Theory and Algorithms for Constructing Discrete Morse Complexes from Grayscale Digital Images

Robins, Vanessa; Wood, Peter; Sheppard, Adrian

doi:10.1109/tpami.2011.95

Cited by 230 publications

(263 citation statements)

References 27 publications

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“…Cubical complexes have been promoted in particular by V. Kovalevsky [33] in order to provide a sound topological basis for image analysis. Recent advances in this framework includes the design of new image processing operators [19,36,48] as well as applications in different fields such as computer graphics [43] or medical imaging [12,13]. In order to ease the reading this article is selfcontained.…”

Section: Introductionmentioning

confidence: 99%

Collapses and Watersheds in Pseudomanifolds of Arbitrary Dimension

et al. 2014

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Section: Introductionmentioning

confidence: 99%

Collapses and Watersheds in Pseudomanifolds of Arbitrary Dimension

et al. 2014

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“…One particularly useful version looks at the so-called sublevelset filtration of an image, which builds up the simplicial complexes using the values at the pixels rather than starting with a collection of points (Kaczynski, Mischaikow, & Mrozek, 2006;Robins, Wood, & Sheppard, 2011). There are also many variants of persistent homology that may be of interest depending on the application.…”

Section: Further Readingmentioning

confidence: 99%

A User’s Guide to Topological Data Analysis

Munch

2017

Learning Analytics

116

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ABSTRACT. Topological data analysis (TDA) is a collection of powerful tools that can quantify shape and structure in data in order to answer questions from the data's domain. This is done by representing some aspect of the structure of the data in a simplified topological signature. In this article, we introduce two of the most commonly used topological signatures. First, the persistence diagram represents loops and holes in the space by considering connectivity of the data points for a continuum of values rather than a single fixed value. The second topological signature, the mapper graph, returns a 1-dimensional structure representing the shape of the data, and is particularly good for exploration and visualization of the data. While these techniques are based on very sophisticated mathematics, the current ubiquity of available software means that these tools are more accessible than ever to be applied to data by researchers in education and learning, as well as all domain scientists.

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“…There are many algorithms available in the literature to compute the 3D Morse-Smale complex. The algorithms are primarily based on either the quasi Morse-Smale complex formulation [61], [62] or Forman's [63] discrete Morse theory [64], [65], [66], [67]. We use a parallel algorithm based on the latter approach [67] resulting in fast computation even for large datasets.…”

Section: Density Estimation and Filament Modelingmentioning

confidence: 99%

Felix: A Topology Based Framework for Visual Exploration of Cosmic Filaments

Shivashankar

Pranav

Natarajan

et al. 2016

IEEE Trans. Visual. Comput. Graphics

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Abstract-The large-scale structure of the universe is comprised of virialized blob-like clusters, linear filaments, sheet-like walls and huge near empty three-dimensional voids. Characterizing the large scale universe is essential to our understanding of the formation and evolution of galaxies. The density range of clusters, walls and voids are relatively well separated, when compared to filaments, which span a relatively larger range. The large scale filamentary network thus forms an intricate part of the cosmic web. In this paper, we describe Felix, a topology based framework for visual exploration of filaments in the cosmic web. The filamentary structure is represented by the ascending manifold geometry of the 2-saddles in the Morse-Smale complex of the density field. We generate a hierarchy of Morse-Smale complexes and query for filaments based on the density ranges at the end points of the filaments. The query is processed efficiently over the entire hierarchical Morse-Smale complex, allowing for interactive visualization. We apply Felix to computer simulations based on the heuristic Voronoi kinematic model and the standard ΛCDM cosmology, and demonstrate its usefulness through two case studies. First, we extract cosmic filaments within and across cluster like regions in Voronoi kinematic simulation datasets. We demonstrate that we produce similar results to existing structure finders. Second, we extract different classes of filaments based on their density characteristics from the ΛCDM simulation datasets. Filaments that form the spine of the cosmic web, which exist in high density regions in the current epoch, are isolated using Felix. Also, filaments present in void-like regions are isolated and visualized. These filamentary structures are often over shadowed by higher density range filaments and are not easily characterizable and extractable using other filament extraction methodologies.Index Terms-Morse-Smale complexes, tessellations, cosmology theory, cosmic web, large-scale structure of the universe.

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Theory and Algorithms for Constructing Discrete Morse Complexes from Grayscale Digital Images

Cited by 230 publications

References 27 publications

Collapses and Watersheds in Pseudomanifolds of Arbitrary Dimension

Collapses and Watersheds in Pseudomanifolds of Arbitrary Dimension

A User’s Guide to Topological Data Analysis

Felix: A Topology Based Framework for Visual Exploration of Cosmic Filaments

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