🍩 Database of Original & Non-Theoretical Uses of Topology
(found 8 matches in 0.002117s)
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Topological Data Analysis of C. Elegans Locomotion and Behavior (2021)
Ashleigh Thomas, Kathleen Bates, Alex Elchesen, Iryna Hartsock, Hang Lu, Peter BubenikAbstract
Video of nematodes/roundworms was analyzed using persistent homology to study locomotion and behavior. In each frame, an organism's body posture was represented by a high-dimensional vector. By concatenating points in fixed-duration segments of this time series, we created a sliding window embedding (sometimes called a time delay embedding) where each point corresponds to a sequence of postures of an organism. Persistent homology on the points in this time series detected behaviors and comparisons of these persistent homology computations detected variation in their corresponding behaviors. We used average persistence landscapes and machine learning techniques to study changes in locomotion and behavior in varying environments. -
Ghrist Barcoded Video Frames. Application in Detecting Persistent Visual Scene Surface Shapes Captured in Videos (2019)
Arjuna P. H. Don, James F. PetersAbstract
This article introduces an application of Ghrist barcodes in the study of persistent Betti numbers derived from vortex nerve complexes found in triangulations of video frames. A Ghrist barcode (also called a persistence barcode) is a topology of data pic- tograph useful in representing the persistence of the features of changing shapes. The basic approach is to introduce a free Abelian group representation of intersecting filled polygons on the barycenters of the triangles of Alexandroff nerves. An Alexandroff nerve is a maximal collection of triangles of a common vertex in the triangulation of a finite, bounded planar region. In our case, the planar region is a video frame. A Betti number is a count of the number of generators is a finite Abelian group. The focus here is on the persistent Betti numbers across sequences of triangulated video frames. Each Betti number is mapped to an entry in a Ghrist barcode. Two main results are given, namely, vortex nerves are Edelsbrunner-Harer nerve complexes and the Betti number of a vortex nerve equals k + 2 for a vortex nerve containing k edges attached between a pair of vortex cycles in the nerve. -
On the Nonlinear Statistics of Optical Flow (2019)
Henry Adams, Johnathan Bush, Brittany Carr, Lara Kassab, Joshua Mirth -
(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. -
Topological Eulerian Synthesis of Slow Motion Periodic Videos (2018)
Christopher Tralie, Matthew BergerAbstract
We consider the problem of taking a video that is comprised of multiple periods of repetitive motion, and reordering the frames of the video into a single period, producing a detailed, single cycle video of motion. This problem is challenging, as such videos often contain noise, drift due to camera motion and from cycle to cycle, and irrelevant background motion/occlusions, and these factors can confound the relevant periodic motion we seek in the video. To address these issues in a simple and efficient manner, we introduce a tracking free Eulerian approach for synthesizing a single cycle of motion. Our approach is geometric: we treat each frame as a point in high-dimensional Euclidean space, and analyze the sliding window embedding formed by this sequence of points, which yields samples along a topological loop regardless of the type of periodic motion. We combine tools from topological data analysis and spectral geometric analysis to estimate the phase of each window, and we exploit the sliding window structure to robustly reorder frames. We show quantitative results that highlight the robustness of our technique to camera shake, noise, and occlusions, and qualitative results of single-cycle motion synthesis across a variety of scenarios. -
Zebrafish Behavior: Opportunities and Challenges (2017)
Michael B. Orger, Gonzalo G. de Polavieja -
Persistent Homology on Grassmann Manifolds for Analysis of Hyperspectral Movies (2016)
Sofya Chepushtanova, Michael Kirby, Chris Peterson, Lori ZiegelmeierAbstract
The existence of characteristic structure, or shape, in complex data sets has been recognized as increasingly important for mathematical data analysis. This realization has motivated the development of new tools such as persistent homology for exploring topological invariants, or features, in large data sets. In this paper, we apply persistent homology to the characterization of gas plumes in time dependent sequences of hyperspectral cubes, i.e. the analysis of 4-way arrays. We investigate hyperspectral movies of Long-Wavelength Infrared data monitoring an experimental release of chemical simulant into the air. Our approach models regions of interest within the hyperspectral data cubes as points on the real Grassmann manifold Gk,ï źn whose points parameterize the k-dimensional subspaces of \$\$\mathbb \R\\textasciicircumn\$\$Rn, contrasting our approach with the more standard framework in Euclidean space. An advantage of this approach is that it allows a sequence of time slices in a hyperspectral movie to be collapsed to a sequence of points in such a way that some of the key structure within and between the slices is encoded by the points on the Grassmann manifold. This motivates the search for topological features, associated with the evolution of the frames of a hyperspectral movie, within the corresponding points on the Grassmann manifold. The proposed mathematical model affords the processing of large data sets while retaining valuable discriminatory information. In this paper, we discuss how embedding our data in the Grassmann manifold, together with topological data analysis, captures dynamical events that occur as the chemical plume is released and evolves.