Image Restoration for Shift Invariant Blurs-A Survey <s> IMAGE DEGRADATION MODEL <s> The article introduces digital image restoration to the reader who is just beginning in this field, and provides a review and analysis for the reader who may already be well-versed in image restoration. The perspective on the topic is one that comes primarily from work done in the field of signal processing. Thus, many of the techniques and works cited relate to classical signal processing approaches to estimation theory, filtering, and numerical analysis. In particular, the emphasis is placed primarily on digital image restoration algorithms that grow out of an area known as "regularized least squares" methods. It should be noted, however, that digital image restoration is a very broad field, as we discuss, and thus contains many other successful approaches that have been developed from different perspectives, such as optics, astronomy, and medical imaging, just to name a few. In the process of reviewing this topic, we address a number of very important issues in this field that are not typically discussed in the technical literature. <s> BIB001 </s> Image Restoration for Shift Invariant Blurs-A Survey <s> IMAGE DEGRADATION MODEL <s> Image Denoising: Past, Present, and Future, X. Li Historical Review of Image Denoising First Episode: Local Wiener Filtering Second Episode: Understanding Transient Events Third Generation: Understanding Nonlocal Similarity Conclusions and Perspectives Fundamentals of Image Restoration, B.K. Gunturk Linear Shift-Invariant Degradation Model Image Restoration Methods Blind Image Restoration Other Methods of Image Restoration Super Resolution Image Restoration Regularization Parameter Estimation Beyond Linear Shift-Invariant Imaging Model Restoration in the Presence of Unknown Spatially Varying Blur, M. Sorel and F. Sroubek Blur models Space-Variant Super Resolution Image Denoising and Restoration Based on Nonlocal Means, P. van Beek, Y. Su, and J. Yang Image Denoising Based on the Nonlocal Means Image Deblurring Using Nonlocal Means Regularization Recent Nonlocal and Sparse Modeling Methods Reducing Computational Cost of NLM-Based Methods Sparsity-Regularized Image Restoration: Locality and Convexity Revisited, W. Dong and X. Li Historical Review of Sparse Representations From Local to Nonlocal Sparse Representations From Convex to Nonconvex Optimization Algorithms Reproducible Experimental Results Conclusions and Connections Resolution Enhancement Using Prior Information, H.M. Shieh, C.L. Byrne, and M.A. Fiddy Fourier Transform Estimation and Minimum L2-Norm Solution Minimum Weighted L2-Norm Solution Solution Sparsity and Data Sampling Minimum L1-Norm and Minimum Weighted L1-Norm Solutions Modification with Nonuniform Weights Summary and Conclusions Transform Domain-Based Learning for Super Resolution Restoration, P.P. Gajjar, M.V. Joshi, and K.P. Upla Introduction to Super Resolution Related Work Description of the Proposed Approach Transform Domain-Based Learning of the Initial HR Estimate Experimental Results Conclusions and Future Research Work Super Resolution for Multispectral Image Classification, F. Li, X. Jia, D. Fraser, and A. Lambert Methodology Experimental Results Color Image Restoration Using Vector Filtering Operators, R. Lukac Color Imaging Basics Color Space Conversions Color Image Filtering Color Image Quality Evaluation Document Image Restoration and Analysis as Separation of Mixtures of Patterns: From Linear to Nonlinear Models, A. Tonazzini, I. Gerace, and F. Martinelli Linear Instantaneous Data Model Linear Convolutional Data Model Nonlinear Convolutional Data Model for the Recto-Verso Case Conclusions and Future Prospects Correction of Spatially Varying Image and Video Motion Blur Using a Hybrid Camera, Y.-W. Tai and M.S. Brown Related Work Hybrid Camera System Optimization Framework Deblurring of Moving Objects Temporal Upsampling Results and Comparisons <s> BIB002
In many imaging application ideal image is degraded version of original image as it is blurred by some function h which is called impulse response, blur or point spread function (PSF). PSF is the smallest image detail an imaging system can form. The wide range of degradations considered are atmospheric distortions, optical aberrations, sensor blur, motion blur resulting from camera shake or the movements of the objects in the scene and noise. Noises may be introduced by the transmission medium, the recording medium, measurement errors due to the limited accuracy of the recording system, quantization of the data for digital storage or any combination of these  . Degradation process is in general nonlinear and space varying but large number of problems could be addressed with a linear and shift-invariant (LSI) model considering every point in image equally blurred. Output of an LSI system is the convolution of the true image with the impulse response of the system as shown in Fig.1 (a) . is the true image that we would like to recover from the degraded measurement where indicates special coordinates. For a linear shift-invariant system, the imaging process can be formulated as BIB001 where "*" is the convolution operation, h(x, y) is the PSF of the imaging system, and n(x, y) is the additive noise. The block diagram of this process is given in fig.1 (b) .The imaging formulation can also be done in matrix-vector form or in frequency domain. Defining g, f, and n as vectorized versions of , , and , respectively, the matrixvector formulation is given by: (2) where H is a two-dimensional sparse matrix with elements taken from h(x, y) to have the proper mapping from f to g. On the other hand, the Fourier-domain version of the imaging model is given by: (3) Where and are the Fourier transforms of and respectively BIB002 .Various restoration methods optimized with respect to different criteria have been introduced in the paper.