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

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, except for a renewal process structure, the sampling locations and the inter-sample distribution are unknown. It is shown that the mean-squared error decreases as $O(1/n)$ where $n$ is the oversampling employed by the mobile sensor. Oversampling is obtained by controlling the mean value of the inter-sample spacing. An algorithm to ascertain spatial field's bandwidth is detailed, which works with high probability as the oversampling increases. This algorithm works in the same setup, i.e., in the presence of measurement-noise and unknown sampling locations.