Subdivision and multiresolution for PUPs

Abstract

Partition of Unity Parametrics (PUPs) is a generalization of NURBS that permits the use of arbitrary basis functions to model parametric curves and surfaces. An interesting problem for PUPs is the identification of subdivision, reverse subdivision, and multiresolution schemes for this recently developed and flexible class of parametric curves and surfaces.In this paper, we introduce a systematic approach to derive uniform subdivision schemes for PUPs curves and tensor-product surfaces. Our approach formulates PUPs subdivision as a least squares problem. This allows us to find exact subdivision filters for refinable basis functions and optimal approximate schemes for irrefinable ones. Additionally, we derive PUPs multiresolution masks based on their subdivision filters. We formulate the problem as a constrained least squares optimization, such that the resulting multiresolution schemes are banded and optimal in terms of minimizing multiresolution reconstruction error.Finally, to illustrate our methods, we provide sample subdivision and multiresolution schemes with different properties. These include specific examples targeted towards applications of PUPs multiresolution schemes for compression, feature transfer, and macroscopic editing.