A transformation method for the reconstruction of functions from nonuniformly spaced samples

The reconstruction of functions from their samples at nonuniformly distributed locations is an important task for many applications. This paper presents a sampling theory which extends the uniform sampling theory of Whittaker et al. [11] to include nonuniform sample distributions. This extension is similar to the analysis of Papoulis [15], who considered reconstructions of functions that had been sampled at positions deviating slightly from a uniform sequence. Instead of treating the sample sequence as deviating from a uniform sequence, we show that a more general result can be obtained by treating the sample sequence as the result of applying a coordinate transformation to the uniform sequence. It is shown that the class of functions reconstructible in this manner generally include nonband-limited functions. The two-dimensional uniform sampling theory of Petersen and Middle ton [16] can be similarly extended as is shown in this paper. A practical algorithm for performing reconstructions of two-dimensional functions from nonuniformly spaced samples is described, as well as examples illustrating the performance of the algorithm.