On Non-Parametric Field Estimation using Randomly Deployed, Noisy, Binary Sensors

We consider the problem of reconstructing a deterministic data field from binary quantized noisy observations of sensors randomly deployed over the field domain. Our focus is on the extremes of lack of control in the sensor deployment, arbitrariness and lack of knowledge of the noise distribution, and low-precision and unreliability in the sensors. These adverse conditions are motivated by possible real-world scenarios where a large collection of low-cost, crudely manufactured sensors are mass-deployed in an environment where little can be assumed about the ambient noise. We propose a simple estimator that reconstructs the entire data field from these unreliable, binary quantized, noisy observations. Under the assumption of a bounded amplitude field, we prove almost sure and mean-square convergence of the estimator to the actual field as the number of sensors tends to infinity. For fields with bounded-variation, Sobolev differentiable, or finite-dimensionality properties, we derive specific mean squared error (MSE) decay rates. The analysis techniques used herein expose the effects of field ";smoothness"; properties, location randomness, and noise on the MSE scaling behavior.