## 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 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.

## Keywords

Orthogonal Polynomial Chebyshev Polynomial Reciprocal Space Gaussian Quadrature Airy Function## Preview

Unable to display preview. Download preview PDF.

## References

- Abramowitz, M. and Stegun, I. A. (1965).
*Handbook of Mathematical Functions*. Dover, New York.Google Scholar - Cormack, A. M. (1963). Representation of a function by its line integrals, with some radiological applications.
*J. Appl. Phys.***34**:2722–2727.CrossRefGoogle Scholar - Cormack, A. M. (1964). Representation of a function by its line integrals, with some radiological applications. II.
*J. Appl. Phys.***35**:2908–2912CrossRefGoogle Scholar - Deans, S. R. (1979). Gegenbauer transforms via Radon transforms.
*SIAM J. Math. Am.***10**:577–585CrossRefGoogle Scholar - DeRosier, D. J. and Klug, A. (1968). Reconstruction of three-dimensional structures from electron micrographs.
*Nature***217**:130–134.CrossRefGoogle Scholar - Helgason, S. (1980).
*The Radon Transform*. Birkhäuser, Boston.Google Scholar - Herman, G. T. (1979).
*Image Reconstruction from Projections*. Springer-Verlag, Berlin.CrossRefGoogle Scholar - Howard, J. (1988). Tomography and reliable information.
*J. Opt. Soc. Am.*5:999–1014.CrossRefGoogle Scholar - Lerche, I. and Zeitler, E. (1976). Projections, reconstructions and orthogonal functions.
*J. Math. Anal. Appl.*56(3 ):634–649.CrossRefGoogle Scholar - Lewitt, R. M. and Bates, R. H. T. (1978a). Image reconstruction from projections. I: General theoretical considerations.
*Optik***50**(1 ):19–33.Google Scholar - Lewitt, R. M. and Bates, R. H. T. (1978b). Image reconstruction from projections. III: Projection completion methods (theory).
*Optik***50**(3):189–204.Google Scholar - Lewitt, R. M., Bates, R. H. T., and Peters, T. M. (1978). Image reconstruction from projections. II: Modified back-projection methods.
*Optk***50**(2):85–109.Google Scholar - Provencher, S. W. and Vogel, R. H. (1988). Three-dimensional reconstruction from electron micrographs of disordered specimens. I: Method.
*Ultramicroscopy***25**:209–222.PubMedCrossRefGoogle Scholar - Radon, J. (1917). ÜÜber die Bestimmung von Funktionen durch ihre Integralwerte längs gewisser Mannigfaltigkeiten.
*Ber. Verh. Sächs. Akad.***69**:262–277.Google Scholar - Ramachandran, G. N. and Lakshiminarayanan, A. V. (1971). Three-dimensional reconstruction from radiographs and electron micrographs: Application of convolutions instead of Fourier transforms.
*Proc. Nat. Acad. Sci. USA***68**(9):2236–2240.PubMedCrossRefGoogle Scholar - Smith, P. R. (1978). An integrated set of computer programs for processing electron micrographs of biological structures.
*Ultramicroscopy***3**:153–160.PubMedCrossRefGoogle Scholar - Smith, P. R., Peters, T. M., and Bates, R. H. T. (1973). Image reconstruction from finite numbers of projections,
*J. Phys.***6**:319–381.Google Scholar - Vogel, R. H. and Provencher, S. W. (1988). Three-dimensional reconstruction from electron micrographs of disordered specimens. II: Implementation and results
*UUtramicroscoppyy***25**:22 3–240Google Scholar - Zeitler, E. (1974). The reconstruction of objects from their projections.
*Optik***39**:396–415.Google Scholar