Multiresolution heterogeneous solid modeling and visualization using trivariate simplex splines

This paper presents a new and powerful heterogeneous solid modeling paradigm for representing, modeling, and rendering of multi-dimensional, physical attributes across any volumetric objects. The modeled solid can be of complicated geometry and arbitrary topology. It is formulated using a trivariate simplex spline defined over a tetrahedral decomposition of any 3D domain. Heterogeneous material attributes associated with solid geometry can be modeled and edited by manipulating the control vectors and/or associated knots of trivariate simplex splines easily. The multiresolution capability is achieved by interactively subdividing any regions of interest and allocating more knots and control vectors accordingly. We also develop a feature-sensitive fitting algorithm that can reconstruct a more compact, continuous trivariate simplex spline from structured or unstructured volumetric grids. This multiresolution representation results from the adaptive and progressive tetrahedralization of the 3D domain. In addition, based on the simplex spline theory, we derive several theoretical formula and propose a fast direct rendering algorithm for interactive data analysis and visualization of the simplex spline volumes. Our experiments demonstrate that the proposed paradigm augments the current modeling and visualization techniques with the new and unique advantages.