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Digital Signal Processing Handbook P34

Iterative Recovery Algorithms 34.3 Spatially Invariant Degradation 34.4 Matrix-Vector Formulation Degradation Model • Basic Iterative Restoration Algorithm • Convergence • Reblurring Basic Iteration • Least-Squares Iteration 34.5 Matrix-Vector and Discrete Frequency Representations 34.6 Convergence Basic Iteration • Iteration with Reblurring 34.7 Use of Constraints The Method of Projecting Onto Convex Sets (POCS) 34.8 Class of Higher Order Iterative Algorithms 34.9 Other Forms of (x) Ill-Posed Problems and Regularization Theory • Constrained Minimization Regularization Approaches • Iteration Adaptive Image Restoration Algorithms Aggelos K. Katsaggelos Northwestern University 34.10 Discussion. | Katsaggelos A.K. Iterative Image Restoration Algorithms Digital Signal Processing Handbook Ed. Vijay K. Madisetti and Douglas B. Williams Boca Raton CRC Press LLC 1999 1999 by CRC Press LLC 34 Iterative Image Restoration Algorithms Aggelos K. Katsaggelos Northwestern University 34.1 Introduction 34.2 Iterative Recovery Algorithms 34.3 Spatially Invariant Degradation Degradation Model Basic Iterative Restoration Algorithm Convergence Reblurring 34.4 Matrix-Vector Formulation Basic Iteration Least-Squares Iteration 34.5 Matrix-Vector and Discrete Frequency Representations 34.6 Convergence Basic Iteration Iteration with Reblurring 34.7 Use of Constraints The Method of Projecting Onto Convex Sets POCS 34.8 Class of Higher Order Iterative Algorithms 34.9 Other Forms of x Ill-Posed Problems and Regularization Theory Constrained Minimization Regularization Approaches Iteration Adaptive Image Restoration Algorithms 34.10 Discussion References 34.1 Introduction In this chapter we consider a class of iterative restoration algorithms. If y is the observed noisy and blurred signal D the operator describing the degradation system x the input to the system and n the noise added to the output signal the input-output relation is described by 3 51 y Dx C n. 34.1 Henceforth boldface lower-case letters represent vectors and boldface upper-case letters represent a general operator or a matrix. The problem therefore to be solved is the inverse problem of recovering x from knowledge of y D and n. Although the presentation will refer to and apply to signals of any dimensionality the restoration of greyscale images is the main application of interest. There are numerous imaging applications which are described by Eq. 34.1 3 5 28 36 52 . D for example might represent a model of the turbulent atmosphere in astronomical observations with ground-based telescopes or a model of the degradation introduced by an out-of-focus imaging device. D might also represent the quantization performed on a .