Subdivision schemes, network flows and linear optimization
For researchers in approximation theory and geometric modeling, this work offers a novel theoretical framework and practical algorithms for analyzing and constructing subdivision schemes with guaranteed smoothness.
The paper establishes a connection between multivariate vector subdivision schemes and network flow theory/linear optimization, proving the existence of optimal difference masks that unify regularity analysis across univariate and multivariate cases, and providing efficient algorithms for their construction.
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.