Reconstruction with Orthogonal Functions
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 mentions some relations 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.” And 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 usefull 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 gometry. Yet it remained unknown to the physicists (also to Cormack, whose paper shared the very same fate for a long time). But the problem of projection and reconstruction, the problem of tomography as we call it today , is so general and ubiquitous that scientists from a variety of fields stumbled on it and looked for a solution, without, however, looking back or looking to other fields. Today there is a vast literature which cannot comprehensively be appreciated in this short contribution. It was Cormack (1963, 1964) who first made use of orthogonal functions for the solution of Radon’s problem. Not only is their application elegant, but it also provides a good understanding about the intrinsic relations of a structure to its projections. The goal of this contribution is to demonstrate these relations.
KeywordsOrthogonal Polynomial Chebyshev Polynomial Reciprocal Space Gaussian Quadrature Airy Function
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