Generalized PolyCube Trivariate Splines

Abstract

This paper develops a new trivariate hierarchical spline scheme for volumetric data representation. Unlike conventional spline formulations and techniques, our new framework is built upon a novel parametric domain called Generalized PolyCube (GPC), comprising a set of regular cubes being glued together. Compared with the conventional PolyCube (PC) that could serve as a ``one-piece'' $3$-manifold domain, GPC has more powerful and flexible representation ability. We develop an effective framework that parameterizes a solid model onto a topologically equivalent GPC domain, and design a hierarchical fitting scheme based on trivariate T-splines. The entire data-spline-conversion modeling framework provides high-accuracy data fitting and greatly reduce the number of superfluous control points. It is a powerful toolkit with broader application appeal in shape modeling, engineering analysis, and reverse engineering.