Topological analysis of scalar functions for scientific data visualization

Scientists attempt to understand physical phenomena by studying various quantities measured over the region of interest. A majority of these quantities are scalar (real-valued) functions. These functions are typically studied using traditional visualization techniques like isosurface extraction, volume rendering etc. As the data grows in size and becomes increasingly complex, these techniques are no longer effective. State of the art visualization methods attempt to automatically extract features and annotate a display of the data with a visualization of its features. In this thesis, we study and extract the topological features of the data and use them for visualization. We have three results: (1) An algorithm that simplifies a scalar function defined over a tetrahedral mesh. In addition to minimizing the error introduced by the approximation of the function, the algorithm improves the mesh quality and preserves the topology of the domain. We perform an extensive set of experiments to study the effect of requiring better mesh quality on the approximation error and the level of simplification possible. We also study the effect of simplification on the topological features of the data. (2) An extension of three-dimensional Morse-Smale complexes to piecewise linear 3-manifolds and an efficient algorithm to compute its combinatorial analog. Morse-Smale complexes partition the domain into regions with similar gradient flows. Letting n be the number of vertices in the input mesh, the running time of the algorithm is proportional to n log(n) plus the total size of the input mesh plus the total size of the output. We develop a visualization tool that displays different substructures of the Morse-Smale complex. (3) A new comparison measure between k functions defined on a common d-manifold. For the case d = k = 2, we give alternative formulations of the definition based on a Morse theoretic point of view. We also develop visualization software that performs local comparison between pairs of functions in datasets containing multiple and sometimes time-varying functions.;We apply our methods to data from medical imaging, electron microscopy, and x-ray crystallography. The results of these experiments provide evidence of the usability of our methods.