Reconstruction With Orthogonal Functions

Abstract

In 1917, Johann Radon posed the question of whether the integral over a function with two variables along an arbitrary line can uniquely define that function such that this functional transformation can be inverted. He also solved this problem as a purely mathematical one, although he mentioned some relationships to the physical potential theory in the plane. Forty-six years later, A. M. Cormack published a paper with a title very similar to that by Radon yet still not very informative to the general reader, namely ‘Representation of a function by its line integrals’—but now comes the point: ‘with some radiological applications’. Another point is that the paper appeared in a journal devoted to applied physics. Says Cormack, ‘A method is given of finding a real function in a finite region of a plane given by its line integrals along all lines intersecting the region. The solution found is applicable to three problems of interest for precise radiology and radiotherapy’. Today we know that the method is useful and applicable to the solution of many more problems, including that which won a Nobel prize in medicine, awarded to A. M. Cormack and G. N. Hounsfield in 1979. Radon’s pioneering paper (1917) initiated an entire mathematical field of integral geometry. Yet it remained unknown to the physicists (also to Cormack, whose paper shared the very same fate for a long time). However, the problem of projection and reconstruction, the problem of tomography as we call it today, is so general and ubiquitous that scientists from all kinds of fields stumbled on it and looked for a solution-without, however, looking back or looking to other fields.