708 research outputs found
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
Learning Probabilistic Topological Representations Using Discrete Morse Theory
Accurate delineation of fine-scale structures is a very important yet
challenging problem. Existing methods use topological information as an
additional training loss, but are ultimately making pixel-wise predictions. In
this paper, we propose the first deep learning based method to learn
topological/structural representations. We use discrete Morse theory and
persistent homology to construct an one-parameter family of structures as the
topological/structural representation space. Furthermore, we learn a
probabilistic model that can perform inference tasks in such a
topological/structural representation space. Our method generates true
structures rather than pixel-maps, leading to better topological integrity in
automatic segmentation tasks. It also facilitates semi-automatic interactive
annotation/proofreading via the sampling of structures and structure-aware
uncertainty.Comment: 16 pages, 11 figure
Persistence barcodes and Laplace eigenfunctions on surfaces
We obtain restrictions on the persistence barcodes of Laplace-Beltrami
eigenfunctions and their linear combinations on compact surfaces with
Riemannian metrics. Some applications to uniform approximation by linear
combinations of Laplace eigenfunctions are also discussed.Comment: Revised version; some references adde
LNCS
We describe an algorithm for segmenting three-dimensional medical imaging data modeled as a continuous function on a 3-manifold. It is related to watershed algorithms developed in image processing but is closer to its mathematical roots, which are Morse theory and homological algebra. It allows for the implicit treatment of an underlying mesh, thus combining the structural integrity of its mathematical foundations with the computational efficiency of image processing
Generating Second Order (Co)homological Information within AT-Model Context
In this paper we design a new family of relations between
(co)homology classes, working with coefficients in a field and starting
from an AT-model (Algebraic Topological Model) AT(C) of a finite cell
complex C These relations are induced by elementary relations of type
“to be in the (co)boundary of” between cells. This high-order connectivity
information is embedded into a graph-based representation model,
called Second Order AT-Region-Incidence Graph (or AT-RIG) of C. This
graph, having as nodes the different homology classes of C, is in turn,
computed from two generalized abstract cell complexes, called primal
and dual AT-segmentations of C. The respective cells of these two complexes
are connected regions (set of cells) of the original cell complex C,
which are specified by the integral operator of AT(C). In this work in
progress, we successfully use this model (a) in experiments for discriminating
topologically different 3D digital objects, having the same Euler
characteristic and (b) in designing a parallel algorithm for computing
potentially significant (co)homological information of 3D digital objects.Ministerio de EconomĂa y Competitividad MTM2016-81030-PMinisterio de EconomĂa y Competitividad TEC2012-37868-C04-0
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
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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