Consistency of Variational Continuous-Domain Quantization via Kinetic Theory

TL;DR

This paper rigorously derives the mean-field kinetic equation for particle systems used in image halftoning, demonstrating the method's consistency and convergence to the prescribed image profile as the number of particles increases.

Contribution

It provides a rigorous derivation of the mean-field limit for particle-based halftoning and proves the system's energy stability and convergence to the desired image.

Findings

Energy functional acts as a Lyapunov functional.

Solutions tend to equilibrium minimizing the energy.

In a special case, equilibrium matches the prescribed image profile.

Abstract

We study the kinetic mean-field limits of the discrete systems of interacting particles used for halftoning of images in the sense of continuous-domain quantization. Under mild assumptions on the regularity of the interacting kernels we provide a rigorous derivation of the mean-field kinetic equation. Moreover, we study the energy of the system, show that it is a Lyapunov functional and prove that in the long time limit the solution tends to an equilibrium given by a local minimum of the energy. In a special case we prove that the equilibrium is unique and is identical to the prescribed image profile. This proves the consistency of the particle halftoning method when the number of particles tends to infinity.