On Minimal Trajectories for Mobile Sampling of Bandlimited Fields

TL;DR

This paper investigates optimal mobile sampling trajectories for bandlimited fields, identifying minimal path densities for parallel lines and establishing bounds when stability margins are specified, addressing the problem's well-posedness.

Contribution

It characterizes minimal path density trajectories for stable sampling of bandlimited fields and addresses the ill-posedness by incorporating stability margins.

Findings

Parallel line trajectories with minimal path density are identified.

The problem is ill-posed without stability margin constraints.

Lower bounds on path density are established with explicit stability margins.

Abstract

We study the design of sampling trajectories for stable sampling and the reconstruction of bandlimited spatial fields using mobile sensors. The spectrum is assumed to be a symmetric convex set. As a performance metric we use the path density of the set of sampling trajectories that is defined as the total distance traveled by the moving sensors per unit spatial volume of the spatial region being monitored. Focussing first on parallel lines, we identify the set of parallel lines with minimal path density that contains a set of stable sampling for fields bandlimited to a known set. We then show that the problem becomes ill-posed when the optimization is performed over all trajectories by demonstrating a feasible trajectory set with arbitrarily low path density. However, the problem becomes well-posed if we explicitly specify the stability margins. We demonstrate this by obtaining a non-trivial lower bound on the path density of an arbitrary set of trajectories that contain a sampling set with explicitly specified stability bounds.