Browse > Article
http://dx.doi.org/10.9717/kmms.2016.19.2.146

Topological Analysis of Spaces of Waveform Signals  

Hahn, Hee Il (Dept. of Information and Communications Eng., Hankuk University of Foreign Studies)
Publication Information
Journal of Korea Multimedia Society / v.19, no.2, 2016 , pp. 146-154 More about this Journal
Abstract
This paper presents methods to analyze the topological structures of the spaces composed of patches extracted from waveform signals, which can be applied to the classification of signals. Commute time embedding is performed to transform the patch sets into the corresponding geometries, which has the properties that the embedding geometries of periodic or quasi-periodic waveforms are represented as closed curves on the low dimensional Euclidean space, while those of aperiodic signals have the shape of open curves. Persistent homology is employed to determine the topological invariants of the simplicial complexes constructed by randomly sampling the commute time embedding of the waveforms, which can be used to discriminate between the groups of waveforms topologically.
Keywords
Topology; Commute Time Embedding; Manifold Learning; Homology; Persistent Homology;
Citations & Related Records
Times Cited By KSCI : 2  (Citation Analysis)
연도 인용수 순위
1 J.B. Tenenbaum, V. DeSilva, and J.C. Langford, “A Global Geometric Framework for Nonlinear Dimensionality Reduction,” Science, Vol. 290, pp. 2319-2323, 2000.   DOI
2 A. Zomorodian and G. Carlsson, "Computing Persistent Homology," Proceeding of Discrete Computational Geometry 33, pp. 249-274, 2005.   DOI
3 M. Belkin and P. Niyogi, “Laplacian Eigenmaps for Dimensionality Reduction and Data Representation,” Neural Computation, Vol. 15, No. 6, pp. 1373-1396, 2003.   DOI
4 H. Hahn, “Proposing a Connection Method for Measuring Differentiation of Tangent Vectors at Shape Manifold,” Journal of Korea Multimedia Society, Vol. 16, No. 2, pp. 160-168, 2013.   DOI
5 H. Edelsbrunner, D. Letscher, and A. Zomorodian, "Topological Persistence and Simplification," Proceeding of Discrete Computational Geometry 28, pp. 511-533, 2002.   DOI
6 H. Chintakunta and H. Krim “Distributed Localization of Coverage Holes Using Topological Persistence,” IEEE Transactions on Signal Processing, Vol. 62, No. 10, pp. 2531-2541, 2014.   DOI
7 F. Chazal, L. Guibas, S.Y. Oudot, and P. Skraba, "Analysis of Scalar Fields Over Point Cloud Data," Proceedings of the Nineteenth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 1021-1030, 2009.
8 D. Cohen-Steiner, H. Edelsbrunner, and J. Harer, "Stability of Persistence Diagrams," Proceeding of 21st ACM Symposium on Computational Geometry, pp. 263-271, 2005.
9 K.M. Taylor, The Geometry of Signal and Image Patch-sets, Doctor's Thesis of University of Colorado, 2011.
10 H. Hahn, “Analysis of Commute Time Embedding Based on Spectral Graph,” Journal of Korea Multimedia Society, Vol. 17, No. 1, pp. 101-109, 2014.   DOI