Three-dimensional signal processing: a new approach in dynamical sampling via tensor products

TL;DR

This paper introduces a novel approach for three-dimensional dynamical sampling using tensor products, providing necessary sampling conditions and an efficient reconstruction method, supported by numerical simulations.

Contribution

It extends dynamical sampling theory to 3D signals with tensor product evolution systems, offering new necessary conditions and an optimization-based reconstruction framework.

Findings

Established necessary sampling conditions for 3D tensor product signals

Reformulated reconstruction as an efficient optimization problem

Numerical simulations demonstrate successful signal recovery

Abstract

The dynamical sampling problem is centered around reconstructing signals that evolve over time according to a dynamical process, from spatial-temporal samples that may be noisy. This topic has been thoroughly explored for one-dimensional signals. Multidimensional signal recovery has also been studied, but primarily in scenarios where the driving operator is a convolution operator. In this work, we shift our focus to the dynamical sampling problem in the context of three-dimensional signal recovery, where the evolution system can be characterized by tensor products. Specifically, we provide a necessary condition for the sampling set that ensures successful recovery of the three-dimensional signal. Furthermore, we reformulate the reconstruction problem as an optimization task, which can be solved efficiently. To demonstrate the effectiveness of our approach, we include some straightforward numerical simulations that showcase the reconstruction performance.