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Theory and Algorithms for Constructing Discrete Morse Complexes from Grayscale Digital Images.

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IEEE Transactions on Pattern Analysis and Machine Intelligence

|May 18, 2011

Summary

This summary is machine-generated.

We developed an algorithm to compute the Morse complex for digital images, simplifying topological analysis. This method identifies critical points, aiding in understanding image structures and persistent homology.

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Area of Science:

Computational Topology
Digital Image Analysis
Computer Vision

Background:

Digital images are often represented as cubical complexes.
Understanding the topology of image level sets is crucial for analysis.
Existing methods may not efficiently capture all topological changes.

Purpose of the Study:

To present a novel algorithm for computing the Morse complex of 2D and 3D grayscale digital images.
To simplify the representation of image topology by identifying essential critical points.
To enable efficient computation of persistent homology.

Main Methods:

Modeling digital images as cubical complexes.
Developing a homotopic algorithm to construct a discrete Morse function.
Utilizing discrete Morse theory and simple homotopy theory for correctness proof.

Main Results:

An algorithm for determining the Morse complex of grayscale digital images.
The Morse complex accurately represents topological changes (critical points) in image level sets.
The resulting Morse complex is significantly simpler than the original cubical complex.

Conclusions:

The proposed algorithm provides an efficient way to compute the Morse complex for digital images.
The Morse complex serves as a simplified topological descriptor of grayscale images.
This approach facilitates the computation of persistent homology for image analysis.