Optimal Measurement under Cost Constraints for Estimation of Propagating Wave Fields

We give a precise mathematical formulation of some measurement problems arising in optics, which is also applicable in a wide variety of other contexts. In essence the measurement problem is an estimation problem in which data collected by a number of noisy measurement probes are combined to reconstruct an unknown realization of a random process f(x) indexed by a spatial variable x epsi R^k for some k ges 1. We wish to optimally choose and position the probes given the statistical characterization of the process f(x) and of the measurement noise processes. We use a model in which we define a cost function for measurement probes depending on their resolving power. The estimation problem is then set up as an optimization problem in which we wish to minimize the mean-square estimation error summed over the entire domain of f subject to a total cost constraint for the probes. The decision variables are the number of probes, their positions and qualities. We are unable to offer a solution to this problem in such generality; however, for the metrical problem in which the number and locations of the probes are fixed, we give complete solutions for some special cases and an efficient numerical algorithm for computing the best trade-off between measurement cost and mean-square estimation error. A novel aspect of our formulation is its close connection with information theory; as we argue in the paper, the mutual information function is the natural cost function for a measurement device. The use of information as a cost measure for noisy measurements opens up several direct analogies between the measurement problem and classical problems of information theory, which are pointed out in the paper.