113 research outputs found
Topological descriptors for 3D surface analysis
We investigate topological descriptors for 3D surface analysis, i.e. the
classification of surfaces according to their geometric fine structure. On a
dataset of high-resolution 3D surface reconstructions we compute persistence
diagrams for a 2D cubical filtration. In the next step we investigate different
topological descriptors and measure their ability to discriminate structurally
different 3D surface patches. We evaluate their sensitivity to different
parameters and compare the performance of the resulting topological descriptors
to alternative (non-topological) descriptors. We present a comprehensive
evaluation that shows that topological descriptors are (i) robust, (ii) yield
state-of-the-art performance for the task of 3D surface analysis and (iii)
improve classification performance when combined with non-topological
descriptors.Comment: 12 pages, 3 figures, CTIC 201
Tuning the Performance of a Computational Persistent Homology Package
In recent years, persistent homology has become an attractive method for data analysis. It captures topological features, such as connected components, holes, and voids from point cloud data and summarizes the way in which these features appear and disappear in a filtration sequence. In this project, we focus on improving the performanceof Eirene, a computational package for persistent homology. Eirene is a 5000-line open-source software library implemented in the dynamic programming language Julia. We use the Julia profiling tools to identify performance bottlenecks and develop novel methods to manage them, including the parallelization of some time-consuming functions on multicore/manycore hardware. Empirical results show that performance can be greatly improved
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
A persistence landscapes toolbox for topological statistics
Topological data analysis provides a multiscale description of the geometry
and topology of quantitative data. The persistence landscape is a topological
summary that can be easily combined with tools from statistics and machine
learning. We give efficient algorithms for calculating persistence landscapes,
their averages, and distances between such averages. We discuss an
implementation of these algorithms and some related procedures. These are
intended to facilitate the combination of statistics and machine learning with
topological data analysis. We present an experiment showing that the
low-dimensional persistence landscapes of points sampled from spheres (and
boxes) of varying dimensions differ.Comment: 24 page
SimBa: An Efficient Tool for Approximating Rips-Filtration Persistence via Simplicial Batch-Collapse
In topological data analysis, a point cloud data P extracted from a metric space is often analyzed by computing the persistence diagram or barcodes of a sequence of Rips complexes built on P indexed by a scale parameter. Unfortunately, even for input of moderate size, the size of the Rips complex may become prohibitively large as the scale parameter increases. Starting with the Sparse Rips filtration introduced by Sheehy, some existing methods aim to reduce the size of the complex so as to improve the time efficiency as well. However, as we demonstrate, existing approaches still fall short of scaling well, especially for high dimensional data. In this paper, we investigate the advantages and limitations of existing approaches. Based on insights gained from the experiments, we propose an efficient new algorithm, called SimBa, for approximating the persistent homology of Rips filtrations with quality guarantees. Our new algorithm leverages a batch collapse strategy as well as a new sparse Rips-like filtration. We experiment on a variety of low and high dimensional data sets. We show that our strategy presents a significant size reduction, and our algorithm for approximating Rips filtration persistence is order of magnitude faster than existing methods in practice
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