Abstract

We study the problem of reconstructing the minimum support object that is consistent with a given image of the object produced by a not necessarily isoplanatic diffraction-limited imaging system. We assume without loss of generality that the object lies entirely in the first quadrant and its support is tangent to both coordinate axes. Furthermore, we assume that the size of the given image is larger than that of the object. We begin by discretizing the imaging equation to obtain a linear system of the form where x and y are vectors made of samples of the object and image, respectively, n is a measurement noise process, and H is a matrix made of samples of the possibly spacevariant impulse response of the imaging system. Next, we apply a sequence of statistical tests to the data to estimate the support of the minimal support object that is closest to the data in a 2-norm sense. The tests are equivalent to projecting the data on subspaces corresponding to the span of a sequence of increasing size combinations of columns of H and comparing to a threshold proportional to the noise variance. The estimated minimal support then specifies a matrix H' made of a subset of columns of H. The pseudoinverse of H' is finally used together with a constraint of the form || Bx || ≤ α, reflecting some a priori knowledge about the object (such as smoothness), to estimate x.

© 1989 Optical Society of America