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Author: Szymczak, Andrzej × End date: 2005 × Start date: 2005 ×

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    Subdomain Aware Contour Trees and Contour Evolution in Time-Dependent Scalar Fields
    (Georgia Institute of Technology, 2005) Szymczak, Andrzej
    For time-dependent scalar fields, one is often interested in topology changes of contours in time. In this paper, we focus on describing how contours split and merge over a certain time interval. Rather than attempting to describe all individual contour splitting and merging events, we focus on the simpler and therefore more tractable in practice problem: describing and querying the cumulative effect of the splitting and merging events over a user-specified time interval. Using our system one can, for example, find all contours at time tº that continue to two contours at time t¹ without hitting the boundary of the domain. For any such contour, there has to be a bifurcation happening to it somewhere between the two times, but, in addition to that, many other events may possibly happen without changing the cumulative outcome (e.g. merging with several contours born after tº or splitting off several contours that disappear before t¹). Our approach is flexible enough to enable other types of queries, if they can be cast as counting queries for numbers of connected components of intersections of contours with certain simply connected domains. Examples of such queries include finding contours with large life spans, contours avoiding certain subset of the domain over a given time interval or contours that continue to two at a later time and then merge back to one some time later. Experimental results show that our method can handle large 3D (2 space dimensions plus time) and 4D (3D+time) datasets. Both preprocessing and query algorithms can easily be parallelized.
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    Coronary Vessel Cores From 3D Imagery: A Topological Approach
    (Georgia Institute of Technology, 2005) Mischaikow, Konstantin ; Tannenbaum, Allen R. ; Szymczak, Andrzej
    We propose a simple method for reconstructing thin, low-contrast blood vessels from three-dimensional greyscale images. Our algorithm first extracts persistent maxima of the intensity on all axis-aligned two-dimensional slices through the input volume. Those maxima tend to concentrate along one-dimensional intensity ridges, in particular along blood vessels. Persistence (which can be viewed as a measure of robustness of a local maximum with respect to perturbations of the data) allows to filter out the 'unimportant' maxima due to noise or inaccuracy in the input volume. We then build a minimum forest based on the persistent maxima that uses edges of length smaller than a certain threshold. Because of the distribution of the robust maxima, the structure of this forest already reflects the structure of the blood vessels. We apply three simple geometric filters to the forest in order to improve its quality. The first filter removes short branches from the forest's trees. The second filter adds edges, longer than the edge length threshold used earlier, that join what appears (based on geometric criteria) to be pieces of the same blood vessel to the forest. Such disconnected pieces often result from non-uniformity of contrast along a blood vessel. Finally, we let the user select the tree of interest by clicking near its root (point from which blood would flow out into the tree). We compute the blood flow direction assuming that the tree is of the correct structure and cut it in places where the vessel's geometry would force the blood flow direction to change abruptly. Experiments on clinical CT scans show that our technique can be a useful tool for segmentation of thin and low contrast blood vessels. In particular, we successfully applied it to extract coronary arteries from heart CT scans. Volumetric 3D models of blood vessels can be obtained from the graph described above by adaptive thresholding.
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    Simplifying the Topology of Volume Datasets: An Opportunistic Approach Authors
    (Georgia Institute of Technology, 2005) Vanderhyde, James ; Szymczak, Andrzej
    Understanding isosurfaces and contours (their connected components) is important for the analysis as well as effective visualization of 3D scalar fields. The topological changes that the contours undergo as the isovalue varies are typically represented using the contour tree, which can be obtained from the input scalar field by collapsing every contour to a single point. Contour trees are known to provide useful information, allowing one to find interesting isovalues and contours, speed up computations involving isosurfaces or contours, or analyze or visualize the scalar field's qualitative structure. However, the applicability of contour trees can, in many cases, be problematic because of their large size. Morse theory relates the contour topology changes to critical points in the underlying scalar fields. We describe a simple algorithm that can decrease the number of critical points in a regularly sampled volume dataset. The procedure produces a perturbed version of the input volume that has fewer critical points but, at the same time, is guaranteed to be less than a user-specified threshold away from the input volume (in the supremum norm sense). Because the input and output volumes are close, the algorithm preserves the most stable topological features of the scalar field. Although we do not guarantee that the number of critical points in the output volume is minimum among all volumes within the threshold away from the input dataset, our experiments demonstrate that the procedure is quite effective for a variety of input data types. Apart from reducing the size of the contour tree, it also reduces the topological complexity of individual isosurfaces.