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.

Digital Image Restoration

The application of regularization to ill-conditioned problems necessitates the choice of a regularization parameter which trades fidelity to the data with smoothness of the solution. The value of the regularization parameter depends on the variance of the noise in the data. The problem of choosing the regularization parameter and estimating the noise variance in image restoration is examined. An error analysis based on an objective mean-square-error (MSE) criterion is used to motivate regularization. Two approaches for choosing the regularization parameter and estimating the noise variance are proposed. The proposed and existing methods are compared and their relationship to linear minimum-mean-square-error filtering is examined. Experiments are presented that verify the theoretical results. >

Methods for choosing the regularization parameter and estimating the noise variance in image restoration and their relation

BLIND IMAGE DECONVOLUTION: PROBLEM FORMULATION AND EXISTING APPROACHES Tom E. Bishop, S. Derin Babacan, Bruno Amizic, Aggelos K. Katsaggelos, Tony Chan, and Rafael Molina Introduction Mathematical Problem Formulation Classification of Blind Image Deconvolution Methodologies Bayesian Framework for Blind Image Deconvolution Bayesian Modeling of Blind Image Deconvolution Bayesian Inference Methods in Blind Image Deconvolution Non-Bayesian Blind Image Deconvolution Models Conclusions References BLIND IMAGE DECONVOLUTION USING BUSSGANG TECHNIQUES: APPLICATIONS TO IMAGE DEBLURRING AND TEXTURE SYNTHESIS Patrizio Campisi, Alessandro Neri, Stefania Colonnese, Gianpiero Panci, and Gaetano Scarano Introduction Bussgang Processes Single-Channel Bussgang Deconvolution Multichannel Bussgang deconvolution Conclusions References BLIND MULTIFRAME IMAGE DECONVOLUTION USING ANISOTROPIC SPATIALLY ADAPTIVE FILTERING FOR DENOISING AND REGULARIZATION Vladimir Katkovnik, Karen Egiazarian, and Jaakko Astola Introduction Observation Model and Preliminaries Frequency Domain Equations Projection Gradient Optimization Anisotropic LPA-ICI Spatially Adaptive Filtering Blind Deconvolution Algorithm Identifiability and Convergence Simulations Conclusions Acknowledgments References BAYESIAN METHODS BASED ON VARIATIONAL APPROXIMATIONS FOR BLIND IMAGE DECONVOLUTION Aristidis Likas and Nikolas P. Galatsanos Introduction Background on Variational Methods Variational Blind Deconvolution Numerical Experiments Conclusions and Future Work APPENDIX A: Computation of the Variational Bound F(q,?) APPENDIX B: Maximization of F(q,?) References DECONVOLUTION OF MEDICAL IMAGES FROM MICROSCOPIC TO WHOLE BODY IMAGES Oleg V. Michailovich and Dan R. Adam Introduction Nonblind Deconvolution Blind Deconvolution in Ultrasound Imaging Blind Deconvolution in SPECT Blind Deconvolution in Confocal Microscopy Summary References BAYESIAN ESTIMATION OF BLUR AND NOISE IN REMOTE SENSING IMAGING Andre Jalobeanu, Josiane Zerubia, and Laure Blanc-Feraud Introduction The Forward Model Bayesian Estimation: Invert the Forward Model Possible Improvements and Further Development Results Conclusions Acknowledgments References DECONVOLUTION AND BLIND DECONVOLUTION IN ASTRONOMY Eric Pantin, Jean-luc Starck, and Fionn Murtagh Introduction The Deconvolution Problem Linear Regularized Methods CLEAN Bayesian Methodology Iterative Regularized Methods Wavelet-Based Deconvolution Deconvolution and Resolution Myopic and Blind Deconvolution Conclusions and Chapter Summary Acknowledgments References MULTIFRAME BLIND DECONVOLUTION COUPLED WITH FRAME REGISTRATION AND RESOLUTION ENHANCEMENT Filip Sroubek, Jan Flusser, and Gabriel Cristobal Introduction Mathematical Model Polyphase Formulation Reconstruction of Volatile Blurs Blind Superresolution Experiments Conclusions Acknowledgments References BLIND RECONSTRUCTION OF MULTIFRAME IMAGERY BASED ON FUSION AND CLASSIFICATION Dimitrios Hatzinakos, Alexia Giannoula, and Jianxin Han Introduction System Overview Recursive Inverse Filtering with Finite Normal-Density Mixtures (RIF-FNM) Optimal Filter Adaptation Effects of Noise The Fusion and Classification Recursive Inverse Filtering Algorithm (FAC-RIF) Experimental Results Final Remarks References BLIND DECONVOLUTION AND STRUCTURED MATRIX COMPUTATIONS WITH APPLICATIONS TO ARRAY IMAGING Michael K. Ng and Robert J. Plemmons Introduction One-Dimensional Deconvolution Formulation Regularized and Constrained TLS Formulation Numerical Algorithms Two-Dimensional Deconvolution Problems Numerical Examples Application: High-Resolution Image Reconstruction Concluding Remarks and Current Work Acknowledgments References INDEX

http://www.dbabacan.info/papers/Campisi_Egiazarian_BIDmain_FOR_CH1.pdf

Blind image deconvolution: theory and applications

The primary difficulty with blind image restoration, or joint blur identification and image restoration, is insufficient information. This calls for proper incorporation of a priori knowledge about the image and the point-spread function (PSF). A well-known space-adaptive regularization method for image restoration is extended to address this problem. This new method effectively utilizes, among others, the piecewise smoothness of both the image and the PSF. It attempts to minimize a cost function consisting of a restoration error measure and two regularization terms (one for the image and the other for the blur) subject to other hard constraints. A scale problem inherent to the cost function is identified, which, if not properly treated, may hinder the minimization/blind restoration process. Alternating minimization is proposed to solve this problem so that algorithmic efficiency as well as simplicity is significantly increased. Two implementations of alternating minimization based on steepest descent and conjugate gradient methods are presented. Good performance is observed with numerically and photographically blurred images, even though no stringent assumptions about the structure of the underlying blur operator is made.

