Subdivision schemes, network flows and linear optimization

TL;DR

This paper establishes a novel connection between subdivision schemes, network flow theory, and linear optimization to analyze regularity and smoothness, providing new methods for constructing optimal masks and proving regularity.

Contribution

It introduces the concept of optimal difference masks and offers efficient algorithms for their construction, unifying regularity analysis across univariate and multivariate schemes.

Findings

Existence of optimal difference masks with key properties.

Efficient algorithms for constructing these masks.

Analytic proofs of $C^k$-regularity based on integrality.

Abstract

We link regularity and smoothness analysis of multivariate vector subdivision schemes with network flow theory and with special linear optimization problems. This connection allows us to prove the existence of what we call optimal difference masks that posses crucial properties unifying the regularity analysis of univariate and multivariate subdivision schemes. We also provide efficient optimization algorithms for construction of such optimal masks. Integrality of the corresponding optimal values leads to purely analytic proofs of $C^k-$regularity of subdivision.