Bandlimited Field Reconstruction from Samples Obtained at Unknown Random Locations on a Grid

We study the sampling of spatial fields using sensors that are location-unaware but deployed according to a known statistical distribution. It has been shown that uniformly distributed location-unaware sensors cannot infer bandlimited fields due to the symmetry and shift-invariance of the field. This work studies asymmetric (nonuniform) distributions on location-unaware sensors that will enable bandlimited field inference. For the sake of analytical tractability, location-unaware sensors are restricted to a discrete grid. Oversampling followed by clustering of the samples using the probability distribution that governs sensor placement on the grid is used to infer the field . Based on this clustering algorithm, the main result of this work is to find the optimal probability distribution on sensor locations that minimizes the detection error-probability of the underlying spatial field. The proposed clustering algorithm is also extended to include the case of signal reconstruction in the presence of sensor noise by treating the distribution of the noisy samples as a mixture model and using clustering to estimate the mixture model parameters.