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Shape Fingerprints, Geodesic Trails and Free Abelian Groups on Skeletal Vortexes

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Computational Geometry, Topology and Physics of Digital Images with Applications

Part of the book series: Intelligent Systems Reference Library ((ISRL,volume 162))

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Abstract

This chapter takes another look at filament skeletons, skeletal vortexes and skeletal nerves in cell complexes. The focus here is on the group theory underlying a Computational Topology of digital images (CTdi). A digital image is an example of what is known as a shape space. A space is any nonempty set of points. A shape space is a collection of sets of points X and each particular configuration (arrangement of the points) in a subset of X defines a shape. A digital image shape space is a collection of digitized optical sensor values that provide a record of the hue angles of pixels in the Hue Saturation Value colour space. There is a 1-to-1 correspondence between the pixel hue angles and the wavelengths of light reflected from the surfaces in a visual scene at a given instant in spacetime. It is this 1-to-1 correspondence that leads to a deeper view of skeletal complexes on triangulated video frames.

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Notes

  1. 1.

    Many thanks to S. Ramanna for providing these Apple iPad \(^{{\textregistered }}\) images.

  2. 2.

    Many thanks to M. Z. Ahmad for supplying the Matlab script used to draw a barycentric filament skeleton.

  3. 3.

    Many thanks to R. Tozzi and A. Tozzi for the Napoli breakfast picture.

  4. 4.

    Many thanks to Alexander Yurkin for supplying this self-portrait of his father.

  5. 5.

    Many thanks to Arjuna P. H. Don for this sample triangulated video frame.

  6. 6.

    Many thanks to Fatemeh Gorgannejad for correcting this example.

  7. 7.

    Many thanks to Enze Cui for the video frame in this example.

  8. 8.

    This is also a video frame from Enze Cui.

  9. 9.

    See F. Haimo’s review MR0067879 at https://mathscinet.ams.org/mathscinet/.

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  1. Department of Electrical and Computer Engineering, Engineering and Information Technology Complex, University of Manitoba, Winnipeg, MB, Canada

    James F. Peters

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Peters, J.F. (2020). Shape Fingerprints, Geodesic Trails and Free Abelian Groups on Skeletal Vortexes. In: Computational Geometry, Topology and Physics of Digital Images with Applications. Intelligent Systems Reference Library, vol 162. Springer, Cham. https://doi.org/10.1007/978-3-030-22192-8_3

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