Algorithms for reproducing objects from their X-rays

A problem of considerable importance in biology is the reconstruction of three-dimensional biological structures from electron micrographs. The problem is mathematically equivalent to that of reconstructing a three-dimensional object from x-rays taken at several angles. Solutions to these problems by Fourier methods have been proposed; unfortunately, the techniques are only applicable to highly symmetric objects; otherwise they require an unrealistic number of electron micrographs. More recently, some Monte Carlo and algebraic reconstruction techniques have been proposed. They are applicable to asymmetric objects with a realistic number of electron micrographs and less computer time; however, their performance with unknown objects is difficult to evaluate. In this eaper we discuss the first steps towards linear-least-mean-squares techniques which can be used to develop rapid reconstruction algorithms that are also applicable to asymmetric objects with any number of electron micrographs. Such algorithms have the advantage of being the best of a class; in addition, their performance can be evaluated.