Covering Point Patterns

TL;DR

This paper investigates the minimal bit rate required to encode point patterns from a Poisson process so that a subset containing all points can be reconstructed with a bounded average measure, extending to a Wyner-Ziv setting.

Contribution

It derives the asymptotic minimal encoding rate for Poisson point patterns under measure constraints and extends the analysis to scenarios with partial point knowledge.

Findings

Minimal bits per second needed is $- ext{lambda} ext{log} D$ for Poisson processes.

Any pattern with up to $ ext{lambda} T$ points can be described at this rate asymptotically.

The problem is extended to a Wyner-Ziv setting with partial point knowledge.

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 $\lambda$, 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 $-\lambda\log D$. It is also shown that, as $T$ tends to infinity, any point pattern on $[0,T]$ containing no more than $\lambda T$ points can be successfully described using $-\lambda \log D$ 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.