Extrapolation of Bandlimited Multidimensional Signals from Continuous Measurements

TL;DR

This paper introduces a new iterative method for extrapolating multidimensional bandlimited signals from continuous measurements over bounded regions, with proven convergence and improved stability through regularization.

Contribution

It develops a theoretical framework and an iterative reconstruction method for multidimensional signals from continuous data, extending classical sampling theory.

Findings

Proves strong convergence of the iterative method for multidimensional signals.

Introduces a regularized iteration linked to Tikhonov regularization.

Demonstrates numerical stability improvements in 2D signal reconstruction.

Abstract

Conventional sampling and interpolation commonly rely on discrete measurements. In this paper, we develop a theoretical framework for extrapolation of signals in higher dimensions from knowledge of the continuous waveform on bounded high-dimensional regions. In particular, we propose an iterative method to reconstruct bandlimited multidimensional signals based on truncated versions of the original signal to bounded regions---herein referred to as continuous measurements. In the proposed method, the reconstruction is performed by iterating on a convex combination of region-limiting and bandlimiting operations. We show that this iteration consists of a firmly nonexpansive operator and prove strong convergence for multidimensional bandlimited signals. In order to improve numerical stability, we introduce a regularized iteration and show its connection to Tikhonov regularization. The method is illustrated numerically for two-dimensional signals.