Three-Dimensional Reconstruction from Projections: A Review of Algorithms

Abstract

This chapter provides an overview of three-dimensional reconstruction from projections. The chapter reviews the algorithms that have been proposed to solve the reconstruction problem. The known reconstruction algorithms are classified into four categories—summation, the use of Fourier transform, analytic solution of the integral equations, and series expansion approaches. For each class of algorithms several points need to be considered—a general intuitive description, a precise mathematical description of a typical reconstruction method of the class, and a brief description of other methods in the class. All algorithms for reconstruction take as input the projection data, and all produce as output an estimate of the original structure based on the available data. The estimate varies from method to method. The relative performance of the various methods depends on the object and how the data are collected. The simplest algorithm for reconstruction is to estimate the density at a point by adding all the ray sums of the rays through that point. The Fourier method depends on transforming the projections into the Fourier space, where they define part of the Fourier transform of the whole object. Each projection may be shown to yield values on a central section of the Fourier space, which is a line or plane through the origin at an angle corresponding to the direction of the projection in real space.