Persistent Homology of Delay Embeddings and its Application to Wheeze Detection

We propose a new approach to detect and quantify the periodic structure of dynamical systems using topological methods. We propose to use delay-coordinate embedding as a tool to detect the presence of harmonic structures by using persistent homology for robust analysis of point clouds of delay-coordinate embeddings. To discover the proper delay, we propose an autocorrelation like (ACL) function of the signals, and apply the introduced topological approach to analyze breathing sound signals for wheeze detection. Experiments have been carried out to substantiate the capabilities of the proposed method.

[1]  F. Takens Detecting strange attractors in turbulence , 1981 .

[2]  Kenneth A. Brown,et al.  Nonlinear Statistics of Human Speech Data , 2009, Int. J. Bifurc. Chaos.

[3]  Trevor Gibbs Auscultation Skills: Breath and Heart Sounds , 2000, BMJ : British Medical Journal.

[4]  Leontios J. Hadjileontiadis,et al.  Lung Sounds: An Advanced Signal Processing Perspective , 2008, Lung Sounds.

[5]  H. Kantz,et al.  Nonlinear time series analysis , 1997 .

[6]  J. D. Farmer,et al.  A Theory of State Space Reconstruction in the Presence of Noise , 1991 .

[7]  Antoni Homs-Corbera,et al.  Time-frequency detection and analysis of wheezes during forced exhalation , 2004, IEEE Transactions on Biomedical Engineering.

[8]  Facundo Mémoli,et al.  Eurographics Symposium on Point-based Graphics (2007) on the Use of Gromov-hausdorff Distances for Shape Comparison , 2022 .

[9]  Herbert Edelsbrunner,et al.  Computational Topology - an Introduction , 2009 .

[10]  Robert L. Wilkins,et al.  Fundamentals of lung and heart sounds , 2015 .

[11]  Mikael Vejdemo-Johansson,et al.  javaPlex: A Research Software Package for Persistent (Co)Homology , 2014, ICMS.

[12]  Leontios J. Hadjileontiadis,et al.  Analysis of Wheezes Using Wavelet Higher Order Spectral Features , 2010, IEEE Transactions on Biomedical Engineering.

[13]  Gunnar E. Carlsson,et al.  Topology and data , 2009 .

[14]  Leontios J. Hadjileontiadis,et al.  Wheeze detection based on time-frequency analysis of breath sounds , 2007, Comput. Biol. Medicine.

[15]  Christopher J. Rozell,et al.  Stable Takens' Embeddings for Linear Dynamical Systems , 2010, IEEE Transactions on Signal Processing.

[16]  Gunnar E. Carlsson,et al.  Topological estimation using witness complexes , 2004, PBG.

[17]  Leonidas J. Guibas,et al.  Gromov‐Hausdorff Stable Signatures for Shapes using Persistence , 2009, Comput. Graph. Forum.

[18]  L. Tsimring,et al.  The analysis of observed chaotic data in physical systems , 1993 .

[19]  James P. Crutchfield,et al.  Geometry from a Time Series , 1980 .