Differentiation and regularity of semi-discrete optimal transport with respect to the parameters of the discrete measure

TL;DR

This paper investigates the conditions under which semi-discrete optimal transport is twice differentiable with respect to the parameters of the discrete measure, with applications in stippling and blue noise problems.

Contribution

It establishes minimal conditions on the background measure for differentiability and demonstrates numerical applications in stippling and blue noise generation.

Findings

Semi-discrete optimal transport is twice differentiable under certain conditions.

Numerical illustrations show practical applications in stippling and blue noise.

Minimal background measure conditions ensure differentiability.

Abstract

This paper aims at determining under which conditions the semi-discrete optimal transport is twice differentiable with respect to the parameters of the discrete measure and exhibits numerical applications. The discussion focuses on minimal conditions on the background measure to ensure differentiability. We provide numerical illustrations in stippling and blue noise problems.