The dead leaves model : general results and limits at small scales

TL;DR

This paper rigorously defines the dead leaves model for natural image modeling, explores its limit as object sizes tend to zero, and analyzes the properties and regularity of the resulting random field.

Contribution

It introduces a limit random field for the dead leaves model at small scales and provides a rigorous mathematical framework and analysis of its properties.

Findings

Derived the probability of n compacts being in distinct visible parts.

Established the limit model as object sizes tend to zero.

Analyzed the regularity properties of the limit random field.

Abstract

In this work, we introduce a random field in view of natural image modeling, obtained as a limit of sequences of dead leaves models, when considering arbitrarily small or big objects. The dead leaves model, introduced by the Mathematical Morphology school, consists in the superposition of random closed sets, and enables to model the occlusion phenomena. When combined with specific sizes distributions for objects, they are known to provide adequate models for natural images. However this framework yields a small scales cutoff and a limit random field is introduced by letting this cutoff tend to zero. We first give a rigorous definition of the dead leaves model, and compute the probability that n compacts are included in distinct visible parts, which characterizes the model. Then, we derive our limit model and some of its property, and study its regularity.