The paper deals with state estimation of a spatially distributed system given noisy measurements from pointwise-in-time-and-space threshold sensors spread over the spatial domain of interest. A Maximum A posteriori Probability (MAP) approach is undertaken and a Moving Horizon (MH) approximation of the MAP cost-function is adopted. It is proved that, under system linearity and log-concavity of the noise probability density functions, the proposed MH-MAP state estimator amounts to the solution, at each sampling interval, of a convex optimization problem. Moreover, a suitable centralized solution for large-scale systems is proposed with a substantial decrease of the computational complexity. The latter algorithm is shown to be feasible for the state estimation of spatially-dependent dynamic fields described by Partial Differential Equations (PDE) via the use of the Finite Element (FE) spatial discretization method. A simulation case-study concerning estimation of a diffusion field is presented in order to demonstrate the effectiveness of the proposed approach. Quite remarkably, the numerical tests exhibit a noise-assisted behavior of the proposed approach in that the estimation accuracy results optimal in the presence of measurement noise with non-null variance.
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