Dynamical Sampling and its Applications

Abstract

Dynamical sampling is a new area in sampling theory that deals with signals that evolve over time under the action of a linear operator. There are lots of studies on various aspects of the dynamical sampling problem. However, they all focus on uniform discrete time-sets $mathcal{T} subset mathbb{N}$. In our first paper, we concentrate on the case $mathcal{T} = [0, L]$. The goal of the present work is to study the frame property of the systems ${A^tg : g inmathcal{G}, t in [0, L]}$. To this end, we also characterize the completeness and Besselness properties of these systems. In our second paper, we consider dynamical sampling when the samples are corrupted by additive noises. The purpose of the second paper is to analyze the performance of the basic dynamical sampling algorithms in the finite dimensional case and study the impact of additive noise. The algorithms are implemented and tested on synthetic and real data sets, and denoising techniques are integrated to mitigate the effect of the noise. We also develop theoretical and numerical results that validate the algorithm for recovering the driving operators, which are defined via a real symmetric convolution.