High resolution coding of point processes and the Boolean model

The thesis High resolution coding of point processes and the Boolean model is a contribution to the field of coding theory, with a special focus on the problem of quantization, entropy constrained coding and random coding. We provide an asymptotic upper bound for the quantization error of point processes on bounded metric spaces with finite upper Minkowski-dimension. Therefore we consider the point process conditioned upon the number of points and construct specific codebooks for these conditional processes. Via the cardinality of these codebooks we get a relation between the quantization error and the given rate. As a special case, we establish upper and lower bounds for the quantization error asymptotics of a stationary Poisson point process on a compact subset of R under Hausdorffdistance. For the lower bound we use the relation between the quantization error and the so called small ball probabilities. Furthermore we compute an asymptotic upper bound of the entropy constrained error and compare the results with the Gaussian case. In the case of one dimension we introduce aD ([0, a], {w1, . . . , wq})-valued random element induced by a point process on the compact interval [0, a] ⊂ R satisfying a certain growth condition and provide an asymptotic upper bound of the quantization error under L1-distance. For a D ([0, 1], {0, 1})-valued random element induced by a stationary Poisson point process on [0, 1] we give asymptotic upper and lower bounds of the quantization error and compare these to the asymptotics of the random coding error and the entropy constrained error. We further discuss the Boolean model, where a random set is constructed as the Minkowski sum of the points of a Poisson point process and a given random set, e.g. a ball with random radius. For an asymptotic upper bound of the quantization error under Hausdorff-distance we consider the corresponding Poisson point process conditioned upon the number of points in a compact set. We use one part of the given rate to code the number and the position of these points and the rest of the rate to code the random compact sets. For the lower bound we use again the relation between the quantization error and the small ball probabilities. Therewith we provide asymptotic upper and lower bounds for the quantization error under Hausdorff-distance and compare these with the asymptotics of the quantization error of the Boolean model under the L1-distance.