Persistent homology of attractors for action recognition

Venkataraman, Vinay; Ramamurthy, Karthikeyan Natesan; Turaga, Pavan

doi:10.1109/icip.2016.7533141

Cited by 45 publications

(29 citation statements)

References 37 publications

(55 reference statements)

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“…Accuracy (%) Time (sec) Chaos [3] 52.44 -VR-Complex [26] 93.68 -DT2 [24] 93.92 -T-VR Complex (L1) [23] 96.48…”

Section: Methodsmentioning

confidence: 99%

“…However, this approach considers only spatial adjacency between the points and ignores the temporal adjacency. We improve upon the existing VR filtration approach by explicitly creating temporal links between x t−1 , x t , and x t+1 in the one-skeleton of S, thereby creating a metric space which encodes adjacency in both space and time [23]. The persistence diagrams for homology groups of dimensions 0 and 1 are then estimated.…”

Section: Algorithmic Detailsmentioning

confidence: 99%

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A Riemannian Framework for Statistical Analysis of Topological Persistence Diagrams

Anirudh

Venkataraman

Ramamurthy³

et al. 2016

2016 IEEE Conference on Computer Vision and Pattern Recognition Workshops (CVPRW)

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Topological data analysis is becoming a popular way to study high dimensional feature spaces without any contextual clues or assumptions. This paper concerns itself with one popular topological feature, which is the number of d−dimensional holes in the dataset, also known as the Betti−d number. The persistence of the Betti numbers over various scales is encoded into a persistence diagram (PD), which indicates the birth and death times of these holes as scale varies. A common way to compare PDs is by a pointto-point matching, which is given by the n-Wasserstein metric. However, a big drawback of this approach is the need to solve correspondence between points before computing the distance; for n points, the complexity grows according to O(n 3 ). Instead, we propose to use an entirely new framework built on Riemannian geometry, that models PDs as 2D probability density functions that are represented in the square-root framework on a Hilbert Sphere. The resulting space is much more intuitive with closed form expressions for common operations. The distance metric is 1) correspondence-free and also 2) independent of the number of points in the dataset. The complexity of computing distance between PDs now grows according to O(K 2 ), for a K × K discretization of [0, 1] 2 . This also enables the use of existing machinery in differential geometry towards statistical analysis of PDs such as computing the mean, geodesics, classification etc. We report competitive results with the Wasserstein metric, at a much lower computational load, indicating the favorable properties of the proposed approach. arXiv:1605.08912v1 [math.AT]

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“…Accuracy (%) Time (sec) Chaos [3] 52.44 -VR-Complex [26] 93.68 -DT2 [24] 93.92 -T-VR Complex (L1) [23] 96.48…”

Section: Methodsmentioning

confidence: 99%

Section: Algorithmic Detailsmentioning

confidence: 99%

A Riemannian Framework for Statistical Analysis of Topological Persistence Diagrams

Anirudh

Venkataraman

Ramamurthy³

et al. 2016

2016 IEEE Conference on Computer Vision and Pattern Recognition Workshops (CVPRW)

Self Cite

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show abstract

“…Recently, combination of delay-coordinate embedding and persistent homology has been proposed for the classification of time series data [5][6][7][8][9]. Persistent homology can count the number of holes in a given shape and estimate the width of each hole.…”

Section: Problem Statementmentioning

confidence: 99%

Criterion for determining the optimal delay of attractor reconstruction using persistent homology

Tsuji

Aihara

2019

NOLTA

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Persistent homology computes the number and the widths of holes of a shape. The delay-coordinate embedding reconstructs an attractor from time series data. Recently the combination of persistent homology and attractor reconstruction is proposed for clustering time series data. We propose a criterion to determine the delay of attractor reconstruction for periodic and chaotically periodic signals. Our criterion chooses the delay that maximizes the hole width of reconstructed attractors. We compare it to the criterion with mutual information and that by Perea and Harer. It is found that the combination of our criterion with Perea and Harer's criterion works well.

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“…Recently in signal analysis tasks, TDA showed great potential for pattern classification and outlier detection tasks [22][23][24][25][26], than the traditional statistical-based methods. Motivated by these works, the main contribution of this work includes:…”

Section: Introductionmentioning

confidence: 99%

Gait Rhythm Dynamics for Neuro-Degenerative Disease Classification via Persistence Landscape- Based Topological Representation

Yan

Ivanov

Omisore

et al. 2020

Sensors

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Neuro-degenerative disease is a common progressive nervous system disorder that leads to serious clinical consequences. Gait rhythm dynamics analysis is essential for evaluating clinical states and improving quality of life for neuro-degenerative patients. The magnitude of stride-to-stride fluctuations and corresponding changes over time—gait dynamics—reflects the physiology of gait, in quantifying the pathologic alterations in the locomotor control system of health subjects and patients with neuro-degenerative diseases. Motivated by algebra topology theory, a topological data analysis-inspired nonlinear framework was adopted in the study of the gait dynamics. Meanwhile, the topological representation–persistence landscapes were used as input of classifiers in order to distinguish different neuro-degenerative disease type from healthy. In this work, stride-to-stride time series from healthy control (HC) subjects are compared with the gait dynamics from patients with amyotrophic lateral sclerosis (ALS), Huntington’s disease (HD), and Parkinson’s disease (PD). The obtained results show that the proposed methodology discriminates healthy subjects from subjects with other neuro-degenerative diseases with relatively high accuracy. In summary, our study is the first attempt to provide a topological representation-based method into the disease classification with gait rhythms measured from the stride intervals to visualize gait dynamics and classify neuro-degenerative diseases. The proposed method could be potentially used in earlier interventions and state monitoring.

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Persistent homology of attractors for action recognition

Cited by 45 publications

References 37 publications

A Riemannian Framework for Statistical Analysis of Topological Persistence Diagrams

A Riemannian Framework for Statistical Analysis of Topological Persistence Diagrams

Criterion for determining the optimal delay of attractor reconstruction using persistent homology

Gait Rhythm Dynamics for Neuro-Degenerative Disease Classification via Persistence Landscape- Based Topological Representation

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