171 research outputs found
Convergence between Categorical Representations of Reeb Space and Mapper
The Reeb space, which generalizes the notion of a Reeb graph, is one of the
few tools in topological data analysis and visualization suitable for the study
of multivariate scientific datasets. First introduced by Edelsbrunner et al.,
it compresses the components of the level sets of a multivariate mapping and
obtains a summary representation of their relationships. A related construction
called mapper, and a special case of the mapper construction called the Joint
Contour Net have been shown to be effective in visual analytics. Mapper and JCN
are intuitively regarded as discrete approximations of the Reeb space, however
without formal proofs or approximation guarantees. An open question has been
proposed by Dey et al. as to whether the mapper construction converges to the
Reeb space in the limit.
In this paper, we are interested in developing the theoretical understanding
of the relationship between the Reeb space and its discrete approximations to
support its use in practical data analysis. Using tools from category theory,
we formally prove the convergence between the Reeb space and mapper in terms of
an interleaving distance between their categorical representations. Given a
sequence of refined discretizations, we prove that these approximations
converge to the Reeb space in the interleaving distance; this also helps to
quantify the approximation quality of the discretization at a fixed resolution
Natural Stratifications of Reeb Spaces and Higher Morse Functions
Both Reeb spaces and higher Morse functions induce natural stratifications.
In the former, we show that the data of the Jacobi set of a function induces stratifications on , and the associated
Reeb space, and give conditions under which maps between these three spaces are
stratified maps. We then extend this type of construction to the codomain of
higher Morse functions, using the singular locus to induce a stratification of
which sub-posets are equivalent to multi-parameter filtrations.Comment: v2: Additional examples/figures, Appendix on notions of criticality
for PL ma
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
Notes on explicit special generic maps into Eulidean spaces whose dimensions are greater than 4
Special generic maps are higher dimensional versions of Morse functions with
exactly two singular points, characterizing spheres topologically except
4-dimensional cases and 4-dimensional standard spheres. The class of such maps
also contains canonical projections of unit spheres.
This class is interesting from the viewpoint of algebraic topology and
differential topology of manifolds. These maps have been shown to restrict the
topologies and the differentiable structures of the manifolds strongly by
Calabi, Saeki and Sakuma before 2010s, and later Nishioka, Wrazidlo and the
author. So-called exotic spheres admit no special generic map in considerable
cases and homology groups and cohomology rings are shown to be strongly
restricted. Moreover, special generic maps into Euclidean spaces whose
dimensions are smaller than or equal to 4 have been studied well. The present
paper mainly concerns cases where the dimensions of target spaces are larger
than or equal to 5.Comment: 11 pages, A short exposition on a special generic map added, this
will be improved, after improvement on the content etc. this will be
submitted to a refereed journal (when the time comes
An edit distance for Reeb graphs
We consider the problem of assessing the similarity of 3D shapes
using Reeb graphs from the standpoint of robustness under
perturbations. For this purpose, 3D objects are viewed as spaces
endowed with real-valued functions, while the similarity between
the resulting Reeb graphs is addressed through a graph edit
distance. The cases of smooth functions on manifolds and piecewise
linear functions on polyhedra stand out as the most interesting
ones. The main contribution of this paper is the introduction of a
general edit distance suitable for comparing Reeb graphs in these
settings. This edit distance promises to be useful for
applications in 3D object retrieval because of its stability
properties in the presence of noise
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