Recent improvements in computational capability have given scientists increased ability to simulate large-scale, complex, real world phenomena. The data sets generated from these simulations vary in structure and organization, and rendering these complex topological connectivities is still a challenging problem. In this thesis, however, we move away from the traditional methods of visualizing simulation data, and present a novel approach that procedurally encodes the simulation data and discards the underlying grid and connectivity information. This encoding enables interactive manipulation and rendering of these large-scale simulations and requires significantly less storage than that needed for the original data representation. This thesis presents the complete process required to procedurally encode volumetric data using radial basis functions (RBFs). Several solution techniques for obtaining parameters needed in the RBF representations are described. These techniques range from low-cost clustering algorithms to computationally expensive nonlinear optimizations. A comparison of these techniques and the different basis functions targeting both low encoding errors and interactive renderings is presented. This functional approximation system is also extended to using more general basis functions, such as ellipsoidal basis functions (EBFs) that provide greater compression and visually more accurate encodings of volumetric scattered datasets. From the procedural encoding, compression ratios between 19:1 and 8075:1 can be obtained. Moreover, with the compact and accurate RBF and EBF functional representations of large-scale complex data, interactive rendering of data can be performed with desktop PCs utilizing commodity graphics hardware. In addition to static data approximation, temporal data is encoded using results from encoding previous timestep to speed the encoding time.
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