Patchwork B-spline refinement

Abstract

Hierarchical splines allow to use representations with varying level of detail in different parts of a geometric model. However, the progression from coarse to fine scale is based on a single sequence of nested spline spaces. More precisely, each space defining a representation of some level must simultaneously be a subspace of all the higher level spaces and contain all the lower level ones. This requirement imposes severe restrictions on the available refinement strategies. We introduce the new framework of Patchwork B-splines (PB-splines), which alleviates these constraints and therefore increases the flexibility of the representations that are available in different parts of a geometric model. We derive the mathematical foundations of multivariate PB-splines, in particular focusing on the construction of a basis that forms a convex partition of unity. This generalizes the concept of truncated hierarchical (TH) B-splines to the novel framework. Moreover, we discuss the application of PB-splines to surface reconstruction with adaptive refinement. It is observed that the increased flexibility of the local representations provides significant advantages.