A regularization approach to joint blur identification and image restoration

We present a novel blind deconvolution technique for the restoration of linearly degraded images without explicit knowledge of either the original image or the point spread function. The technique applies to situations in which the scene consists of a finite support object against a uniformly black, grey, or white background. This occurs in certain types of astronomical imaging, medical imaging, and one-dimensional (1-D) gamma ray spectra processing, among others. The only information required are the nonnegativity of the true image and the support size of the original object. The restoration procedure involves recursive filtering of the blurred image to minimize a convex cost function. We prove convexity of the cost function, establish sufficient conditions to guarantee a unique solution, and examine the performance of the technique in the presence of noise. The new approach is experimentally shown to be more reliable and to have faster convergence than existing nonparametric finite support blind deconvolution methods. For situations in which the exact object support is unknown, we propose a novel support-finding algorithm.

A novel blind deconvolution scheme for image restoration using recursive filtering

In this paper, the problem of identifying the image and blur parameters and restoring a noisy blurred image is addressed. Specifying the blurring process by its point spread function (PSF), the blur identification problem is formulated as the maximum likelihood estimation (MLE) of the PSF. Modeling the original image and the additive noise as zeromean
Gaussian processes, the MLE of their covariance matrices is also computed. An iterative approach, called the EM (expectation-maximization) algorithm, is used to find the maximum likelihood estimates ofthe relevant unknown parameters. In applying the EM algorithm, the original image is chosen to be part of the complete data; its estimate is computed in the
E-step of the EM iterations and represents the restored image. Two algorithms for identification/restoration, based on two different choices of complete data, are derived and compared. Simultaneous blur identification and restoration is performed by the first algorithm, while the identification of the blur results from a separate minimization in the second algorithm. Experiments with simulated and photographically blurred images
are shown.

Image identification and restoration based on the expectation-maximization algorithm

We propose algorithms for estimating the phase of a deterministic signal using its bispectrum. The bispectrum of a signal is the (discrete) Fourier transform of its triple correlation. While second-order statistics (e.g., correlation function, power spectrum, etc.) do not provide any information about the phase of the signal, third-order statistics (e.g., triple correlation, bispectrum, etc.) allow the recovery of the phase of the signal. We showthatthe applicability oftwo commonly used algorithms for phase estimation using the bispectrum is restricted to signals with simple phase characteristics. We propose algorithms for estimating the phase of arbitrary signals such as images, by taking into account the ambiguity due to the use of the principal value of the phase component. The resulting estimated phase is incorporated into a restoration filter. Image lines and images are used in our experiments to test the effectiveness of the proposed algorithms.

Phase estimation using the bispectrum and its application to image restoration

Previous work has demonstrated the effectiveness of the expectation-maximization algorithm to restore noisy and blurred single- channel images and simultaneously identify its blur. In addition, a gen- eral framework for processing multichannel images using single-channel techniques has been developed. The authors combine and extend the two approaches to the simultaneous blur identification and restoration of multichannel images. Explicit equations for that purpose are developed for the general case when cross-channel degradations are present. An important difference from the single-channel problem is that the cross power spectra are complex quantities, which further complicates the analysis of the algorithm. The proposed algorithm is very effective at restoring multichannel images, as is demonstrated experimentally.

Multichannel image identification and restoration using the expectation-maximization algorithm

Algorithms for the simultaneous identification of the blur and the restoration of a noisy blurred image are presented in this paper. The original image and the additive noise are modeled as zero-mean Gaussian random processes, which are characterized by their covariance matrices. The covariance matrices are unknown parameters. The blurring process is specified by its point spread function, which is also unknown. Maximum likelihood estimation is used to find these unknown parameters. In turn, the EM algorithm is exploited to find the maximum likelihood estimates. In applying the EM algorithm, the original image is chosen to be part of the complete data; its estimate, which represents the restored image, is computed in the E-step of the EM iterations. Explicit iterative expressions are derived for the estimation of relevant parameters. Experiments with simulated and photographically blurred images are shown.

Simultaneous Blur Identification And Image Restoration Using The EM Algorithm

In this paper we propose two algorithms for the restoration of images based on the bispectrum. The bispectrum of a signal is the Fourier transform of its triple correlation. While second- order statistics (e.g., correlation function, power spectrum, etc.) do not provide any information about the phase of the signal, third-order statistics (e.g., triple correlation, bispectrum, etc.) allow the recovery of the phase of the signal. We propose two algorithms for estimating the magnitude and the phase of the image, where the ambiguity due to the use of the principal value of the phase component is taken into account. Image lines are used in our experiments to test the effectiveness of the proposed algorithms.

Kuen-Tsair Lay

Papers

Image identification and restoration based on the expectation-maximization algorithm

Phase estimation using the bispectrum and its application to image restoration

Multichannel image identification and restoration using the expectation-maximization algorithm

Simultaneous Blur Identification And Image Restoration Using The EM Algorithm

Image restoration algorithms based on the bispectrum