Resolution in Diffraction-limited Imaging, a Singular Value Analysis

Previous work in this series, in which the theory of singular systems has been used to discuss the problem of diffraction-limited imaging, is generalized to allow the reconstruction of an object over a region with ‘soft’ edges. The generalization is introduced to deal with gaussian-beam or other illumination or uncertain a priori knowledge of position. The solutions are developed in a weighted L 2 space. Examples of singular functions and vectors with both gaussian and sinc illumination are given. The analysis is applied to determine the amplitude response functions of the scanning optical or acoustic microscope systems proposed by Bertero and Pike in the first paper of the series and it is found that the performance of a scanning microscope of the new type should approach exactly that of a conventional coherent microscope of twice the resolving power.