EUROGRAPHICS 2006 / E. Gröller and L. Szirmay-Kalos (Guest Editors) Volume 25 (2006), Number 3 Enhancing the Interactive Visualization of Procedurally Encoded Multifield Data with Ellipsoidal Basis Functions Yun Jang † , Ralf P. Botchen ‡ , Andreas Lauser ‡ , David S. Ebert † , Kelly P. Gaither § , Thomas Ertl ‡ Purdue University Rendering and Perceptualization Lab, Purdue University, USA † Visualization and Interactive Systems, University of Stuttgart, Germany ‡ Texas Advanced Computing Center, University of Texas at Austin, TX, USA § Abstract Functional approximation of scattered data is a popular technique for compactly representing various types of datasets in computer graphics, including surface, volume, and vector datasets. Typically, sums of Gaussians or similar radial basis functions are used in the functional approximation and PC graphics hardware is used to quickly evaluate and render these datasets. Previously, researchers presented techniques for spatially-limited spherical Gaussian radial basis function encoding and visualization of volumetric scalar, vector, and multifield datasets. While truncated radially symmetric basis functions are quick to evaluate and simple for encoding op- timization, they are not the most appropriate choice for data that is not radially symmetric and are especially problematic for representing linear, planar, and many non-spherical structures. Therefore, we have developed a volumetric approximation and visualization system using ellipsoidal Gaussian functions which provides greater compression, and visually more accurate encodings of volumetric scattered datasets. In this paper, we extend previous work to use ellipsoidal Gaussians as basis functions, create a rendering system to adapt these basis func- tions to graphics hardware rendering, and evaluate the encoding effectiveness and performance for both spherical Gaussians and ellipsoidal Gaussians. Categories and Subject Descriptors (according to ACM CCS): I.3.3 [Computer Graphics]: Scientific Visualization, Ellipsoidal Basis Functions, Functional Approximation, Texture Advection 1. Introduction Most current visualization techniques are datagrid specific and do not allow scientists and researchers to interactively visualize various unstructured and scattered datasets in a single system on their desktop computers. As one solution, Jang et al. [JWH * 04] and Weiler et al. [WBS * 05] developed an interactive visualization and feature detection system for scalar, vector and multifield data using radial basis func- tions (RBFs). RBFs have been widely used as basis func- tions to approximate datasets, and the function is constructed as sums of RBFs. Mathematically, for given data samples P j =(  x j , Y j ), j = 1, ..., N, where the values  x j are the spatial † {jangy|ebertd}@purdue.edu ‡ {botchen|lauseras|ertl}@vis.uni-stuttgart.de § kelly@tacc.utexas.edu locations and the values Y j are the data values that exist at the corresponding spatial locations, the data values can be approximated with a function f (  x) defined as f (  x)= L ∑ i=1 λ i φ   x, m i , σ 2 i  , (1) where L is the number of basis functions, λ i the weight, m i the center, and σ 2 i the variance of a single basis function. With this mathematical formula, λ , m and σ 2 are optimized to find the best approximation of the original data values. Jang et al. [JWH * 04] and Weiler et al. [WBS * 05] used spatially-limited spherical Gaussian RBFs, since the trun- cated radially symmetric basis functions are quick to eval- uate and simple for encoding optimization. However, spher- ical Gaussians are not the most appropriate choice for volu- metric data that is not radially symmetric and they are espe- c  The Eurographics Association and Blackwell Publishing 2006. Published by Blackwell Publishing, 9600 Garsington Road, Oxford OX4 2DQ, UK and 350 Main Street, Malden, MA 02148, USA.