Exact Reconstruction of Spatially Undersampled Signals in Evolutionary Systems

TL;DR

This paper addresses the problem of reconstructing initial signals in evolution systems from coarse, multi-time samples, introducing a novel dynamical sampling approach that enables exact recovery under certain conditions.

Contribution

It develops a framework for exact reconstruction of signals from undersampled data across multiple time levels in evolution systems, expanding classical sampling theory.

Findings

Exact reconstruction formulas derived for undersampled signals.

Conditions identified for perfect recovery in shift-invariant spaces.

Applicability demonstrated for signals in () and shift-invariant spaces.

Abstract

We consider the problem of spatiotemporal sampling in which an initial state $f$ of an evolution process $f_t=A_tf$ is to be recovered from a combined set of coarse samples from varying time levels $\{t_1,\dots,t_N\}$. This new way of sampling, which we call dynamical sampling, differs from standard sampling since at any fixed time $t_i$ there are not enough samples to recover the function $f$ or the state $f_{t_i}$. Although dynamical sampling is an inverse problem, it differs from the typical inverse problems in which $f$ is to be recovered from $A_Tf$ for a single time $T$. In this paper, we consider signals that are modeled by $\ell^2(\mathbb Z)$ or a shift invariant space $V\subset L^2(\mathbb R)$.