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Chapter 6:
The Stabilizing-Functional Approach to Regularization

6.1 Introduction

Signal formation is often represented by the standard linear model:

y = Hx + n, (6.1)

where x and y are the original and observed signals, respectively, and n is the additive noise due to the measuring device. Signal restoration is the process of inferring the best estimate for the target signal x given the observed signal y and some prior knowledge, if available, about the target.

An inverse problem is said to be ill posed when direct inversion does not ensure the existence, uniqueness, and stability of a solution. Signal restoration generally belongs to this class of problems, and regularization theory formulates how solutions may be found for such ill-posed problems. One method for developing such solutions is the stabilizing-functional approach wherein the ill-posed problem is recast as a constrained minimization of a chosen functional, which is called a stabilizing functional.

The first regularization techniques for signal restoration were often based on mean-square norms. It has been shown that such constrained least-squares approaches are related to the stabilizing-functional approach via quadratic functionals of a special form. Here, we will be concerned with using the nonquadratic functionals typically encountered in information theory.

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