Skeletonization and Partitioning of Digital Images Using Discrete Morse Theory
Date
2015-02-13
Authors
Delgado-Friedrichs, Olaf
Robins, Vanessa
Sheppard, Adrian
Journal Title
Journal ISSN
Volume Title
Publisher
Institute of Electrical and Electronics Engineers (IEEE Inc)
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.
Description
Keywords
Curve skeleton, surface skeleton, medial axis transform, watershed transform, discrete Morse theory, persistent homology
Citation
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Source
IEEE Transactions on Pattern Analysis and Machine Intelligence
Type
Journal article