity and/or poor prior selections. To mitigate some of these problems, a number of authors have recently proposed substituting a variational Bayesian (VB) strategy that marginalizes over the high-dimensional image space leading to better estimates of the blur kernel. However, the underlying cost function now involves both integrals with no closed-form solution and complex, function-valued arguments, thus losing the transparency of MAP. Beyond standard Bayesian-inspired intuitions, it thus remains unclear by exactly what mechanism these methods are able to operate, rendering understanding, improvements and extensions more difficult. To elucidate these issues, we demonstrate that the VB methodology can be recast as an unconventional MAP problem with a very particular penalty/prior that conjoins the image, blur kernel, and noise level in a principled way. This unique penalty has a number of useful characteristics pertaining to relative concavity, local minima avoidance, normalization, and scale-invariance that allow us to rigorously explain the success of VB including its existing implementational heuristics and approximations. It also provides strict criteria for learning the noise level and choosing the optimal image prior that, perhaps counter-intuitively, need not reflect the statistics of natural scenes. In so doing we challenge the prevailing notion of why VB is successful for blind deconvolution while providing a transparent platform for introducing enhancements and extensions. Moreover, the underlying insights carry over to a wide variety of other bilinear models common in the machine learning literature such as independent component analysis, dictionary learning/sparse coding, and non-negative matrix factorization. <s> BIB003
The bottom-up approach, first proposed by Babacan et al. BIB002 , refers to formulating a weight update scheme φ(ν) = ρ (ν)/ν for the Gaussian prior approximation (without knowing explicitly the penalty function) provided that φ(ν) is decreasing on (0, +∞). A crucial and very challenging question is how to choose a good penalty function ρ or φ for VB blind deconvolution. Wipf and Zhang BIB003 state that the preferred distribution is not the one reflecting the accurate statistics of the latent image, but the one that is most likely to guide VB iterations to high quality global solutions by strongly differentiating between blurry and sharp images. This implies choosing a φ that strongly discriminates sharp and blurry images. To that end, φ should be strongly sparsity promoting and also very edge preserving, such as φ(ν) = ν −p with (p ≥ 2). We believe that a trade-off between preserving edges and promoting sparsity should be achieved when dealing with noisy images. If noise is high, a very edge preserving φ(ν) cannot suppress it. Babacan et al. BIB002 also suggest a more general form φ(ν) = (F ν) −p , where F is a linear operator (e.g., nonlocal mean filter BIB001 ). A variety of heuristics can easily be embedded through F to combat noise and increase robustness. Finally, we emphasize that given a φ, the value of α γ should be chosen properly, as we will show in the experimental section.