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How to Use a Computer Algebra System for Reconstruction of Functions from Parallel-Line Projections

Joseph M. Winograd
James H. McClellan
Organization: Georgia Tech
Journal / Anthology

Computers in Physics
Year: 1995
Volume: 9
Issue: 2
Page range: 156-163

Computers have long been used by scientists, engineers, and mathematicians for the solution of numerical problems requiring more calculations than can be reasonably performed by hand. Computer-algebra systems enable computers to solve problems involving extensive manipulation of symbolic expressions, thereby facilitating a class of investigations that were previously impractical. Symbolic analysis offers a level of precision not possible with numerical computation because mathematical expressions are stored and manipulated in an unevaluated symbolic form, and so they maintain infinite precision throughout processing. Not only is computer algebra useful for problem solving, but it has also been recognized as an important tool for education. Computer algebra facilitates exploring and demonstrating nonintuitive mathematical relationships. One such nonintuitive, yet extraordinarily useful, mathematical relationship is the Radon inversion formula for reconstruction from parallel projections. Radon inversion is the mathematical process by which a multidimensional function is found exactly when only straight-line integrals across it are known. The physical analogue of this process is tomography, the process of determining the internal structure of an object by the analysis of radiation that has passed through it. Many computed-tomography systems based upon a discretization of this reconstruction problem have been successfully built and used. However, fundamental constraints on realizable systems require that the conditions for perfect reconstruction be compromised. Consequently, much research effort has been expended in attempts to bring the performance of these systems as close as possible to the mathematical ideal. Numerical simulations of the techniques available for approximating ideal reconstruction have been presented in the literature, but analytic derivations have generally been omitted because of their algebraic complexity. Computer-algebra systems present an environment in which such analytic formulations can be generated, allowing ideal mathematical reconstructions to be compared with proposed implementations. Furthermore, when a computer-algebra system is used, the results at each step of the reconstruction may be inspected and displayed graphically, so that the effects of the various operations (such as projection or discretization) can be readily demonstrated for educational purposes. The article presents the application of Georgia Tech Signal Processing Packages (SPP) for Mathematica to the reconstruction of functions from parallel-line projections. We demonstrate the utility of this software, and that of computer-algebra systems in general, for the exploration and demonstration of advanced mathematical relations.

*Applied Mathematics
*Engineering > Signal Processing
*Mathematics > Calculus and Analysis