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The reconstruction of a bounded deterministic field from binary-quantized observations of sensors which are randomly deployed over the field domain is studied. The sensor observations are corrupted by bounded additive noise. The study focuses on the extremes of lack of deterministic control in the sensor deployment, lack of knowledge of the noise distribution, and lack of sensing precision and reliability. Such 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. A simple estimator that reconstructs the entire field from these unreliable, binary-quantized, noisy observations is proposed. Technical conditions for the almost sure and mean squared error (MSE) convergence of the estimate to the field, as the number of sensors tends to infinity, are derived and their implications are discussed. For finite-dimensional, bounded-variation, and Sobolev-differentiable function classes, specific MSE decay rates are derived.
Date of Publication: March 2009