One of the most intriguing questions in image processing is the problem of recovering the desired or perfect image from a degraded version. In many instances one has the feeling that the degradations in the image are such that relevant information is close to being recognizable, if only the image could be sharpened just a little. This monograph discusses the two essential steps by which this can be achieved, namely the topics of image identification and restoration. More specifically the goal of image identifi­ cation is to estimate the properties of the imperfect imaging system (blur) from the observed degraded image, together with some (statistical) char­ acteristics of the noise and the original (uncorrupted) image. On the basis of these properties the image restoration process computes an estimate of the original image. Although there are many textbooks addressing the image identification and restoration problem in a general image processing setting, there are hardly any texts which give an indepth treatment of the state-of-the-art in this field. This monograph discusses iterative procedures for identifying and restoring images which have been degraded by a linear spatially invari­ ant blur and additive white observation noise. As opposed to non-iterative methods, iterative schemes are able to solve the image restoration problem when formulated as a constrained and spatially variant optimization prob­ In this way restoration results can be obtained which outperform the lem. results of conventional restoration filters.

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1 The Image Identification and Restoration Problem.- 1.1 Introduction.- 1.2 Restoration Methods.- 1.3 Identification Methods.- 1.4 Scope of the Monograph.- 2 Image Formation Models.- 2.1 Blur Models.- 2.1.1 Linear Image Formation.- 2.1.2 State-Space Representation.- 2.1.3 Boundary Value Problem.- 2.2 Image Models.- 2.2.1 2-D AR Modeling.- 2.2.2 State-Space Representation.- 2.2.3 Model Fitting Problem.- 2.3 Common Point-spread Functions.- 2.3.1 Linear Motion Blur.- 2.3.2 Uniform Out-of-Focus Blur.- 2.3.3 Atmospheric Turbulence Blur.- 2.3.4 Scatter Blur.- 3 Regularized Image Restoration.- 3.1 Ill-Conditionedness of the Image Restoration Problem.- 3.2 Stochastic Restoration.- 3.2.1 Bayesian Methods.- 3.2.2 Wiener Filtering.- 3.2.3 Kalman Filtering.- 3.3 Algebraic Restoration.- 3.3.1 Tikhonov-Miller Regularization.- 3.3.2 Choice of the Regularizing Operator.- 3.3.3 Formal Relation with Stochastic Restoration.- 3.4Multiple Constraints Restoration.- 3.4.1 Deterministic A Priori Constraints.- 3.4.2 Projections onto Convex Sets.- 4 Iterative Image Restoration.- 4.1 VanCittert's Iteration.- 4.1.1 Formulation of the Algorithm.- 4.1.2 Convergence Analysis.- 4.1.3 Reblurring Procedure.- 4.2 Regularization via Truncated Iterations.- 4.3 Iterative Tikhonov-Miller Solution.- 4.4 Implementations with Faster Convergence.- 4.4.1 Analysis of Convergence Speed.- 4.4.2 Method of Conjugate Gradients.- 4.4.3 Iteration Method with Higher Convergence Order.- 5 Image Restoration with Ringing Reduction.- 5.1 Analysis of Ringing Artifacts.- 5.1.1 The Error Spectrum.- 5.1.2 Relation between the Error Spectrum and Ringing Artifacts.- 5.1.3 Ringing Reduction Methods.- 5.2 Constrained Adaptive Iterative Restoration.- 5.2.1 Introduction.- 5.2.2 A Priori Knowledge.- 5.2.3 Formulation of the Algorithm.- 5.3 Conjugate Gradients-based Implementation.- 5.4 Experimental Restoration Results.- 6 Maximum Likelihood Image Identification.- 6.1 Conventional Identification Methods.- 6.2 Maximum Likelihood Estimator.- 6.2.1 Introduction.- 6.2.2 Definition of the Likelihood Function.- 6.2.3 Properties of the Estimator.- 6.2.4 Analytic Solutions.- 6.3Implementations for Noiseless Data.- 6.3.1 Least-Squares Solution.- 6.3.2 Parallel 1-D Least-Squares Solution.- 6.4Implementations for Noisy Data.- 6.4.1 Gradient-based Iterative Optimization.- 6.4.2 Prediction Error Based Solution.- 7 Imáge Identification Using the EM-Algorithm.- 7.1 Review of the EM-Algorithm.- 7.2 EM-Algorithm Applied to Image Identification.- 7.3 The E-step of the Algorithm.- 7.4 The M-step of the Algorithm.- 7.4.1 Image Model Identification.- 7.4.2 Blur Model Identification.- 7.5 Experimental Results.- 7.5.1 Linear Motion Blur.- 7.5.2 Defocusing Blur.- 8 Methods for Improved Image Identification.- 8.1 Parametric Image Identification.- 8.1.1 Parametric Modeling.- 8.1.2 Image Model.- 8.1.3 Blur Model.- 8.2 Experimental Results Using Parametric Models.- 8.2.1 Linear Motion Blur.- 8.2.2 Atmospheric Turbulence Blur.- 8.2.3 Photographic Motion Blur.- 8.3 Hierarchical Image Identification.- 8.3.1 Use of Resolution Pyramids.- 8.3.2 Downsampling of Blurred Images.- 8.3.3 Image and Parameter Interpolation.- 8.3.4 Decision Tree for PSF Support Size.- 8.4 Experimental Results Using the Hierarchical Method.- 8.4.1 Linear Motion Blur.- 8.4.2 Defocusing Blur.- 8.4.3 Photographic Out-of-Focus Blur.- A Eigenvalue Analysis for 2-D Systems.- B Properties of the Iteration (5.21).- C Derivation of Equation (7.14).