Time-varying contour topology

Abstract

The contour tree has been used to compute the topology of isosurfaces, generate a minimal seed set for accelerated isosurface extraction, and provide a user interface to segment individual contour components in a scalar field. In this paper, we extend the benefits of the contour tree to time-varying data visualization. We define temporal correspondence of contour components and describe an algorithm to compute the correspondence information in 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 labeled on the graph. This topology change graph helps users to detect significant topological and geometric changes in timevarying 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.

Isocontour based Visualization of Time-varying Scalar Fields

“…Two algorithms extend the Reeb graph to time-varying functions to aid the user in selecting interesting time and isovalue parameters for visualization. The first algorithm, by Bajaj & Sohn [58], uses a overlap heuristic to connect the Reeb graph for each time slice; it works well when the time sampling rate is high relative to the phenomenon under study so that there is good temporal coherence. The second algorithm, by Edelsbrunner et al [24], determines the actual dynamics of the Reeb graph over all time slices, under a chosen interpolation function.…”

“…Another problem is detecting when and how these components change topology. Sohn & Bajaj [58] address these problems by computing the correspondence of contour trees over time. They assume that the scalar field can change unpredictably between two successive time steps, and define temporal correspondence of contour tree arcs for successive time steps using a notion of an overlap between an isocontour at time t with an isocontour at time t + 1.…”

“…When a lower object in X ≤s overlaps one in Y ≤s we can create the mapping between the arcs of JTt+1 with the corresponding arcs of JTt. For a more detailed description of this step see [58]. 4.…”

Computing and Visualizing Time-Varying Merge Trees for High-Dimensional Data

“…Sohn and Bajaj [25] track contours using a similarity measure that considers spatial overlap of the inside and outside of contours. Ji and Shen [16] use the earth mover's distance to determine correspondence among contours.…”

“…Sup-porting threshold changes over time or on a per-feature basis depends on specifying a large number of parameters a priori. Moreover, many existing methods correlate features via spatial overlap in consecutive time steps-an approach that can lead to ambiguities [25] and becomes computationally intractable in higher dimensions. Time-varying Reeb graphs [11,19] require complicated distinction of cases for topological events; their number increases with the dimensionality and no case tables have been presented beyond the 3-D case.…”