Journal ArticleDOI
Theory and Algorithms for Constructing Discrete Morse Complexes from Grayscale Digital Images
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TLDR
An algorithm for determining the Morse complex of a two or three-dimensional grayscale digital image 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 is presented.Abstract:
We present an algorithm for determining the Morse complex of a two or three-dimensional 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.read more
Citations
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X-ray imaging and analysis techniques for quantifying pore-scale structure and processes in subsurface porous medium systems
TL;DR: X-ray microtomographic imaging is a non-destructive technique for quantifying these processes in three dimensions within individual pores, and as reported here, with rapidly increasing spatial and temporal resolution.
Journal ArticleDOI
Morse Theory for Filtrations and Efficient Computation of Persistent Homology
TL;DR: An efficient preprocessing algorithm is introduced to reduce the number of cells in a filtered cell complex while preserving its persistent homology groups through an extension of combinatorial Morse theory from complexes to filtrations.
Journal ArticleDOI
The Topology ToolKit
TL;DR: An algorithm for the construction of a discrete gradient that complies to the critical points extracted in the piecewise-linear setting is presented, which guarantees a combinatorial consistency across the topological abstractions supported by TTK, and importantly, a unified implementation of topological data simplification for multi-scale exploration and analysis.
Journal ArticleDOI
A survey of topology-based methods in visualization
Christian Heine,Heike Leitte,Mario Hlawitschka,Federico Iuricich,L. De Floriani,Gerik Scheuermann,Hans Hagen,Christoph Garth +7 more
TL;DR: The process and results of an extensive annotation for generating a definition and terminology for topology‐based visualization are described, which enabled a typology for topological models which is used to organize research results and the state of the art.
Journal ArticleDOI
Porous Media Characterization Using Minkowski Functionals: Theories, Applications and Future Directions
Ryan T. Armstrong,James E. McClure,Vanessa Robins,Zhishang Liu,Christoph H. Arns,Steffen Schlüter,Steffen Berg,Steffen Berg +7 more
TL;DR: The theoretical basis of the Minkowski functionals, mathematical theorems and methods necessary for porous media characterization, common measurement errors when using micro-CT data and recent findings relating the MF to macroscale porous media properties are reviewed.
References
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Journal ArticleDOI
Watersheds in digital spaces: an efficient algorithm based on immersion simulations
Luc Vincent,Pierre Soille +1 more
TL;DR: A fast and flexible algorithm for computing watersheds in digital gray-scale images is introduced, based on an immersion process analogy, which is reported to be faster than any other watershed algorithm.
Book
Elements of Algebraic Topology
TL;DR: Elements of Algebraic Topology provides the most concrete approach to the subject with coverage of homology and cohomology theory, universal coefficient theorems, Kunneth theorem, duality in manifolds, and applications to classical theorem of point-set topology.
Journal ArticleDOI
Topological Persistence and Simplification
Edelsbrunner,Letscher,Zomorodian +2 more
TL;DR: Fast algorithms for computing persistence and experimental evidence for their speed and utility are given for topological simplification within the framework of a filtration, which is the history of a growing complex.
Journal ArticleDOI
Morse Theory for Cell Complexes
TL;DR: In this article, a discrete Morse theory for CW complexes is presented, which can be used to give a Morse theoretic proof of the Poincare conjecture in dimension 5, along the lines of the proof in [Mi2] along with discrete analogues of such intrinsically smooth notions as the gradient vector field and the gradient flow associated to a Morse function.