Abel integral equation inversion and deconvolution by multi-Gaussian approximation

It is shown that the numerical inversion of the Abel integral equation relating projected and spatial densities, and the deconvolution of an observed profile from its point-spread function, becomes, by means of Gabor expansions in series of elementary Gaussians, nonlinear fitting problems when the involved distributions are characterized by radial symmetry and smooth behavior. Thus, both these ill-posed inverse problems, relevant in all research fields, can be easily solved by a robust estimation procedure based on the Newton-Gauss method, regularized in the Tikhonov sense, using only a personal computer. Further, this approach allows us to reach satisfactory automatic determinations from indirect measurements made under standard physical conditions. As examples of astronomical applications, the deconvolution of an undersampled CCD image and the spatial density in a King-reduced brightness profile are presented.