Rate-distortion problem for physics based distributed sensing [temperature measurement]

We consider the rate-distortion problem for sensing the continuous space-time physical temperature in a circular ring on which a heat source is applied over space and time, and which is allowed to cool by radiation or convection. The heat source is modelled as a continuous space-time stochastic process which is bandlimited over space and time. The temperature field is the result of a certain continuous space-time convolution of the heat source with the Green´s function corresponding to the heat equation, which is space and time invariant. The temperature field is sampled at uniform spatial locations by a set of sensors and it has to be reconstructed at a base station. The goal is to minimize the mean-square-error per second, for a given number of nats per second, assuming ideal communication channels between sensors and base station. We find a) the centralized R^c(D) function of the temperature field, where the base station can optimally encode all the space-time samples jointly. Then, we obtain b) the R^s-i(D) function, where each sensor, independently, encodes its samples optimally over time, and c) the R^st-i(D) function, where each sensor is constrained to encode also independently over time. We also study two distributed prediction-based approaches: a) with perfect feedback from the base station, where temporal prediction is performed at the base station and each sensor performs differential encoding; and b) without feedback, where each sensor locally performs temporal prediction.