FEM-based scattered data modeling and visualization

Abstract

A critical challenge in visualizing scattered data is to correctly model the sample data so that data variation throughout the volume of interest can be accurately rendered. The commonly used interpolation-based approach is unsatisfactory, as it often generates physically impossible data values in the modeling process. In addition, it does not provide a systematic way of estimating errors. The interpolation methods used for modeling are usually different from those used for rendering, which causes inconsistency and misrepresentation. Furthermore, interpolation methods cannot handle discontinuities, due to their inherent assumption that the data are continuous. To eliminate these and other problems, we construct an alternative approach to scattered data visualization. Based on the finite element method (FEM), this FEM-based approach incorporates the governing equations of the data into the modeling process to ensure the modeled data to be physically meaningful. It provides error estimates that can guide the refinement of the finite element network to obtain the desired accuracy. It allows the selection of basis functions in the modeling process to match with the interpolation functions used by the rendering process so that consistency can be achieved. It handles discontinuities with the help of the double-layer scheme. Furthermore, it converts the data-modeling problem from an interpolation problem into a boundary-value problem, and therefore reduces the requirement on the density of the input sample data, a feature which is very valuable to applications where sample data are hard to obtain. This paper presents the framework and a sample implementation of the FEM-based approach along with some examples.