Task-Based Augmented Contour Trees with Fibonacci Heaps

Abstract

This paper presents a new algorithm for the fast, shared memory, multi-core computation of augmented contour trees on triangulations. In contrast to most existing parallel algorithms our technique computes augmented trees, enabling the full extent of contour tree based applications including data segmentation. Our approach completely revisits the traditional, sequential contour tree algorithm to re-formulate all the steps of the computation as a set of independent local tasks. This includes a new computation procedure based on Fibonacci heaps for the join and split trees, two intermediate data structures used to compute the contour tree, whose constructions are efficiently carried out concurrently thanks to the dynamic scheduling of task parallelism. We also introduce a new parallel algorithm for the combination of these two trees into the output global contour tree. Overall, this results in superior time performance in practice, both in sequential and in parallel thanks to the OpenMP task runtime. We report performance numbers that compare our approach to reference sequential and multi-threaded implementations for the computation of augmented merge and contour trees. These experiments demonstrate the run-time efficiency of our approach and its scalability on common workstations. We demonstrate the utility of our approach in data segmentation applications.

Wasserstein Distances, Geodesics and Barycenters of Merge Trees

“…This section presents experimental results obtained on a computer with two Xeon CPUs (3.2 GHz, 2x10 cores, 96GB of RAM). The input merge trees were computed with FTM [48] and pre-processed to discard noisy features (persistence simplification threshold: 0.25% Fig. 13.…”

“…Efficient algorithms have also been documented for its variants, the merge and contour trees [25,108] (Sec. 2.3), and parallel algorithms have also been described [2,27,48,69]. The Morse-Smale complex [22,33,34], which depicts the global behaviour of integral lines, is another popular topological data abstraction in visualization [29].…”

An Overview of the Topology ToolKit

“…Critical points [5] extract points of interest. Merge/contour trees and Reeb graphs [18,19,20,21] estimate skeletons and meaningfully segment the input data along level sets. Persistence diagrams [10] visually represent the population of points of interest (critical points) as well as their salience (topological persistence).…”

Topological Analysis of High Velocity Turbulent Flow

“…2). In particular, vortices were identified as local minima of ρ and their region of influence given by a merge tree based segmentation [8]. The rotation direction of each vortex is estimated by the sign of the orthogonal component The two trajectories which maximize lateral movement (blue and green) correspond to 2 clockwise vortices which merge together.…”