A diffusion-driven Characteristic Mapping method for particle management

TL;DR

This paper introduces a diffusion-driven characteristic mapping method to improve particle management by maintaining uniform sampling on evolving curves and surfaces, leveraging heat equation properties for efficiency.

Contribution

It proposes a novel particle management technique using characteristic mapping and diffusion processes to ensure even distribution during surface evolution.

Findings

Uniform redistribution of marker points achieved.

Efficient handling of extensive deformations.

Improved sampling quality on evolving surfaces.

Abstract

We present a novel particle management method using the Characteristic Mapping framework. In the context of explicit evolution of parametrized curves and surfaces, the surface distribution of marker points created from sampling the parametric space is controlled by the area element of the parametrization function. As the surface evolves, the area element becomes uneven and the sampling, suboptimal. In this method we maintain the quality of the sampling by pre-composition of the parametrization with a deformation map of the parametric space. This deformation is generated by the velocity field associated to the diffusion process on the space of probability distributions and induces a uniform redistribution of the marker points. We also exploit the semigroup property of the heat equation to generate a submap decomposition of the deformation map which provides an efficient way of maintaining evenly distributed marker points on curves and surfaces undergoing extensive deformations.