Precision bounds in superresolution processing

Abstract

A rather large class of problems involving the determination of an object function from observation is linear-inversion problems for which unique solutions exist but that have the property that any signal-processing algorithm designed to approximate the exact solution too precisely is unstable. This is because the problems are ill posed. The precision attainable in a class of such problems is treated here abstractly in terms of a concept called a linear-precision gauge, which essentially involves an ordered family of linear estimators. Fundamental properties of linear-precision gauges are demonstrated and discussed. A major portion of the paper is given over to applying the linear-precision gauge concept and results to Fourier imaging problems that can occur, for example, in radar and tomography.