  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.
