Technical Report HCIS-2003-06

This paper deals with modeling heterogeneous volumetric objects as point sets with attributes using trivariate B-splines. In contrast to homogeneous volumes with uniform distribution of material and other properties, a heterogeneous volumetric object has a number of variable attributes assigned at each point. An attribute is a mathematical model of an object property of arbitrary nature (material, photometric, physical, statistical, etc.). In our approach, the function representation (FRep) is used as the basic model for both object geometry and attributes represented independently using real-valued scalar functions of point coordinates. While FRep directly defines object geometry, for an attribute it specifies a space partition used to define the attribute function. We propose a volume sculpting scheme with multiresolution capability based on trivariate B-spline functions to define both object geometry and its attributes. A new trivariate B-spline primitive is proposed and can be used as a leave in a FRep constructive tree. Additional deformations can be applied to the constructed object by moving arbitrary control points in space. An interactive volume modeler based on trivariate B-splines and other primitives is described. It supports the proposed sculpting, constructive modeling, deformations by space mappings, and real time re-polygonization of the surface during interactive modeling. We illustrate that the space partition obtained in the modeling process can be applied to define attributes for the objects with an arbitrary geometry model such as B-rep or homogeneous volume models. 1. INTRODUCTION Volume modeling has become an important research topic. In contrast to homogeneous volumes with uniform distribution of properties, a heterogeneous volu-metric object has a number of attributes assigned at each point and varying in 3D space. It is not necessary for an attribute to be continuous. Heterogeneous volu-metric objects can be modeled as 3D point sets with non-uniform distribution of attributes of an arbitrary nature (photometric, material, physical, statistical, etc.). Heterogeneous objects are considered in such different areas as CAD/CAM and rapid prototyping of objects with multiple materials and varying material distribution , representing results of physical simulations, geological and medical modeling, volume modeling and rendering. Different models based on the boundary representation [10, 9], volumetric [3, 19, 14], and function-based constructive techniques [20] have been proposed for heterogeneous objects. As it was shown in the mentioned publications, real-valued functions serve well for modeling both geometry and attributes. We follow in this paper a generalized constructive hypervolume model [20, 21], which supports uniform constructive modeling of …