Optimal Transport Approximation of 2-Dimensional Measures

TL;DR

This paper introduces a scalable algorithm for projecting 2D measures onto structured sets, enabling efficient applications in sampling, rendering, and path planning with improved convergence and acceleration techniques.

Contribution

It generalizes previous blue-noise generation methods to structured measures and develops new algorithms for curve projection with bounded geometric properties.

Findings

Algorithm is fast and scalable for 2D measure projection.

Effective in applications like sampling, rendering, and path planning.

Shows improved convergence and acceleration over previous methods.

Abstract

We propose a fast and scalable algorithm to project a given density on a set of structured measures defined over a compact 2D domain. The measures can be discrete or supported on curves for instance. The proposed principle and algorithm are a natural generalization of previous results revolving around the generation of blue-noise point distributions, such as Lloyd's algorithm or more advanced techniques based on power diagrams. We analyze the convergence properties and propose new approaches to accelerate the generation of point distributions. We also design new algorithms to project curves onto spaces of curves with bounded length and curvature or speed and acceleration. We illustrate the algorithm's interest through applications in advanced sampling theory, non-photorealistic rendering and path planning.