🍩 Database of Original & Non-Theoretical Uses of Topology
(found 7 matches in 0.00126s)
Topological Skeletonization and Tree-Summarization of Neurons Using Discrete Morse Theory (2018)Suyi Wang, Xu Li, Partha Mitra, Yusu Wang
RGB Image-Based Data Analysis via Discrete Morse Theory and Persistent Homology (2018)Chuan Du, Christopher Szul, Adarsh Manawa, Nima Rasekh, Rosemary Guzman, Ruth Davidson
AbstractUnderstanding and comparing images for the purposes of data analysis is currently a very computationally demanding task. A group at Australian National University (ANU) recently developed open-source code that can detect fundamental topological features of a grayscale image in a computationally feasible manner. This is made possible by the fact that computers store grayscale images as cubical cellular complexes. These complexes can be studied using the techniques of discrete Morse theory. We expand the functionality of the ANU code by introducing methods and software for analyzing images encoded in red, green, and blue (RGB), because this image encoding is very popular for publicly available data. Our methods allow the extraction of key topological information from RGB images via informative persistence diagrams by introducing novel methods for transforming RGB-to-grayscale. This paradigm allows us to perform data analysis directly on RGB images representing water scarcity variability as well as crime variability. We introduce software enabling a a user to predict future image properties, towards the eventual aim of more rapid image-based data behavior prediction.
Skeletonization and Partitioning of Digital Images Using Discrete Morse Theory (2015)Olaf Delgado-Friedrichs, Vanessa Robins, Adrian Sheppard
AbstractWe 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.
Morse Theory and Persistent Homology for Topological Analysis of 3D Images of Complex Materials (2014)O. Delgado-Friedrichs, V. Robins, A. Sheppard
AbstractWe 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 For-man's discrete Morse theory and provides a filtration that can be used as input to persistent homology algorithms. Efficient implementations allow us to process large-scale x-ray micro-CT 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. Sheppard
AbstractWe 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.
The Persistent Cosmic Web and Its Filamentary Structure – II. Illustrations (2011)T. Sousbie, C. Pichon, H. Kawahara
AbstractAbstract. The recently introduced discrete persistent structure extractor (DisPerSE, Sousbie, Paper I) is implemented on realistic 3D cosmological simulations