Generalized Sampling: A Variational Approach. Part I: Theory

Abstract

We consider the problem of reconstructing a multidimensional vector function fin: Rm→Rn from a finite set of linear measures. These can be irregularly sampled responses of several linear filters. Traditional approaches reconstruct in an a priori given space, e.g., the space of bandlimited functions. Instead, we have chosen to specify a reconstruction that is optimal in the sense of a quadratic plausibility criterion J. First, we present the solution of the generalized interpolation problem. Later, we also consider the approximation problem, and we show that both lead to the same class of solutions. Imposing generally desirable properties on the reconstruction largely limits the choice of the criterion J. Linearity leads to a quadratic criterion based on bilinear forms. Specifically, we show that the requirements of translation, rotation, and scale-invariance restrict the form of the criterion to essentially a one-parameter family. We show that the solution can be obtained as a linear combination of generating functions. We provide analytical techniques to find these functions and the solution itself. Practical implementation issues and examples of applications are treated in a companion paper.