A tutorial on art (algebraic reconstruction techniques)

Abstract

Algebraic Reconstruction Techniques (ART) were introduced by Gordon, Bender & Herman (1970) for solving the problem of three dimensional reconstruction from projections in electron microscopy and radiology. This is a deconvolution problem of a particular type: an estimate of a function in a higher dimensional space is deconvolved from its experimentally measured projections to a lower dimensional space. For instance, an x-ray photograph represents the projection of the three-dimensional distribution of x-ray densities within the body onto a two-dimensional plane. A finite number of such photographs taken at different angles allows us to reconstruct an estimate of the original 3-D densities. (Density refers to optical density.) The ART algorithms for solving this problem have a simple intuitive basis. Each projected density is thrown back across the higher dimensional region from whence it came, with repeated corrections to bring each projection of the estimate into agreement with the corresponding measured projection. In order to discuss the ART algorithms, one must first carefully consider the representation of space in digital computers (Sections II and III). Subsequent sections will survey the original ART algorithm (Section IV), convergence criteria (Section V), variations on the ART algorithm (Section VI), reliability of reconstructions (Section VII) and computing efficiency (Section VIII). All symbols used are summarized in Table 1.