Radon transform on algebraic Cormack α-curves inversion formula

Abstract

Abstract Radon transforms (RTs) in R 2 , defined on simple integration manifolds such as straight lines or circles, have attracted extensive interest since they provide the reconstruction of physical density functions in various applicative inverse problems, via a closed form inversion formula. In an attempt to see whether RT on more involved integration manifolds would still admit closed form inversion formulas, A M Cormack has sought to invert RT defined on a class of non-trivial curves in the plane, called α-curves. They are generated by application of a specially conceived geometric transform on straight lines, so that the RT inversion process remains as close as possible to that of the RT on the lines. It was hoped that the inversion of RT on lines would show the way to invert the RT on α-curves. However Cormack could only establish a closed form inversion formula for RT on 1 m ′ —curves, with m ′ = 1 , 2 , … . In this paper, we generalize his formula to RT on algebraic α-curves, i.e. for α = m m ′ with ( m , m ′ ) , a pair of positive relatively prime integers and thereby give support to his initial conjecture.