Reconstruction from Partial Information with Applications to Tomography

Abstract

The problem of reconstruction from projections in Hilbert space is treated. An axiomatic basis is considered which leads to techniques which provide improved reconstructions by incorporating prior knowledge to tailor the Hilbert space to the problem at hand. When applied to reconstruction of the Fourier transform of a function sampled at finitely many discrete points, the procedures lead to previously derived optimal estimation techniques. When applied to x-ray tomography, the procedures lead to new reconstruction techniques which are shown to include as a special case the minimum energy reconstruction of Logan and Shepp [Duke Math. J., 42 (1975), pp. 645–659].