Covering Point Patterns
Abstract
An encoder observes a point pattern---a finite number of points in the interval [0,T]---which is to be described to a reconstructor using bits. Based on these bits, the reconstructor wishes to select a subset of [0,T] that contains all the points in the pattern. It is shown that, if the point pattern is produced by a homogeneous Poisson process of intensity λ, and if the reconstructor is restricted to select a subset of average Lebesgue measure not exceeding DT, then, as T tends to infinity, the minimum number of bits per second needed by the encoder is -λ D. It is also shown that, as T tends to infinity, any point pattern on [0,T] containing no more than λ T points can be successfully described using -λ D bits per second in this sense. Finally, a Wyner-Ziv version of this problem is considered where some of the points in the pattern are known to the reconstructor.