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Chapter 2:
The Inverse Problem
Abstract

2.1 Introduction

Given a list of effects, the problem of determining cause has intrigued philosophers, mathematicians and engineers throughout recorded history. Problems of this type are formally referred to as inverse problems. Inverse problems pose a particularly difficult challenge: no solution is guaranteed to be unique or stable. The solution is unique only if for some reason known to the observer the given list of effects can be due to one and only one cause.

We are concerned here with the inverse problem as it relates to signal and image restoration. In this context of linear time-invariant (LTI) systems, it is common to use the terms inverse problem and deconvolution interchangeably. The problem here may be stated as that of estimating the true signal given a distorted and noisy version of the true signal.

2.2 Signal Restoration

In general, the goal of signal recovery is to find the best estimate of a signal that has been distorted. Although the mathematics is the same, we would like to distinguish between signal restoration and signal reconstruction. In the first problem, the research is concerned with obtaining a signal that has been distorted by a measuring device whose transfer function is available. Such a problem arises in image processing, wherein the distorting apparatus could be a lens or an image grabber. In the second problem, the scientist is faced with the challenge of reconstructing a signal from a set of its projections, generally corrupted by noise. This problem arises in spectral estimation, tomography, and image compression. In the image-compression problem, a finite subset of projections of the original signal are given, perhaps on the orthonormal cosine basis, and the original signal is desired.

Generally, to go about the problem of signal recovery, a mathematical model of the signal-formation system is needed. Different models are available; simple linear models are easy to work with but do not reflect the real world. More realistic models are complex and may be used at some additional computational cost.