🍩 Database of Original & Non-Theoretical Uses of Topology
(found 6 matches in 0.001741s)
-
-
Sliding Windows and Persistence: An Application of Topological Methods to Signal Analysis (2015)
Jose A. Perea, John HarerAbstract
We develop in this paper a theoretical framework for the topological study of time series data. Broadly speaking, we describe geometrical and topological properties of sliding window embeddings, as seen through the lens of persistent homology. In particular, we show that maximum persistence at the point-cloud level can be used to quantify periodicity at the signal level, prove structural and convergence theorems for the resulting persistence diagrams, and derive estimates for their dependency on window size and embedding dimension. We apply this methodology to quantifying periodicity in synthetic data sets and compare the results with those obtained using state-of-the-art methods in gene expression analysis. We call this new method SW1PerS, which stands for Sliding Windows and 1-Dimensional Persistence Scoring. -
(Quasi)Periodicity Quantification in Video Data, Using Topology (2018)
Christopher J. Tralie, Jose A. PereaAbstract
This work introduces a novel framework for quantifying the presence and strength of recurrent dynamics in video data. Specifically, we provide continuous measures of periodicity (perfect repetition) and quasiperiodicity (superposition of periodic modes with noncommensurate periods), in a way which does not require segmentation, training, object tracking, or 1-dimensional surrogate signals. Our methodology operates directly on video data. The approach combines ideas from nonlinear time series analysis (delay embeddings) and computational topology (persistent homology) by translating the problem of finding recurrent dynamics in video data into the problem of determining the circularity or toroidality of an associated geometric space. Through extensive testing, we show the robustness of our scores with respect to several noise models/levels; we show that our periodicity score is superior to other methods when compared to human-generated periodicity rankings; and furthermore, we show that our quasiperiodicity score clearly indicates the presence of biphonation in videos of vibrating vocal folds, which has never before been accomplished quantitatively end to end. -
Chatter Classification in Turning Using Machine Learning and Topological Data Analysis (2018)
Firas A. Khasawneh, Elizabeth Munch, Jose A. PereaAbstract
Chatter identification and detection in machining processes has been an active area of research in the past two decades. Part of the challenge in studying chatter is that machining equations that describe its occurrence are often nonlinear delay differential equations. The majority of the available tools for chatter identification rely on defining a metric that captures the characteristics of chatter, and a threshold that signals its occurrence. The difficulty in choosing these parameters can be somewhat alleviated by utilizing machine learning techniques. However, even with a successful classification algorithm, the transferability of typical machine learning methods from one data set to another remains very limited. In this paper we combine supervised machine learning with Topological Data Analysis (TDA) to obtain a descriptor of the process which can detect chatter. The features we use are derived from the persistence diagram of an attractor reconstructed from the time series via Takens embedding. We test the approach using deterministic and stochastic turning models, where the stochasticity is introduced via the cutting coefficient term. Our results show a 97% successful classification rate on the deterministic model labeled by the stability diagram obtained using the spectral element method. The features gleaned from the deterministic model are then utilized for characterization of chatter in a stochastic turning model where there are very limited analysis methods. -
A Klein-Bottle-Based Dictionary for Texture Representation (2014)
Jose A. Perea, Gunnar CarlssonAbstract
A natural object of study in texture representation and material classification is the probability density function, in pixel-value space, underlying the set of small patches from the given image. Inspired by the fact that small \$\$n\times n\$\$n×nhigh-contrast patches from natural images in gray-scale accumulate with high density around a surface \$\$\fancyscript\K\\subset \\mathbb \R\\\textasciicircum\n\textasciicircum2\\$\$K⊂Rn2with the topology of a Klein bottle (Carlsson et al. International Journal of Computer Vision 76(1):1–12, 2008), we present in this paper a novel framework for the estimation and representation of distributions around \$\$\fancyscript\K\\$\$K, of patches from texture images. More specifically, we show that most \$\$n\times n\$\$n×npatches from a given image can be projected onto \$\$\fancyscript\K\\$\$Kyielding a finite sample \$\$S\subset \fancyscript\K\\$\$S⊂K, whose underlying probability density function can be represented in terms of Fourier-like coefficients, which in turn, can be estimated from \$\$S\$\$S. We show that image rotation acts as a linear transformation at the level of the estimated coefficients, and use this to define a multi-scale rotation-invariant descriptor. We test it by classifying the materials in three popular data sets: The CUReT, UIUCTex and KTH-TIPS texture databases. -
Revealing Brain Network Dynamics During the Emotional State of Suspense Using TDA (2025)
Astrid A. Olave, Jose A. Perea, Francisco GómezAbstract
Suspense is an affective state that is ubiquitous in human life, from art to quotidian events. However, little is known about the behavior of large-scale brain networks during suspenseful experiences. To address this question, we examined the continuous brain responses of participants watching a suspenseful movie, along with reported levels of suspense from an independent set of viewers. We employ sliding window analysis and Pearson correlation to measure functional connectivity states over time. Then, we use Mapper, a topological data analysis tool, to obtain a graphical representation that captures the dynamical transitions of the brain across states; this representation enables the anchoring of the topological characteristics of the combinatorial object with the measured suspense. Our analysis revealed changes in functional connectivity within and between the salience, fronto-parietal, and default networks associated with suspense. In particular, the functional connectivity between the salience and fronto-parietal networks increased with the level of suspense. In contrast, the connections of both networks with the default network decreased. Together, our findings reveal specific dynamical changes in functional connectivity at the network level associated with variation in suspense, and suggest topological data analysis as a potentially powerful tool for studying dynamic brain networks.Suspense is a complex emotional experience whose neural bases remain poorly understood. This study analyzes fMRI data from participants watching a suspenseful Alfred Hitchcock film to track time-evolving brain connectivity patterns. The Mapper algorithm represented moment-to-moment functional connectivity, transforming high-dimensional data into a graph that retains local detail and global shape. During heightened suspense, connectivity between salience and fronto-parietal control networks increases, whereas links with the default-mode network diminish. These shifts trace recurring paths through a finite set of connectivity states, offering a topological view of large-scale neural dynamics and clarifying how distributed networks coordinate during emotionally engaging experiences.Community Resources