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Linear inverse problems with discrete data. I. General formulation and singular system analysis

The authors discuss linear methods for the solution of linear inverse problems with discrete data. Such problems occur frequently in instrumental science, e.g. tomography, radar, sonar, optical imaging, particle sizing and so on. They give a general formulation of the problem by extending the approach of Backus and Gilbert (1968) and by defining a mapping from an infinite-dimensional function space into a finite-dimensional vector space. The singular system of this mapping is introduced and used to define natural bases both in the solution and in the data space. They analyse in this context normal solutions, least-squares solutions and generalised inverses. They illustrate the wide applicability of the singular system technique by discussing several examples in detail. Particular attention is devoted to showing the many connections between this method and techniques developed in other topics like the extrapolation of band-limited signals and the interpolation of functions specified on a finite set of points. For example, orthogonal polynomials for least-squares approximation, spline functions and discrete prolate spheroidal functions are particular cases of the singular functions introduced. The problem of numerical stability is briefly discussed, but the investigation of the method developed for overcoming this difficulty, like truncated expansions in the singular bases, regularised solutions, iterative methods, and so on, is deferred to a second part of this work.