Prof James F. PetersUniversity of Manitoba, Canada and Adiyaman University, Turkey
Computational Geometry and Topology of Data Views of Brain Activation Regions in rs-fMRI Videos. Betti Numbers, Ghrist Pictographs in Tracking the Persistence of Brain Activations
In this talk, we consider the efficacy of the use of computational geometry and the topology of data in approximating, analyzing, comparing and measuring brain activation regions that appear in rs-fMRI videos.
Varying shapes of rain activation regions commonly occur during resting state (rs)-fMRI videos. In this paper, we demonstrate the utility of computational geometry combined with a topology of data approach in tracking the persistence of recurring blood oxygen level dependent (BOLD) signals in triangulated rs-fMRI video frames. This is accomplished by considering vortexes (nested, filled 1-cycles in a Whitehead Closure-finite Weak Topological Space) that cover brain activation regions with collections of path-connected vertexes with no end vertex. Measure of persistence of vortex shapes in BOLD signal propagation is carried out in terms of Betti numbers that rise and fall over time during spontaneous activity of the brain. Each vortex has a Rotman free group presentation, resulting from the presence of vertexes that serve as a basis for a free group (each vortex cycle vertex is represented by a linear combination of the basis elements. Consequently, each brain activation region vortex is represented by a Betti number equally the rank of its free group.
Betti numbers of free group presentations of brain activation region vortexes appear, disappear and sometimes reappear in the frames during a typical rs-fMRI video. It has been observed in 1999 by Edelsbrunner that a topology of data circumvents noise in data and focuses on those data that persist over time. Computational topology of data provides a practical method in isolating, measuring, and classifying persistent lag structures BOLD signals in each rs-fMRI video frame that are mapped to point clouds in a finite, bounded region in an n-dimensional Euclidean metric space. Brain activation region vortexes in triangulated rs-fMRI video frames make it possible to approximate, measure, track and compare brain activation region shapes. As stepping stones to understanding the proposed approach to tracking rs-fMRI brain activation regions, basic results and examples are given.
James F. Peters, B.Sc.(Math), M.Sc.(Math), Ph.D. (Proof Theory), Postdoctoral Fellow, Syracuse University and Researcher in the Rome AI Laboratory, Griffiss Air Force Base, New York (1991), Asst. Prof., University of Arkansas, 1991-1994, and Researcher in the Mission Sequencing and Deep Space Telecommunications Divisions at the Jet Propulsion Laboratory/Caltech, Pasadena, California (1991-1994), Full Professor, ECE Department, University of Manitoba, 1995-present He is now a Full Professor in the Department of Electrical and Computer Engineering (ECE) at the University of Manitoba and a Professor in the Department of Mathematics at Adiyaman University, Turkey.
He started working on descriptively near sets in 2002 and formally introduced near sets in 2007, tolerance near sets in 2009, descriptive proximity spaces in 2011, topology and proximity spaces in 2012, computational proximity in 2016, computational geometry, topology and physics of digital images in 2017-2021.
He is author of books entitled
Computational Geometry, Topology and Physics of Digital Images with Applications. Shape Complexes, Optical Vortex Nerves and Proximities, 2020, Springer Nature, Switzerland AG.
Foundations of Computer Vision. Computational Geometry, Visual Image Structures and Object Shape Recognition}, 2017, Springer Int. Publishing, Switzerland.
Computational Proximity. Excursions in the Topology of Digital Images, 2016, Springer Int. Publishing, Switzerland.
Topology of Digital Images: Visual Pattern Discovery via Proximity Spaces, 2014, Springer, Berlin. He is co-author (with Prof. Som Naimpally),
Topology with Applications: Topological Spaces via Near and Far, 2013, published by World Scientific, Singapore.
He is co-Editor with S. Pal, Indian Statistical Institute, Kolkata, India, Rough Fuzzy Image Analysis: Foundations and Methodologies, published in 2010 by Chapman \& Hall/CRC Press, co-Editor with E. Orlowska, G. Rozenberg, A. Skowron, New Frontiers in Scientific Discovery Commemorating the Life and Work of Zdzislaw Pawlak, published in 2007 by IOS Press, author of over 100 articles published in refereed journals during the past 5 years. He is co-Editor-in-Chief of the Springer Transactions on Rough Sets, Associate Editor of two journals and Editorial Board member of a number of other journals. His main research interests are computational geometry, topology and proximity spaces in the mathematics of near sets and visual surface shape detection, presentation, and analysis.
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