Title: Proximal Planar Shape Signatures. Homology Nerves and Descriptive Proximity,
Abstract: This article introduces planar shape signatures derived from homology nerves,
which are intersecting 1-cycles in a collection of homology groups endowed with
a proximal relator (set of nearness relations) that includes a descriptive
proximity. A 1-cycle is a closed, connected path with a zero boundary in a
simplicial complex covering a finite, bounded planar shape. The signature of a
shape sh A (denoted by sig(sh A)) is a feature vector that describes sh A. A
signature sig(sh A) is derived from the geometry, homology nerves, Betti
number, and descriptive CW topology on the shape sh A. Several main results are
given, namely, (a) every finite, bounded planar shape has a signature derived
from the homology group on the shape, (b) a homology group equipped with a
proximal relator defines a descriptive Leader uniform topology and (c) a
description of a homology nerve and union of the descriptions of the 1-cycles
in the nerve have same homotopy type.