# Topological analysis of level sets and its use in data visualization

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Data visualization techniques use computational modeling and rendering methods to aid scientific discovery. The data is often volumetric and arises from various 3D imaging modalities. Time-varying volumetric data also arises as a result of various time-varying computational simulations. The data analysis involves identification, extraction, and quantitative analysis of features present in data, which are often represented as isosurfaces (i.e. level sets). This dissertation is focused on analyzing level sets topology in each of the processes to augment accuracy and functionality of visualization. We use contour trees as our main topological tool. The contour tree has been used to compute the topology of isosurfaces, generate a minimal seed set for accelerated isosurface extraction, and additionally provides a user interface to segment individual contour components in a scalar field. As one of the main contributions of our dissertation, we extend the benefits of contour trees to the analysis of time-varying data. We define temporal correspondence of contour components, and describe an algorithm to compute the correspondence information with time dependent contour trees. A graph representing the topology changes of time-varying isosurfaces is constructed in real-time for any selected isovalue using the precomputed correspondence information. Quantitative properties such as surface area and volume of contour components are computed and labelled on the graph. This topology change graph helps users to detect significant topological and geometric changes in time-varying isosurfaces. The graph is also used as an interactive user interface to segment, track and visualize the evolution of any selected contour components over time. The accurate construction of contour trees usually requires the data to be defined over a tetrahedral mesh of the domain. Most scalar volumetric data are very often defined over a rectilinear grid. Based on an analysis of level set topology of trilinear functions, the second contribution of my thesis is a procedure to decompose a rectilinear grid cell into a set of tetrahedra with the property that the level sets topology is preserved through the decomposition. General visualization algorithms that require scalar data to be defined on a tetrahedral grid utilize this technique to process trilinear functions on 3D rectilinear data, with topological preservation. The final part of my dissertation addresses the problem of triangular and tetrahedral mesh extraction from volumetric data. I focus specifically on generating a manifold mesh with correct trilinear topology. That is directly applicable to multiresolution meshing of level sets of 3D imaging and time-varying simulation data.