Tomography, approximate reconstruction, and continuous wavelet transforms

Abstract

It has been recognized for some time now that certain high-frequency information concerning planar densities f in a neighborhood of a point can be recovered from data which consist of averages of f over lines that are relatively close to that point. The wavelet transform of f is a classical tool for analyzing local frequency content. In this article we introduce continuous wavelet transforms which are particularly well suited to producing high-resolution local reconstructions from local data of the type described above. We also show how such transforms can be realized numerically via simple modifications of well-established convolution backprojection-type algorithms. As part of our development we review the concepts of "local tomography" and "pseudolocal tomography" introduced by several authors and indicate that, in effect, these notions basically involve the computation of a wavelet transform. The results in this paper are based on the observation that Radon's classical inversion formula is a summability formula with an integrable convolution-type summability kernel. © 1999 Academic Press.