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

This paper is the first part of a work which is concerned with linear methods for the solution of linear inverse problems with discrete data. Such problems occur frequently in instrumental science, for example tomography, radar, sonar, optical imaging, particle sizing and so on. We give a general formulation of the problem by extending the approach of Backus and Gilbert 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. We analyse in this context normal solutions, least-squares solutions and generalised inverses. We 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 in this paper. The problem of numerical stability is briefiy discussed but the investigation ofthe methods 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.

[1]  Mario Bertero,et al.  Resolution in Diffraction-limited Imaging, a Singular Value Analysis: I. The Case of Coherent Illumi , 1982 .

[2]  J. Cullum.  Numerical Differentiation and Regularization , 1971 .

[3]  C. Byrne,et al.  Reconstruction from Partial Information with Applications to Tomography , 1982 .

[4]  G. Wahba,et al.  Generalized Inverses in Reproducing Kernel Spaces: An Approach to Regularization of Linear Operator Equations , 1974 .

[5]  S Twomey.  Information content in remote sensing. , 1974, Applied optics.

[6]  Mario Bertero,et al.  On the recovery and resolution of exponential relaxation rates from experimental data: a singular-value analysis of the Laplace transform inversion in the presence of noise , 1982, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[7]  G. Backus,et al.  The Resolving Power of Gross Earth Data , 1968 .

[8]  A. Papoulis.  A new method of inversion of the Laplace transform , 1957 .

[9]  A Tikhonov,et al.  Solution of Incorrectly Formulated Problems and the Regularization Method , 1963 .

[10]  Tomaso A. Poggio,et al.  On Edge Detection , 1984, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[11]  C. Byrne,et al.  Spectral Estimators that Extend the Maximum Entropy and Maximum Likelihood Methods , 1984 .

[12]  Robert Todd Gregory,et al.  A collection of matrices for testing computational algorithms , 1969 .

[13]  M. Teague.  Image analysis via the general theory of moments , 1980 .

[14]  Mario Bertero,et al.  Resolution in Diffraction-limited Imaging, a Singular Value Analysis: III. The Effect of Sampling an , 1982 .

[15]  G. Backus,et al.  Uniqueness in the inversion of inaccurate gross Earth data , 1970, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences.

[16]  A. Balakrishnan.  Applied Functional Analysis , 1976 .

[17]  Mario Bertero,et al.  On the recovery and resolution of exponential relaxation rates from experimental data II. The optimum choice of experimental sampling points for Laplace transform inversion , 1984, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[18]  D. Slepian.  Prolate spheroidal wave functions, fourier analysis, and uncertainty — V: the discrete case , 1978, The Bell System Technical Journal.