🍩 Database of Original & NonTheoretical Uses of Topology
(found 7 matches in 0.001996s)


TopologyBased Signal Separation (2004)
V. Robins, N. Rooney, E. Bradley 
Morse Theory and Persistent Homology for Topological Analysis of 3D Images of Complex Materials (2014)
O. DelgadoFriedrichs, V. Robins, A. SheppardAbstract
We develop topologically accurate and compatible definitions for the skeleton and watershed segmentation of a 3D digital object that are computed by a single algorithm. These definitions are based on a discrete gradient vector field derived from a signed distance transform. This gradient vector field is amenable to topological analysis and simplification via Forman's discrete Morse theory and provides a filtration that can be used as input to persistent homology algorithms. Efficient implementations allow us to process largescale xray microCT data of rock cores and other materials. 
Theory and Algorithms for Constructing Discrete Morse Complexes From Grayscale Digital Images (2011)
V. Robins, P. J. Wood, A. P. SheppardAbstract
We present an algorithm for determining the Morse complex of a two or threedimensional grayscale digital image. Each cell in the Morse complex corresponds to a topological change in the level sets (i.e., a critical point) of the grayscale image. Since more than one critical point may be associated with a single image voxel, we model digital images by cubical complexes. A new homotopic algorithm is used to construct a discrete Morse function on the cubical complex that agrees with the digital image and has exactly the number and type of critical cells necessary to characterize the topological changes in the level sets. We make use of discrete Morse theory and simple homotopy theory to prove correctness of this algorithm. The resulting Morse complex is considerably simpler than the cubical complex originally used to represent the image and may be used to compute persistent homology. 
The Extended Persistent Homology Transform of Manifolds With Boundary (2022)
Katharine Turner, Vanessa Robins, James MorganAbstract
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. 
Topological Persistence for Relating Microstructure and Capillary Fluid Trapping in Sandstones (2019)
A. L. Herring, V. Robins, A. P. SheppardAbstract
Results from a series of twophase fluid flow experiments in Leopard, Berea, and Bentheimer sandstones are presented. Fluid configurations are characterized using laboratorybased and synchrotron based 3D Xray computed tomography. All flow experiments are conducted under capillarydominated conditions. We conduct geometrytopology analysis via persistent homology and compare this to standard topological and watershedpartitionbased porenetwork statistics. Metrics identified as predictors of nonwetting fluid trapping are calculated from the different analytical methods and are compared to levels of trapping measured during drainageimbibition cycles in the experiments. Metrics calculated from pore networks (i.e., pore bodythroat aspect ratio and coordination number) and topological analysis (Euler characteristic) do not correlate well with trapping in these samples. In contrast, a new metric derived from the persistent homology analysis, which incorporates counts of topological features as well as their length scale and spatial distribution, correlates very well (R2 = 0.97) to trapping for all systems. This correlation encompasses a wide range of porous media and initial fluid configurations, and also applies to data sets of different imaging and image processing protocols. 
Skeletonization and Partitioning of Digital Images Using Discrete Morse Theory (2015)
Olaf DelgadoFriedrichs, Vanessa Robins, Adrian SheppardAbstract
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 realvalued 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 Morsetheoretic 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 microCT images of porous materials, an application area where accurate topological models of pore connectivity are vital for fluidflow modelling.