CARGO: Multi-Scale Topological Analysis of Time-Evolving Shapes

Georgia Tech Research Corporation, Atlanta GA

Investigators

Abstract

DMS-0138420 Jarek Rossignac We propose to develop the theoretical foundations and a set of practical computing tools for the automatic analysis of time-evolving shapes. Given a family of surfaces that represent the boundary of a 3D shape whose geometry and topology change with time, we propose to construct a higher-dimensional multiresolution representation, which we have named Atlas Transition Diagram, abbreviated ATD, that will identify and track the morphological and topological features of the 3D shape as they evolve with time and with resolution. Our ATD will also associate a chart to each feature, thus providing a surface-parameterization that naturally follows the branches and handles of each shape in the family. The charts evolve smoothly with time and resolution and are topologically glued together at their common boundaries to provide a continuous mapping, S(r,t,u,v;f); f), that, given a feature ff, a resolution rr, a time tt, and two parameters u and v, will identify a unique point on the surface and will allow us to trace its evolution with rr and tt. The theoretical underpinnings and algorithmic designs that will lead to a practical implementation of an efficient system for building and querying ATDs go far beyond simple extensions of Morse theory, of surface segmentation approaches, and of multi-resolution techniques, which have so far been mainly explored for static surfaces in 3D. Evolving surfaces are important to many scientific and engineering disciplines, including medicine, developmental biology, cell biology, computational fluid dynamics and computer aided design. They may for example represent the growth of a tumor, the shifting in position of a vortex over an airplane wing, or the budding of fingers on the hand of a human embryo. We propose to develop and integrate a collection of theoretical and algorithmic tools for the analysis and automated visualization of such evolutions. These tools will allow us to partition the evolving surface into features upon which a high-level description of the shape of the surface and of its evolution will be based. Furthermore, they will allow us to track their points, and thus surface properties, through time and to better visualize their evolutions through texture maps that continuously evolve with the features and highlight their boundaries and natural orientation. Finally, these tools will help us support queries about the time and nature of topological changes in the evolving surface, which may be important for the automatic analysis and retrieval of scientific datasets. To achieve these results, we propose to build a surface representation that is controlled independently or simultaneously by time and resolution and to decompose the time/resolution domain into cells where the surface topology (number of components and through holes) and its partition into features remain constant. To validate our theoretical contributions and to increase their impact on the community, we plan to develop a prototype implementation for animated objects with triangulated boundaries. We expect to make the source code of this implementation and its programming interface publicly available. We envision exploring collaborations with application developers in Science, Engineering, Medicine and Biology to help us refine and validate this approach.