A Sheaf-Theoretic Construction of Shape Space

We present a sheaf-theoretic construction of shape space—the space of all shapes. We do this by describing a homotopy sheaf on the poset category of constructible sets, where each set is mapped to its Persistent Homology Transforms (PHT). Recent results that build on fundamental work of Schapira hav...

Full description

Saved in:
Bibliographic Details
Published in:Foundations of computational mathematics 2024-05
Main Authors: Arya, Shreya, Curry, Justin, Mukherjee, Sayan
Format: Article
Language:English
Citations: Items that this one cites
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary:We present a sheaf-theoretic construction of shape space—the space of all shapes. We do this by describing a homotopy sheaf on the poset category of constructible sets, where each set is mapped to its Persistent Homology Transforms (PHT). Recent results that build on fundamental work of Schapira have shown that this transform is injective, thus making the PHT a good summary object for each shape. Our homotopy sheaf result allows us to “glue” PHTs of different shapes together to build up the PHT of a larger shape. In the case where our shape is a polyhedron we prove a generalized nerve lemma for the PHT. Finally, by re-examining the sampling result of Smale-Niyogi-Weinberger, we show that we can reliably approximate the PHT of a manifold by a polyhedron up to arbitrary precision.
ISSN:1615-3375
1615-3383
DOI:10.1007/s10208-024-09650-1