Progressive Simplification of Tetrahedral Meshes Preserving All Isosurface Topologies

Abstract In this paper, we propose a novel technique for constructing multiple levels of a tetrahedral volume dataset whilepreserving the topologies of all isosurfaces embedded in the data. Our simplification technique has two majorphases. In the segmentation phase, we segment the volume data into topological‐equivalence regions, that is, thesub‐volumes within each of which all isosurfaces have the same topology. In the simplification phase, we simplifyeach topological‐equivalence region independently, one by one, by collapsing edges from the smallest to the largesterrors (within the user‐specified error tolerance, for a given error metrics), and ensure that we do not collapseedges that may cause an isosurface‐topology change. We also avoid creating a tetrahedral cell of negative volume(i.e., avoid the fold‐over problem). In this way, we guarantee to preserve all isosurface topologies in the entiresimplification process, with a controlled geometric error bound. Our method also involves several additionalnovel ideas, including using the Morse theory and the implicit fully augmented contour tree, identifying typesof edges that are not allowed to be collapsed, and developing efficient techniques to avoid many unnecessary orexpensive checkings, all in an integrated manner. The experiments show that all the resulting isosurfaces preservethe topologies, and have good accuracies in their geometric shapes. Moreover, we obtain nice data‐reductionrates, with competitively fast running times.