Topological data analysis and topology-based visualization Leila - - PowerPoint PPT Presentation

Topological data analysis and topology-based visualization

Leila De Floriani

Topology-based methods

v Why topology?

v topology deals with qualitative geometric information, thus providing a structural description of data v topological invariants are robust under continuous and stretching deformations v compact topology-based shape descriptors are available (e.g., Reeb graphs, Morse complexes, persistent diagrams, etc.)

v Topology-based visualization [Heine et al.,2016]

v uses topological concepts (e.g., topological space, cell complex, homotopy, homology, etc.) to describe, reduce,

• rganize, or segment data to be used in visualization

v e.g., highlight data subsets, provide structural overviews, guide interactive exploration

Topology-based methods

v Topological Data Analysis (TDA): subfield of computational topology

v methods rooted in algebraic topology to infer and analyze the structure of point clouds in a metric space v points are connected though complexes to form shapes

v Applications to neuroscience, medical imaging, genetics, biological aggregations, sensor networks, social networks, etc. v TDA combined with machine learning 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…

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

1 connected component 2 1-cycles (a through-hole) 1 cavity

Discrete Morse theory [Forman 1998]

v The main results of smooth Morse theory is transposed to a combinatorial setting: v relations between homology changes and critical points (Morse inequalities) v Scalar function values associated with all the cells

• f the input complex (grid or a mesh) - by extending

the values given at its vertices v Discrete Morse gradient: v collection of adjacent pairs of cells v consecutive cells form paths which do not contain cycles v Critical cells: unpaired cells (minima occur at vertices, maxima at maximal cells, etc.)

Discrete Morse complex

6

Input complex (scalar values @ vertices) Discrete Morse gradient Morse complex

Discrete Morse theory in topological data analysis

v Discrete Morse Complex (DMC): same homology (persistent homology) as the input (filtered) complex v DMC has much less cells than the input complex:

v reduction in the complexity of persistent homology computation (O(n3) process, where n = number of cells)

v Computing a DMC on simplicial complexes (built from high-dimensional point data) [Nanda, 2011;

Fugacci et al., 2015] v number of cells in the input complex from 6 to 105 times the number of critical cells

Discrete Morse theory in data visualization

v DMC as a derivative-free tool for computing segmenting scalar fields

v similar approach to digital topology for image analysis and computer vision [Rosenfeld, 1973]

v Data simplification and noise removal through

v topological persistence v discrete gradient simplification [Forman, 1998]

v Efficient computation of Morse (Morse-Smale) 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

v Multifields data equipped with a multi-valued function F = (f1,f2,…, fk) v Topology-based multifield visualization: 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

v Topological data analysis: multi-parameter persistent homology v multidimensional filtration induced by the partial

• rder defined by multivalued function F

Pareto sets

A tool for multipersistent homology and beyond

v A discrete gradient V (plus critical cells) compatible with multidimensional filtration of function F v Discrete Morse complex associated with V: same multipersistent homology as the input complex endowed with F [Landi et al., 2017] v Parallel discrete gradient computation algorithm [Iuricich et al., 2017] -

v efficient computation of descriptors for multipersistent homology for shape analysis [Scaramuccia, 2018] v Implementation available on Github

v On-going work: discrete Pareto sets and critical cells

Clusters of critical cells larger than 10000 cells color coded according to their size – Hurricane Isabel data set

Critical cells f1=z and f2 =y