204 research outputs found
Computing morse-smale complexes with accurate geometry
pre-printTopological techniques have proven highly successful in analyzing and visualizing scientific data. As a result, significant efforts have been made to compute structures like the Morse-Smale complex as robustly and efficiently as possible. However, the resulting algorithms, while topologically consistent, often produce incorrect connectivity as well as poor geometry. These problems may compromise or even invalidate any subsequent analysis. Moreover, such techniques may fail to improve even when the resolution of the domain mesh is increased, thus producing potentially incorrect results even for highly resolved functions. To address these problems we introduce two new algorithms: (i) a randomized algorithm to compute the discrete gradient of a scalar field that converges under refinement; and (ii) a deterministic variant which directly computes accurate geometry and thus correct connectivity of the MS complex. The first algorithm converges in the sense that on average it produces the correct result and its standard deviation approaches zero with increasing mesh resolution. The second algorithm uses two ordered traversals of the function to integrate the probabilities of the first to extract correct (near optimal) geometry and connectivity. We present an extensive empirical study using both synthetic and real-world data and demonstrates the advantages of our algorithms in comparison with several popular approaches
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
Conforming Morse-Smale complexes
pre-printMorse-Smale (MS) complexes have been gaining popularity as a tool for feature-driven data analysis and visualization. However, the quality of their geometric embedding and the sole dependence on the input scalar field data can limit their applicability when expressing application-dependent features. In this paper we introduce a new combinatorial technique to compute an MS complex that conforms to both an input scalar field and an additional, prior segmentation of the domain. The segmentation constrains the MS complex computation guaranteeing that boundaries in the segmentation are captured as separatrices of the MS complex. We demonstrate the utility and versatility of our approach with two applications. First, we use streamline integration to determine numerically computed basins/mountains and use the resulting segmentation as an input to our algorithm. This strategy enables the incorporation of prior flow path knowledge, effectively resulting in an MS complex that is as geometrically accurate as the employed numerical integration. Our second use case is motivated by the observation that often the data itself does not explicitly contain features known to be present by a domain expert. We introduce edit operations for MS complexes so that a user can directly modify their features while maintaining all the advantages of a robust topology-based representation
Algebraic Topology
The chapter provides an introduction to the basic concepts of Algebraic
Topology with an emphasis on motivation from applications in the physical
sciences. It finishes with a brief review of computational work in algebraic
topology, including persistent homology.Comment: This manuscript will be published as Chapter 5 in Wiley's textbook
\emph{Mathematical Tools for Physicists}, 2nd edition, edited by Michael
Grinfeld from the University of Strathclyd
GPU Parallel Computation of Morse-Smale Complexes
The Morse-Smale complex is a well studied topological structure that
represents the gradient flow behavior of a scalar function. It supports
multi-scale topological analysis and visualization of large scientific data.
Its computation poses significant algorithmic challenges when considering large
scale data and increased feature complexity. Several parallel algorithms have
been proposed towards the fast computation of the 3D Morse-Smale complex. The
non-trivial structure of the saddle-saddle connections are not amenable to
parallel computation. This paper describes a fine grained parallel method for
computing the Morse-Smale complex that is implemented on a GPU. The
saddle-saddle reachability is first determined via a transformation into a
sequence of vector operations followed by the path traversal, which is achieved
via a sequence of matrix operations. Computational experiments show that the
method achieves up to 7x speedup over current shared memory implementations
Parallel Computation of Piecewise Linear Morse-Smale Segmentations
This paper presents a well-scaling parallel algorithm for the computation of
Morse-Smale (MS) segmentations, including the region separators and region
boundaries. The segmentation of the domain into ascending and descending
manifolds, solely defined on the vertices, improves the computational time
using path compression and fully segments the border region. Region boundaries
and region separators are generated using a multi-label marching tetrahedra
algorithm. This enables a fast and simple solution to find optimal parameter
settings in preliminary exploration steps by generating an MS complex preview.
It also poses a rapid option to generate a fast visual representation of the
region geometries for immediate utilization. Two experiments demonstrate the
performance of our approach with speedups of over an order of magnitude in
comparison to two publicly available implementations. The example section shows
the similarity to the MS complex, the useability of the approach, and the
benefits of this method with respect to the presented datasets. We provide our
implementation with the paper.Comment: Journal: IEEE Transactions on Visualization and Computer Graphics /
Submitted: 22-Jun-2022 / Accepted: 13-Mar-202
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