How complex is a random picture?

TL;DR

This paper investigates the informational complexity of random pictures modeled by Boolean models, focusing on the probability of needing many samples for reconstruction and the quantization error measured by Hausdorff distance.

Contribution

It provides new insights into the informational content and complexity of Boolean model samples, linking reconstruction probability and quantization error.

Findings

High probability of requiring many balls for full reconstruction

Quantization error analysis with respect to Hausdorff distance

Quantitative bounds on information content in Boolean models

Abstract

We study the amount of information that is contained in "random pictures", by which we mean the sample sets of a Boolean model. To quantify the notion "amount of information", two closely connected questions are investigated: on the one hand, we study the probability that a large number of balls is needed for a full reconstruction of a Boolean model sample set. On the other hand, we study the quantization error of the Boolean model w.r.t. the Hausdorff distance as a distortion measure.