The TOPORRERY: computation and presentation of multi-resolution topology

Summary

The Contour Tree of a scalar field is the graph obtained by contracting all the con- nected components of the level sets of the field into points. This is a powerful ab- straction 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 inter- face 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 al- lowing 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 topo- logical persistence as the main metric for constructing the tree hierarchy, and using geometric position as a secondary metric for adaptive refinements. The result is pre- sented in linked views of the abstract toporrery and the geometric embedding of the input data.