Bandlimited Spatial Field Sampling with Mobile Sensors in the Absence of Location Information

TL;DR

This paper investigates the problem of estimating spatially bandlimited fields using mobile sensors that collect samples at unknown locations governed by a renewal process, addressing challenges like noise and unknown bandwidth.

Contribution

It introduces a framework for field estimation without known sampling locations, providing an $O(1/n)$ error bound and an algorithm to determine the field's bandwidth.

Findings

Mean-squared error decreases as $O(1/n)$ with the number of samples.

Proposed algorithm accurately estimates the field's bandwidth with high probability.

Field estimation is feasible despite unknown sampling locations and measurement noise.

Abstract

Sampling of physical fields with mobile sensor is an emerging area. In this context, this work introduces and proposes solutions to a fundamental question: can a spatial field be estimated from samples taken at unknown sampling locations? Unknown sampling location, sample quantization, unknown bandwidth of the field, and presence of measurement-noise present difficulties in the process of field estimation. In this work, except for quantization, the other three issues will be tackled together in a mobile-sampling framework. Spatially bandlimited fields are considered. It is assumed that measurement-noise affected field samples are collected on spatial locations obtained from an unknown renewal process. That is, the samples are obtained on locations obtained from a renewal process, but the sampling locations and the renewal process distribution are unknown. In this unknown sampling location setup, it is shown that the mean-squared error in field estimation decreases as $O(1/n)$ where $n$ is the average number of samples collected by the mobile sensor. The average number of samples collected is determined by the inter-sample spacing distribution in the renewal process. An algorithm to ascertain spatial field's bandwidth is detailed, which works with high probability as the average number of samples $n$ increases. This algorithm works in the same setup, i.e., in the presence of measurement-noise and unknown sampling locations.