Reconstruction of Multidimensional Signals from Irregular Noisy Samples

TL;DR

This paper analyzes the quality of reconstructing multidimensional signals from irregular noisy samples, deriving a tight MSE approximation by studying eigenvalue distributions and their asymptotic behavior.

Contribution

It introduces a novel approach to approximate the MSE of signal reconstruction using eigenvalue distribution moments and asymptotic analysis, applicable to high-dimensional fields.

Findings

Eigenvalue distribution tends to Marcenko-Pastur distribution in high dimensions

Derived a closed-form expression for distribution moments

Provided a tight MSE approximation for irregular noisy sampling

Abstract

We focus on a multidimensional field with uncorrelated spectrum, and study the quality of the reconstructed signal when the field samples are irregularly spaced and affected by independent and identically distributed noise. More specifically, we apply linear reconstruction techniques and take the mean square error (MSE) of the field estimate as a metric to evaluate the signal reconstruction quality. We find that the MSE analysis could be carried out by using the closed-form expression of the eigenvalue distribution of the matrix representing the sampling system. Unfortunately, such distribution is still unknown. Thus, we first derive a closed-form expression of the distribution moments, and we find that the eigenvalue distribution tends to the Marcenko-Pastur distribution as the field dimension goes to infinity. Finally, by using our approach, we derive a tight approximation to the MSE of the reconstructed field.