Representation and extraction of volumetric attributes using trivariate splines: a mathematical framework

Abstract

Our goal in this paper is to leverage traditional strengths from the geometric design and scientific visualization communities to produce a tool valuable to both. We present a method for representing and specifying attribute data across a trivariate NURBS volume. Some relevant attribute quantities include material composition and density, optical indices of refraction and dispersion, and data from medical imaging. The method is independent of the granularity of the physical geometry, allowing for a decoupling of the resolution of the carried data from that of the volume. Volume attributes can be modeled or fit to data. A method is presented for efficient evaluation of trivariate NURBS. We incorporate methods for data analysis and visualization including isosurface extraction, planar slicing, volume ray tracing, and optical path tracing, all of which are grounded in refinement theory for splines. The applications for these techniques are diverse, including such fields as optics, fluid dynamics, and medical visualization.