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Skeletonization and Partitioning of Digital Images Using Discrete Morse Theory.

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    Discrete Morse theory offers a robust framework for image analysis, defining image skeletons and partitions using critical cells. This method ensures topological validity and simplifies complex image data for applications like fluid-flow modeling.

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    Area of Science:

    • Computational topology
    • Digital image processing
    • Scientific visualization

    Background:

    • Grayscale digital images are often analyzed using topological methods.
    • Existing methods for image partitioning and skeletonization can lack a rigorous mathematical foundation.
    • Understanding the topology of porous materials is crucial for fluid-flow modeling.

    Purpose of the Study:

    • To establish a unifying and rigorous foundation for defining image skeletons and partitions using discrete Morse theory.
    • To develop a method for simplifying these topological structures through critical cell cancellation.
    • To demonstrate the topological validity of the proposed constructions.

    Main Methods:

    • Modeling grayscale images as cubical complexes with vertex functions.
    • Extending image functions to discrete gradient vector fields.
    • Defining basins and skeleton segments using stable and unstable sets of critical cells.
    • Employing Morse-theoretic cancellation informed by persistent homology for simplification.

    Main Results:

    • A rigorous framework for discrete image skeletons and partitions based on Morse theory.
    • Proof of topological validity, including skeleton homotopy to the original object.
    • Demonstration of simplification techniques for basins and skeletons.
    • Inclusion of Python code for efficient vector field traversal.

    Conclusions:

    • Discrete Morse theory provides a powerful and unifying foundation for digital image topology.
    • The developed methods offer accurate topological models for applications like fluid-flow analysis in porous media.
    • The simplification strategies enhance the practical utility of topological image analysis.