10,736 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
Subsampling Methods for Persistent Homology
Persistent homology is a multiscale method for analyzing the shape of sets
and functions from point cloud data arising from an unknown distribution
supported on those sets. When the size of the sample is large, direct
computation of the persistent homology is prohibitive due to the combinatorial
nature of the existing algorithms. We propose to compute the persistent
homology of several subsamples of the data and then combine the resulting
estimates. We study the risk of two estimators and we prove that the
subsampling approach carries stable topological information while achieving a
great reduction in computational complexity
Geometric Inference on Kernel Density Estimates
We show that geometric inference of a point cloud can be calculated by
examining its kernel density estimate with a Gaussian kernel. This allows one
to consider kernel density estimates, which are robust to spatial noise,
subsampling, and approximate computation in comparison to raw point sets. This
is achieved by examining the sublevel sets of the kernel distance, which
isomorphically map to superlevel sets of the kernel density estimate. We prove
new properties about the kernel distance, demonstrating stability results and
allowing it to inherit reconstruction results from recent advances in
distance-based topological reconstruction. Moreover, we provide an algorithm to
estimate its topology using weighted Vietoris-Rips complexes.Comment: To appear in SoCG 2015. 36 pages, 5 figure
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