Resolution in Diffraction-limited Imaging, a Singular Value Analysis

The familiar theory of information transfer in imaging using the prolate spheroidal functions and their eigenvalue spectrum is extended to allow the object and image domains to differ. The appropriate theory becomes one of singular functions and singular values, and in this paper we give a description of coherent imaging in these terms. Super-resolution in the sense of improving on previous criteria in the presence of noise can then be achieved, particularly at very low Shannon numbers, using the physical continued image, and we give quantitative estimates of such improvements for the linear, square and circular pupil cases. The theory is also shown to provide an efficient proof of the theorem that the number of degrees of freedom of a coherent isoplanatic imaging system is unaffected by phase aberrations in the pupil. Applications of these results in microscopy are outlined, and a practical method of implementation is proposed based on a generalization of numerical inversion techniques developed recently in the field of laser scattering.