G. Carlsson, A. Zomorodian, A. Collins, and L. Guibas. Persistence Barcodes for Shapes. Proc. Symp. Geometry Processing (2004), pp. 127-138.
In this paper, we initiate a study of shape description and
classification via the application of persistent homology to two
tangential constructions on geometric objects.
Our techniques combine the differentiating power of geometry with the
classifying power of topology.
The homology of our first construction, the
tangent complex, can distinguish between topologically identical
shapes with different "sharp'' features, such as corners.
To capture "soft'' curvature-dependent features, we define a second
complex, the filtered tangent complex, obtained by parametrizing a
family of increasing subcomplexes of the tangent complex.
Applying persistent homology, we obtain a shape descriptor, called a
barcode, that is a finite union of intervals.
We define a metric over the space of such
intervals, arriving at a continuous invariant that reflects the
geometric properties of shapes.
We illustrate the power of our methods through a number of detailed studies
of parametrized families of mathematical shapes.
title = "Persistence Barcodes for Shapes",
author = "Carlsson, G. and Zomorodian, A. and Collins, A. and Guibas, L.",
booktitle = "Proc. Geometry Process.",
year = 2004,
pages = "127--138",