A stable numerical inversion of Abel's integral equation using almost Bernstein operational matrix

Abstract

Many problems in physics like reconstruction of the radially distributed emissivity from the line-of-sight projected intensity, the 3-D image reconstruction from cone-beam projections in computerized tomography, etc. lead naturally, in the case of radial symmetry, to the study of Abel's type integral equation. The aim of this communication is to modify the stable algorithm proposed in [Singh VK, Pandey RK, Singh OP. New stable numerical solution of singular integral equations of Abel type by using normalized Bernstein polynomials. Applied Mathematical Sciences 2009;3(5):241–255] which is based on normalized Bernstein polynomial approximation of the projected intensity profile. So, first we construct an orthonormal family { b i 5 } i = 0 5 of polynomials of degree 5 from the 5th degree Bernstein polynomials B i 5 and use them as a basis to approximate the projected intensity profile. Then, a 6×6 matrix P, named as almost Bernstein operational matrix of integration is constructed and used to reduce the integral equation to a system of algebraic equation which can be solved easily. The method is quite accurate and stable even though the approximations are performed by polynomials of degree 5, as illustrated by applying the method to intensity data with and without random noise to invert and compare it with those obtained by the other methods or with the known analytical inverse. Thus it is good method for applying to experimental intensities distorted by noise.