A study of fourier space methods for “limited angle” image reconstruction*

The problem of recovering a function f(x 1,x 2) from a limited number of its one-dimensional projections is an ill-conditioned inverse problem arising in areas which include radio astronomy, electron microscopy, and X-ray tomography. The ill-conditioning of the problem is related to the availability of data only for angles 0 ⩽θ ⩽ α < π. In this paper we make a detailed study of small scale models of the practical implementation of some Fourier methods for the reconstruction of f(x 1 x 2). We concentrate on explaining the source of the ill-conditioning, as well as trying to give a qualitative connection between the amount of “angular data” a and the degree of well-posedness of the problem. Our study leads one naturally to the study of the detailed structure of the spectral properties of a certain selfadjoint positive definite operator, similar to the one encountered in the study of prolate spheroidal functions by Sepian, Pollak, and Landau. A careful look at these spectral properties as a function of the parameter a constitutes the heart of the paper.