Abstract
This chapter explores what nerve structures in cell complexes tell us in approximating shapes revealed by light reflected from curved surfaces. Recall that a nerve, in its simplest form, is a collection of nonempty sets that overlap. That is, the parts of a nerve have nonempty intersection. This simplest form of a nerve was introduced by H. Edelsbrunner and J. L. Harer in their monograph on Computational Topology [1, Sect. 3.2, p. 59]. Of great interest here are the two forms of nerve complexes introduced by Alexandroff [2], namely.
- Alexandroff nerve :
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Collections of triangles with vertices that are various forms of seed points and with a common vertex in a cell complex on a triangulated bounded surface region.
- Alexandroff star nerve :
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Collections of triangles with vertices that are barycenters and with a common vertex in a cell complex on a triangulated bounded surface region (also called a barycentric star nerve).
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Notes
- 1.
Many thanks to Enze Cui for the drone video containing this video frame.
- 2.
Many thanks to M.Z. Ahmad for pointing this out.
- 3.
Many thanks to S. Ramanna for capturing this caustic during a sunny afternoon in Manitoba.
- 4.
- 5.
Observed by M. Z. Ahmad.
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Peters, J.F. (2020). What Nerve Complexes Tell Us About Image Shapes. 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_4
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