A Sheaf-Theoretic Construction of Shape Space

Foundations of Computational Mathematics https://doi.org/10.1007/s10208-024-09650-1 A Sheaf-Theoretic Construction of Shape Space Shreya Arya1 · Justin Curry2,4 · Sayan Mukherjee3,4 Received: 7 September 2022 / Revised: 22 June 2023 / Accepted: 4 September 2023 © The Author(s) 2024 Abstract 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 SmaleNiyogi-Weinberger, we show that we can reliably approximate the PHT of a manifold by a polyhedron up to arbitrary precision. Keywords Statistical shape analysis · Applied sheaf theory · Topological transforms Mathematics Subject Classification 55N31 · 55U99 · 18F20 · 18G80 · 49Q10 Communicated by Herbert Edelsbrunner. B Justin Curry Shreya Arya Sayan Mukherjee 1 Department of Mathematics, Duke University, Durham, NC, USA 2 Department of Mathematics and Statistics, University at Albany SUNY, Albany, NY, USA 3 Departments of Statistical Science, Mathematics, Computer Science, Biostatistics and Bioinformatics, Duke University, Durham, NC, USA 4 Center for Scalable Data Analytics and Artificial Intelligence Universität Leipzig and the Max Planck Institute for Mathematics in the Natural Sciences, Leipzig, Germany 123 Foundations of Computational Mathematics Contents 1 Introduction and Main Results . . . . . . . . . . . . . . . . . . . . 1.1 Prior Work on Shape Space . . . . . . . . . . . . . . . . . . . 1.2 Our Contribution to Shape Space Theory . . . . . . . . . . . . 2 Background on Constructibility, Persistent Homology and Sheaves 2.1 O-Minimality . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Euler Characteristic and the Radon Transform . . . . . . . . . 2.3 Homology and the Betti Curve Transform . . . . . . . . . . . 2.4 Functoriality and the Persistent Homology Transform . . . . . 2.5 Cohomology and Sheaf Theory . . . . . . . . . . . . . . . . . 2.6 Derived Sheaf Theory . . . . . . . . . . . . . . . . . . . . . . 2.7 Constructible Sheaves and their Functions . . . . . . . . . . . 3 A Homotopy Sheaf on Shape Space . . . . . . . . . . . . . . . . . 3.1 Inclusion–Exclusion for the ECT . . . . . . . . . . . . . . . . 3.2 Sites and Homotopy Sheaves . . . . . . . . . . . . . . . . . . 3.3 Gluing Results for the PHT . . . . . . . . . . . . . . . . . . . 3.4 Relative PHT and ECT . . . . . . . . . . . . . . . . . . . . . 4 Metrics, Stability and Approximation Theory for the PHT . . . . . 4.1 Distances on PHTs . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Interleaving-Type Distances on the PHT . . . . . . . . . 4.1.2 Wasserstein-Type Distances on the PHT . . . . . . . . . 4.2 Comparison with Other Distances on PHTs and Shape Spaces . 4.2.1 Distances Between Point Clouds . . . . . . . . . . . . . 4.3 The Stability Theorem . . . . . . . . . . . . . . . . . . . . . 4.4 Point Samples for Approximating the PHT . . . . . . . . . . . 5 Discussion and Future Work . . . . . . . . . . . . . . . . . . . . . Appendix A. Infinity Category Version . . . . . . . . . . . . . . . . . A.1 Simplicial Sets and Infinity Categories . . . . . . . . . . . . . A.2 Homological Algebra in Infinity categories . . . . . . . . . . . A.3 The PHT Viewed as an Object of the Derived ∞-Category. . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Introduction and Main Results Shape spaces are intended to provide a single framework for comparing shapes. Different shapes are rendered as different points in shape space and comparisons of shapes can be formalized in terms of distances between these different points; see Fig. 1. One of the first examples of a shape space was pioneered in [46, 47] and takes the perspective that two shapes can be compared by first placing landmarks down on each shape. If these landmarks are related by a similarity transformation—a rotation or dilation— then they are regarded as the same shape. Non-identical shapes are then compared in an appropriately defined quotient space, assuming the number of landmarks are the same. Another popular model of shape space dispenses with the landmark selection process and considers shapes as immersed submanifolds [13] and then tries to compare them via diffeomorphisms of the ambient space, possibly generated by some optimal transport or control problem [26]. However, common to both the landmark and diffeomorphism-based approaches to shape space is a quotient operation that naturally lends a fiber bundle structure to these data representations and comparisons. Fiber bundles provide a convenient mathematical language for shape comparison, but previous work [43] also illustrates their 123 Foundations of Computational Mathematics Fig. 1 Previous constructions of shape space imply a fiber bundle perspective on shape space. The fibers encode similarity transformations or reparameterizations of a shape and the base space records the shape as an equivalence class. The PHT-based shape space introduced here uses a more algebraic construction where the base space is replaced by a base poset and the “fibers” are unique sheaf-theoretic representations of the shape insufficiency for general morphological comparison. Indeed, the implicit assumption for both of these models—and many more models for shape space discussed below in Sect. 1.1—is that the shapes under consideration have (...truncated)