Multi-Resolution computation and presentation of Contour Trees

The Contour Tree of a scalar field is the graph obtained by contracting all the connected components of the level sets of the field into points. This is a powerful abstraction for representing the structure of the field with explicit description of the topological changes of its level sets. It has proven effective as a data-structure for fast extraction of isosurfaces and its application has been advocated as a user interface component guiding interactive data exploration sessions. In practice, this use has been very limited due the problem of presenting a graph that may be overwhelming in size and in which a planar embedding may be confusing due to self-intersections. Topological simplification techniques have helped in relieving this ∗This work was performed under the auspices of the U.S. Department of Energy by University of California Lawrence Livermore National Laboratory under contract No. W-7405-Eng-48. problem since they allow reducing the size of the graph. We present a multi-resolution data-structure for representing contour trees and an algorithm for its construction. Moreover, we provide a hierarchical layout that allows coarse-to-fine rendering of the tree in a progressive user interface. Construction of our multi-resolution model is only slightly more expensive than the standard tree, but introduces far greater flexibility when filtering, both uniformly and adaptively, the topology of the data by importance with respect to different metrics. We have tested the approach using topological persistence (that is the difference in function value between a pair of critical points that are simplified) as the main metric for constructing the topological hierarchy, and using geometric position (containment in a bounding box) as a secondary metric for adaptive refinement.