Image Restoration for Shift Invariant Blurs-A Survey <s> Iterative Reweighted Least Squares (IRLS) Method <s> We present a nonparametric algorithm for finding localized energy solutions from limited data. The problem we address is underdetermined, and no prior knowledge of the shape of the region on which the solution is nonzero is assumed. Termed the FOcal Underdetermined System Solver (FOCUSS), the algorithm has two integral parts: a low-resolution initial estimate of the real signal and the iteration process that refines the initial estimate to the final localized energy solution. The iterations are based on weighted norm minimization of the dependent variable with the weights being a function of the preceding iterative solutions. The algorithm is presented as a general estimation tool usable across different applications. A detailed analysis laying the theoretical foundation for the algorithm is given and includes proofs of global and local convergence and a derivation of the rate of convergence. A view of the algorithm as a novel optimization method which combines desirable characteristics of both classical optimization and learning-based algorithms is provided. Mathematical results on conditions for uniqueness of sparse solutions are also given. Applications of the algorithm are illustrated on problems in direction-of-arrival (DOA) estimation and neuromagnetic imaging. <s> BIB001 </s> Image Restoration for Shift Invariant Blurs-A Survey <s> Iterative Reweighted Least Squares (IRLS) Method <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
The IRLS method BIB001 is designed to minimize ℓ norm with rewritten as weighted ℓ norm as where is diagonal of matrix with the current estimate is f (i) & as the pseudo-inverse of . The ℓ norm minimization problem becomes; (47) The idea has been used in total variation-based restoration BIB002 , where the regularizer is . 