Topological data analysis and topology-based visualization Leila De Floriani
Topology-based methods v Why topology? topology deals with qualitative geometric information , thus v providing a structural description of data topological invariants are robust under continuous and v stretching deformations compact topology-based shape descriptors are available (e.g., v Reeb graphs, Morse complexes, persistent diagrams, etc.) v Topology-based visualization [Heine et al.,2016] uses topological concepts (e.g., topological space, cell v complex, homotopy, homology, etc.) to describe , reduce , organize , or segment data to be used in visualization e.g., highlight data subsets, provide structural overviews, v guide interactive exploration
Topology-based methods Topological Data Analysis (TDA): subfield of v computational topology methods rooted in algebraic topology to infer and v analyze the structure of point clouds in a metric space points are connected though complexes to form shapes v Applications to neuroscience, medical imaging, v genetics, biological aggregations, sensor networks, social networks, etc. TDA combined with machine learning v techniques to improve data classification and understanding
Tools from algebraic topology v Homology v formalizes the concept of topological features in an algebraic way (Betti numbers) v characterizes a shape in terms of connected components, holes (1-cyles), cavities… 1 connected component 2 1-cycles (a through-hole) 1 cavity v Persistent homology v describes the changes in homology occurring during a filtration provided by the sublevel sets of a scalar function on a manifold domain
Discrete Morse theory [Forman 1998] The main results of smooth Morse theory is v transposed to a combinatorial setting: v relations between homology changes and critical points ( Morse inequalities ) Scalar function values associated with all the cells v of the input complex (grid or a mesh) - by extending the values given at its vertices Discrete Morse gradient: v v collection of adjacent pairs of cells v consecutive cells form paths which do not contain cycles Critical cells: unpaired cells ( minima occur at v vertices , maxima at maximal cells, etc.)
Discrete Morse complex Input complex Discrete Morse Morse complex (scalar values @ vertices) gradient 6
Discrete Morse theory in topological data analysis Discrete Morse Complex (DMC): same v homology ( persistent homology ) as the input (filtered) complex DMC has much less cells than the input complex: v v reduction in the complexity of persistent homology computation ( O(n 3 ) process, where n = number of cells) Computing a DMC on simplicial complexes (built v from high-dimensional point data) [Nanda, 2011; Fugacci et al., 2015] v number of cells in the input complex from 6 to 10 5 times the number of critical cells
Discrete Morse theory in data visualization DMC as a derivative-free tool for computing v segmenting scalar fields v similar approach to digital topology for image analysis and computer vision [Rosenfeld, 1973] Data simplification and noise removal v through v topological persistence v discrete gradient simplification [Forman, 1998] Efficient computation of Morse (Morse-Smale) v complex for 3D scalar fields v based on piece-wise linear interpolant not computationally feasible [Edelsbrunner et al. 2003] v 3D grids [Gyulassy wt al, 2008; Robins et al., 2011; Gunther et al. 2011; Natarajan, 2012] v Unstructured tetrahedral meshes [Weiss et al., 2013]
From scalar fields to multifields Multifields data equipped with a multi-valued v function F = (f 1 ,f 2 ,…, f k ) Topology-based multifield visualization: v v Fiber surfaces - isosurfaces v Reeb spaces- Reeb graphs v Generalization of critical points: v Pareto sets [Huettenberger et al., 2013]: points where gradients are opposite v Jacobi sets [Edelbrunner et al., 2004]: points where gradients are linearly dependent Pareto sets Topological data analysis: multi-parameter v persistent homology v multidimensional filtration induced by the partial order defined by multivalued function F
A tool for multipersistent homology and beyond A discrete gradient V (plus critical cells) compatible v with multidimensional filtration of function F Discrete Morse complex associated with V: same v multipersistent homology as the input complex endowed with F [Landi et al., 2017] Parallel discrete gradient computation algorithm v [Iuricich et al., 2017] - v efficient computation of descriptors for multipersistent homology for shape analysis [Scaramuccia, 2018] v Implementation available on Github On-going work: discrete Pareto sets and critical cells v Critical cells f 1 =z Clusters of critical cells larger than 10000 cells color coded according to their size – and f 2 =y Hurricane Isabel data set
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