The toporrery: Computation and presentation of multiresolution topology

Abstract

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 limited to trivial examples due to the problem of presenting a graph that may be overwhelming in size and in which a planar embedding may have self-intersections. We propose a new metaphor for visualizing the Contour Tree borrowed from the classical design of a mechanical orrery – see Fig. 1a – reproducing a hierarchy of orbits of the planets around the sun or moons around a planet. In the toporrery – see Fig. 1b – the hierarchy of stars, planets and moons is replaced with a hierarchy of maxima, minima and saddles that can be interactively filtered, both uniformly and adaptively, by importance with respect to a given metric. The implementation of the system is based on (1) a hierarchical graph model allowing coarse-to-fine traversal for selective refinements and (2) a new algorithm for constructing a multiresolution Contour Tree with guaranteed topological correctness independently of the simplification metric. We have tested the approach using topological persistence as the main metric for constructing the tree hierarchy, and using geometric position as a secondary metric for adaptive refinements. The result is presented in linked views of the abstract toporrery and the geometric embedding of the input data.