Sampling and Reconstruction of Polyhedra Observed in Noise

The problem of data collection in a sensor network is formulated in this work as one of distributed analog-to-digital conversion of signals observed in noise. The sampling literature is extensive, going back far in time. But, in the formulation considered here, there are two new elements. One is that the signals of interest are polyhedra in Rⁿ. The other is that the analog samples from which these polyhedra are to be recovered are measurements of a wave contained therein, not bandlimited, and subject to random phase perturbations. The formulation comes directly from a real sensor networking system, in which it is necessary to recover the shape of a room out of acoustic measurements collected by the network. Sufficient conditions are given, and a reconstruction algorithm is developed, that guarantee recovery of any polyhedra from samples collected at a finite number of points in space. The performance of this algorithm is also characterized in the presence of random phase perturbations in the samples.