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Chapter 8:
Special Cases of the GMF
Abstract

8.1 Introduction

In Chapter 3, various methods for solving the inverse problem were visited. Among those, the iterative methods have gained the greatest popularity, especially since the advent of digital computers. Conventional methods present major difficulties in implementation. Typically, a matrix needs to be inverted. Large matrices call for storage and computer time in order to be handled. Singular and ill-conditioned matrices make the problem very sensitive to noise and therefore unstable or impossible to solve. Iterative methods are attractive; they avoid the matrix inversion and approach the solution by successive approximations. Also, iterative methods have the advantage of allowing the scientist to control the false information in the estimate by interaction with the solution as it evolves. This may be achieved either automatically in the algorithm or by the exercise of human judgement.

Early work in the deconvolution by successive approximations resulted in various linear methods. These methods have relatively poor performance, especially with bandlimited data. However, when fast computation is desired, linear methods proved useful.

Usually, in signal recovery, some form of knowledge about the system is available, such as the statistics of the corrupting noise or some information about the anticipated solution. Therefore, a good method for the inverse problem is one that uses all of the available information about the system. This leads us to the socalled modern-constrained method or, simply, nonlinear methods. These methods are inherently more robust than the linear ones because they must find a solution that is consistent with both data and physical reality. Sometimes, physical reality is referred to as the prior information about the system. The prior information is imposed as constraints on the mathematical model that describes the method.