Image Restoration for Shift Invariant Blurs-A Survey <s> Bayesian Approach <s> An iterative method of restoring degraded images was developed by treating images, point spread functions, and degraded images as probability-frequency functions and by applying Bayes’s theorem. The method functions effectively in the presence of noise and is adaptable to computer operation. <s> BIB001 </s> Image Restoration for Shift Invariant Blurs-A Survey <s> Bayesian Approach <s> The total variation regularizer is well suited to piecewise smooth images. If we add the fact that these regularizers are convex, we have, perhaps, the reason for the resurgence of interest on TV-based approaches to inverse problems. This paper proposes a new TV-based algorithm for image deconvolution, under the assumptions of linear observations and additive white Gaussian noise. To compute the TV estimate, we propose a majorization-minimization approach, which consists in replacing a difficult optimization problem by a sequence of simpler ones, by relying on convexity arguments. The resulting algorithm has O(N) computational complexity, for finite support convolutional kernels. In a comparison with state-of-the-art methods, the proposed algorithm either outperforms or equals them, with similar computational complexity. <s> BIB002 </s> Image Restoration for Shift Invariant Blurs-A Survey <s> Bayesian Approach <s> Iterative shrinkage/thresholding (1ST) algorithms have been recently proposed to handle a class of convex unconstrained optimization problems arising in image restoration and other linear inverse problems. This class of problems results from combining a linear observation model with a nonquadratic regularizer (e.g., total variation or wavelet-based regularization). It happens that the convergence rate of these 1ST algorithms depends heavily on the linear observation operator, becoming very slow when this operator is ill-conditioned or ill-posed. In this paper, we introduce two-step 1ST (TwIST) algorithms, exhibiting much faster convergence rate than 1ST for ill-conditioned problems. For a vast class of nonquadratic convex regularizers (lscrP norms, some Besov norms, and total variation), we show that TwIST converges to a minimizer of the objective function, for a given range of values of its parameters. For noninvertible observation operators, we introduce a monotonic version of TwIST (MTwIST); although the convergence proof does not apply to this scenario, we give experimental evidence that MTwIST exhibits similar speed gains over IST. The effectiveness of the new methods are experimentally confirmed on problems of image deconvolution and of restoration with missing samples. <s> BIB003
Statistical perspective to the image restoration problem is provided by Bayesian approach where unknown image, noise, and PSF are viewed as random variables. The solution of the problem is obtained by maximum likelihood (ML) estimator or maximum a posteriori (MAP) estimator or minimum mean squared error (MMSE) estimator. ML solution maximizes the probability while the maximum a posteriori (MAP) estimator maximizes . Using the Bayes rule, the MAP estimator can be written in terms of the conditional probability and the prior probability of ; (29) where is dropped from the last term as it is constant with respect to the argument f .The last term reveals that the MAP solution reduces to the ML solution when is a uniform distribution. The minimum mean squared error (MMSE) estimator aims to find the expected value of f, defined as; E[ ] (30) In image restoration problems image noise is mostly modelled to be a zero-mean independent identically distributed (iid) Gaussian random variable so the conditional probability of the observed image is; BIB002 Where the noise standard deviation and M is is the total number of pixels in the observed in image. Well known iterative Richardson-Lucy algorithm is based on Poisson noise model BIB001  . Conditional probability of the observed image & an iteration of method are given by following equations; BIB003 Moacir P.et. al.have presented a restoration approach through band extrapolation  . An extrapolation algorithm using constraints on both spatial and frequency domains with a smoothing operator were combined with the Richardson-Lucy iterative algorithm.