论文标题
形状空间的捆绑理论结构
A Sheaf-Theoretic Construction of Shape Space
论文作者
论文摘要
我们呈现形状空间的融学理论结构 - 所有形状的空间。我们通过描述可构造集的POSET类别的同质纸条来做到这一点，其中每个集合都映射到其持续的综合转换（PHT）。基于Schapira的基本工作的最新结果表明，这种转换是注入性的，因此使PHT成为每种形状的良好摘要对象。我们的同质捆捆结果使我们能够将不同形状的PHT“胶合”在一起，以构建更大形状的PHT。在我们的形状是多面体的情况下，我们证明了PHT的普遍神经引理。最后，通过重新检查Smale-Niyogi-Weinberger的采样结果，我们表明我们可以可靠地近似多面体的多面体将歧管的PHT近似于任意精度。
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 Transform (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.
