Analysis of error in reconstruction of two-dimensional signals from irregularly spaced samples

We consider the problem of estimating a band-limited two-dimensional (2-D) signal based on a finite set of irregularly spaced samples. We derive the minimum mean-squared error estimator of the form of a sum of weighted interpolating functions that are identical in shape but are centered at the irregularly spaced sample points, and show that this estimator is identical to the minimum energy band-limited interpolator that has been previously obtained by others. The relation between the mean-squared error and the set of irregularly spaced sampling points is studied using the minimax principle. The maximum of the mean-squared error over a class of signals is minimized by proper choice of the set of sampling points. Three criteria for evaluating the performance of the sampling point set are derived. From a candidate group of sets of irregularly spaced sampling points, these criteria may be used to select that point set which is optimal for recovery over a class of signals.