Computing contour trees in all dimensions

AbstractWe show that contour trees can be computed in all dimensions by a simple algorithm that merges two trees. Our algorithm extends, simplifies, and improves work of Tarasov and Vyalyi and of van Kreveld et al

Avoiding the Global Sort: A Faster Contour Tree Algorithm

We revisit the classical problem of computing the \emph{contour tree} of a scalar field $f:\mathbb{M} \to \mathbb{R}$, where $\mathbb{M}$ is a triangulated simplicial mesh in $\mathbb{R}^d$. The contour tree is a fundamental topological structure that tracks the evolution of level sets of $f$ and has numerous applications in data analysis and visualization. All existing algorithms begin with a global sort of at least all critical values of $f$, which can require (roughly) $\Omega(n\log n)$ time. Existing lower bounds show that there are pathological instances where this sort is required. We present the first algorithm whose time complexity depends on the contour tree structure, and avoids the global sort for non-pathological inputs. If $C$ denotes the set of critical points in $\mathbb{M}$, the running time is roughly $O(\sum_{v \in C} \log \ell_v)$, where $\ell_v$ is the depth of $v$ in the contour tree. This matches all existing upper bounds, but is a significant improvement when the contour tree is short and fat. Specifically, our approach ensures that any comparison made is between nodes in the same descending path in the contour tree, allowing us to argue strong optimality properties of our algorithm. Our algorithm requires several novel ideas: partitioning $\mathbb{M}$ in well-behaved portions, a local growing procedure to iteratively build contour trees, and the use of heavy path decompositions for the time complexity analysis

The Topology ToolKit

This system paper presents the Topology ToolKit (TTK), a software platform designed for topological data analysis in scientific visualization. TTK provides a unified, generic, efficient, and robust implementation of key algorithms for the topological analysis of scalar data, including: critical points, integral lines, persistence diagrams, persistence curves, merge trees, contour trees, Morse-Smale complexes, fiber surfaces, continuous scatterplots, Jacobi sets, Reeb spaces, and more. TTK is easily accessible to end users due to a tight integration with ParaView. It is also easily accessible to developers through a variety of bindings (Python, VTK/C++) for fast prototyping or through direct, dependence-free, C++, to ease integration into pre-existing complex systems. While developing TTK, we faced several algorithmic and software engineering challenges, which we document in this paper. In particular, we present an algorithm for the construction of a discrete gradient that complies to the critical points extracted in the piecewise-linear setting. This algorithm guarantees a combinatorial consistency across the topological abstractions supported by TTK, and importantly, a unified implementation of topological data simplification for multi-scale exploration and analysis. We also present a cached triangulation data structure, that supports time efficient and generic traversals, which self-adjusts its memory usage on demand for input simplicial meshes and which implicitly emulates a triangulation for regular grids with no memory overhead. Finally, we describe an original software architecture, which guarantees memory efficient and direct accesses to TTK features, while still allowing for researchers powerful and easy bindings and extensions. TTK is open source (BSD license) and its code, online documentation and video tutorials are available on TTK's website

Task-based Augmented Contour Trees with Fibonacci Heaps

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