Image Restoration for Shift Invariant Blurs-A Survey <s> Other Methods <s> The updated 2nd edition of this book presents a variety of image analysis applications, reviews their precise mathematics and shows how to discretize them. For the mathematical community, the book shows the contribution of mathematics to this domain, and highlights unsolved theoretical questions. For the computer vision community, it presents a clear, self-contained and global overview of the mathematics involved in image procesing problems. The second edition offers a review of progress in image processing applications covered by the PDE framework, and updates the existing material. The book also provides programming tools for creating simulations with minimal effort. <s> BIB001 </s> Image Restoration for Shift Invariant Blurs-A Survey <s> Other Methods <s> As a powerful statistical image modeling technique, sparse representation has been successfully used in various image restoration applications. The success of sparse representation owes to the development of the l1-norm optimization techniques and the fact that natural images are intrinsically sparse in some domains. The image restoration quality largely depends on whether the employed sparse domain can represent well the underlying image. Considering that the contents can vary significantly across different images or different patches in a single image, we propose to learn various sets of bases from a precollected dataset of example image patches, and then, for a given patch to be processed, one set of bases are adaptively selected to characterize the local sparse domain. We further introduce two adaptive regularization terms into the sparse representation framework. First, a set of autoregressive (AR) models are learned from the dataset of example image patches. The best fitted AR models to a given patch are adaptively selected to regularize the image local structures. Second, the image nonlocal self-similarity is introduced as another regularization term. In addition, the sparsity regularization parameter is adaptively estimated for better image restoration performance. Extensive experiments on image deblurring and super-resolution validate that by using adaptive sparse domain selection and adaptive regularization, the proposed method achieves much better results than many state-of-the-art algorithms in terms of both PSNR and visual perception. <s> BIB002 </s> Image Restoration for Shift Invariant Blurs-A Survey <s> Other Methods <s> Total least squares(TLS)is a basic estimation method to account for random errors in functional models and has found a wide variety of applications in different areas of science and engineering,including signal and image processing,computer vision,communication engineering and our own subject area(geodesy,photogrammetry,geomatics and GIS).The purpose of this paper is to briefly review TLS methods and algorithms,including a discussion of the accuracy of TLS estimates.Since reliability is of interest in our subject area,we will also briefly touch the reliability issue of TLS.Finally,we will outline some topicsfor further investigations in the future. <s> BIB003
PDE Approach: Solutions to the image restoration uses theories on partial differential equations (PDEs) to develop new models representing an image f(x, y, t) as a function of space and time, and starting with the original image at t = 0. the evolution of the image in time can be written by; (49) where the functional F takes spatial position, the image, and its first and second order spatial derivatives as the input. With the choice of the functional, it is possible to impose smoothness or sharpness on the restored image BIB001 . Total Least Squares: TLS approach is an extension of the least squares method which only allows perturbations in the observation g. The Total Least Squares (TLS) approach allows errors in both the measured (or estimated) PSF and the observation. The TLS problem is formulated as finding f , g and H minimizing the Frobenius norm subject to the imaging constraint g + g = (H+ H) f , where H and g are the error components in the PSF and observation BIB003 . ARMA Parametric Approach: Auto regressive moving average (ARMA) approach is ARMA parameter estimation problem used for blind deconvolution which has been studied extensively  . ARMA model the true image as a twodimensional autoregressive (AR) process and the PSF as a two-dimensional moving average (MA) process. Critical issue with this approach is that global AR parameters may not accurately represent the entire image. W. Dong introduced adaptive regularization terms into the sparse representation framework. Autoregressive (AR) models are learned from the dataset of example image patches to find best fitted AR models to a given patch are adaptively selected to regularize the image local structures than the image non-local selfsimilarity is introduced as another adaptive regularization term BIB002 .