Resolution in Diffraction-limited Imaging, a Singular Value Analysis

Abstract

In two previous papers we have considered the problem of object restoration in diffraction-limited imaging problems where the object and image domains are allowed to differ. The investigations were performed in terms of singular functions and singular values, instead of the usual eigenfunctions and eigenvalues which apply to geometrical imaging only. We deduced the number of degrees of freedom of the image and obtained new resolution limits. More realistically, in an actual diffraction experiment the image data are known only at a set of sampled points rather than as a continuous function. The power of the singular value analysis is such, however, that we may modify the previous work to consider the mapping of continuous object functions into vector data functions as a singular system. This is carried out in the present paper, and new singular values, functions and vectors appropriate to a sampled image and continuous object are found. We prove that when the number of image points tends to infinity, the singular system of the discrete problem converges to the singular system of the continuous problem.