🍩 Database of Original & NonTheoretical Uses of Topology
(found 5 matches in 0.008111s)


(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 1dimensional 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 humangenerated 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. 
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 pointcloud 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 stateoftheart methods in gene expression analysis. We call this new method SW1PerS, which stands for Sliding Windows and 1Dimensional Persistence Scoring. 
A KleinBottleBased 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 pixelvalue space, underlying the set of small patches from the given image. Inspired by the fact that small \$\$n\times n\$\$n×nhighcontrast patches from natural images in grayscale 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 Fourierlike 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 multiscale rotationinvariant descriptor. We test it by classifying the materials in three popular data sets: The CUReT, UIUCTex and KTHTIPS texture databases.