A new method of deconvolution and its application to lunar occultations

Abstract

A new method of deconvolution is described which uses our prior knowledge about the solution to derive some of the information obscured in the data because of the smoothing nature of convolution and the presence of noise. It uses a regularized least-squares criterion of agreement with the data, according to which the computed solution will lead to a minimum variance of noise and also be smooth in the sense of minimum variance of its second-differences. In addition, the present optimum deconvolution method (ODM) also constrains this solution to satisfy our prior knowledge about it by using a combination of a new algorithm for incorporating bounds on the solution like positivity, and the Lagrange multiplier method for equality-constraints. The new algoritham is a rapidly converging sequence of iterations for minimizing a weighted sum of squares of the deviation of the solution from the specified bounds. For the sake of illustration, ODM is compare with the conventional method of Scheuer for deconvolving the lunar occultation data to derive the brightness distribution of a radio source. The required occultation data have been obtained both from computer stimulations and from the observations of occultation with the Ooty radio telescope. A comparison of the restorations using the two methods indicates that a) ODM can be effectively applied even in very noisy situations; b) it leads to superresolution, implying an improvement in resolution by about a factor of two over the conventional method; and c) ODM provides a "clean" output leaving all the effects of noise to the residuals. A practical procedure has also been discussed for obtaining the effective resolution and restoring errors from an analysis of the residuals, perticularly their variance and power spectrum.