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arXiv · 1102.3080

Covering Point Patterns

Amos Lapidoth·Andreas Malär·Ligong Wang
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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 $-λ\log D$. It is also shown that, as $T$ tends to infinity, any point pattern on $[0,T]$ containing no more than $λT$ points can be successfully described using $-λ\log D$ 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.

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Amos Lapidoth, Andreas Malär, Ligong Wang. 2011-02-15. Covering Point Patterns. https://arxiv.org/abs/1102.3080

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