434 research outputs found
The Reeb Graph Edit Distance Is Universal
We consider the setting of Reeb graphs of piecewise linear functions and study distances between them that are stable, meaning that functions which are similar in the supremum norm ought to have similar Reeb graphs. We define an edit distance for Reeb graphs and prove that it is stable and universal, meaning that it provides an upper bound to any other stable distance. In contrast, via a specific construction, we show that the interleaving distance and the functional distortion distance on Reeb graphs are not universal
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
Stability of Reeb graphs under function perturbations: the case of closed curves
Reeb graphs provide a method for studying the shape of a manifold by encoding
the evolution and arrangement of level sets of a simple Morse function defined
on the manifold. Since their introduction in computer graphics they have been
gaining popularity as an effective tool for shape analysis and matching. In
this context one question deserving attention is whether Reeb graphs are robust
against function perturbations. Focusing on 1-dimensional manifolds, we define
an editing distance between Reeb graphs of curves, in terms of the cost
necessary to transform one graph into another. Our main result is that changes
in Morse functions induce smaller changes in the editing distance between Reeb
graphs of curves, implying stability of Reeb graphs under function
perturbations.Comment: 23 pages, 12 figure
Book embeddings of Reeb graphs
Let be a simplicial complex with a piecewise linear function
. The Reeb graph is the quotient of , where we
collapse each connected component of to a single point. Let the
nodes of be all homologically critical points where any homology of
the corresponding component of the level set changes. Then we can
label every arc of with the Betti numbers
of the corresponding -dimensional
component of a level set. The homology labels give more information about the
original complex than the classical Reeb graph. We describe a canonical
embedding of a Reeb graph into a multi-page book (a star cross a line) and give
a unique linear code of this book embedding.Comment: 12 pages, 5 figures, more examples will be at http://kurlin.or
Statistical Analysis and Parameter Selection for Mapper
In this article, we study the question of the statistical convergence of the
1-dimensional Mapper to its continuous analogue, the Reeb graph. We show that
the Mapper is an optimal estimator of the Reeb graph, which gives, as a
byproduct, a method to automatically tune its parameters and compute confidence
regions on its topological features, such as its loops and flares. This allows
to circumvent the issue of testing a large grid of parameters and keeping the
most stable ones in the brute-force setting, which is widely used in
visualization, clustering and feature selection with the Mapper.Comment: Minor modification
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
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