🍩 Database of Original & Non-Theoretical Uses of Topology

1. Pore Configuration Landscape of Granular Crystallization (2017)

   Mohammad Saadatfar, Hiroshi Takeuchi, Vanessa Robins, Nicolas Francois, Yisuaki Hiraoka

   Abstract

   Emergence and growth of crystalline domains in granular media remains under-explored. Here, the authors analyse tomographic snapshots from partially recrystallized packings of spheres using persistent homology and find agreement with proposed transitions based on continuous deformation of octahedral and tetrahedral voids.

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   • Crystal structure
   • Material physics
   • Pore configuration
   • Granular packings
   • Persistence diagrams

2. The Extended Persistent Homology Transform of Manifolds With Boundary (2022)

   Katharine Turner, Vanessa Robins, James Morgan

   Abstract

   The Extended Persistent Homology Transform (XPHT) is a topological transform which takes as input a shape embedded in Euclidean space, and to each unit vector assigns the extended persistence module of the height function over that shape with respect to that direction. We can define a distance between two shapes by integrating over the sphere the distance between their respective extended persistence modules. By using extended persistence we get finite distances between shapes even when they have different Betti numbers. We use Morse theory to show that the extended persistence of a height function over a manifold with boundary can be deduced from the extended persistence for that height function restricted to the boundary, alongside labels on the critical points as positive or negative critical. We study the application of the XPHT to binary images; outlining an algorithm for efficient calculation of the XPHT exploiting relationships between the PHT of the boundary curves to the extended persistence of the foreground.

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   • AI Security
   • Convolutional Neural Networks
   • Machine learning
   • Point cloud
   • Persistent homology
   • Persistent homology:RIps

3. Skeletonization and Partitioning of Digital Images Using Discrete Morse Theory (2015)

   Olaf Delgado-Friedrichs, Vanessa Robins, Adrian Sheppard

   Abstract

   We show how discrete Morse theory provides a rigorous and unifying foundation for defining skeletons and partitions of grayscale digital images. We model a grayscale image as a cubical complex with a real-valued function defined on its vertices (the voxel values). This function is extended to a discrete gradient vector field using the algorithm presented in Robins, Wood, Sheppard TPAMI 33:1646 (2011). In the current paper we define basins (the building blocks of a partition) and segments of the skeleton using the stable and unstable sets associated with critical cells. The natural connection between Morse theory and homology allows us to prove the topological validity of these constructions; for example, that the skeleton is homotopic to the initial object. We simplify the basins and skeletons via Morse-theoretic cancellation of critical cells in the discrete gradient vector field using a strategy informed by persistent homology. Simple working Python code for our algorithms for efficient vector field traversal is included. Example data are taken from micro-CT images of porous materials, an application area where accurate topological models of pore connectivity are vital for fluid-flow modelling.

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   • Image analysis
   • Images
   • images
   • images:grayscale
   • Algorithm
   • Morse theory
   • Persistent homology