Gaussian Random Particles with Flexible Hausdorff Dimension

TL;DR

This paper introduces a flexible Gaussian particle model with adjustable Hausdorff dimension for simulating 3D star-shaped sets, using kernel smoothing of isotropic random fields on the sphere.

Contribution

It develops a novel framework linking kernel choices to the Hausdorff dimension of particle surfaces, enabling boundary control between 2 and 3.

Findings

Closed-form correlation functions for specific kernels

Power kernels allow boundary Hausdorff dimension adjustment

Model facilitates realistic 3D shape simulations

Abstract

Gaussian particles provide a flexible framework for modelling and simulating three-dimensional star-shaped random sets. In our framework, the radial function of the particle arises from a kernel smoothing, and is associated with an isotropic random field on the sphere. If the kernel is a von Mises--Fisher density, or uniform on a spherical cap, the correlation function of the associated random field admits a closed form expression. The Hausdorff dimension of the surface of the Gaussian particle reflects the decay of the correlation function at the origin, as quantified by the fractal index. Under power kernels we obtain particles with boundaries of any Hausdorff dimension between 2 and 3.