In this work, we study the perception problem for garments using tools from
computational topology: the identification of their geometry and position in
space from point-cloud samples, as obtained e.g. with 3D scanners. We present a
reconstruction algorithm based on a direct topological study of the sampled
textile surface that allows us to obtain a cellular decomposition of it via a
Morse function. No intermediate triangulation or local implicit equations are
used, avoiding reconstruction-induced artifices. No a priori knowledge of the
surface topology, density or regularity of the point-sample is required to run
the algorithm. The results are a piecewise decomposition of the surface as a
union of Morse cells (i.e. topological disks), suitable for tasks such as
noise-filtering or mesh-independent reparametrization, and a cell complex of
small rank determining the surface topology. This algorithm can be applied to
smooth surfaces with or without boundary, embedded in an ambient space of any
dimension.

使用计算拓扑的工具研究服装的感知问题，通过直接拓扑研究样本化纺织物表面的重建算法，得到适用于降噪或网格独立参数化等任务的表面分块分解和确定表面拓扑的小秩细胞复合结构。

基于莫尔斯理论的采样纺织品表面重建

Surface reconstruction of sampled textiles via Morse theory

ChatGPT represents a significant milestone in the field of artificial
intelligence (AI), finding widespread applications across diverse domains.
However, its effectiveness in mathematical contexts has been somewhat
constrained by its susceptibility to conceptual errors. Concurrently,
topological data analysis (TDA), a relatively new discipline, has garnered
substantial interest in recent years. Nonetheless, the advancement of TDA is
impeded by the limited understanding of computational algorithms and coding
proficiency among theoreticians. This work endeavors to bridge the gap between
theoretical topological concepts and their practical implementation in
computational topology through the utilization of ChatGPT. We showcase how a
pure theoretician, devoid of computational experience and coding skills, can
effectively transform mathematical formulations and concepts into functional
code for computational topology with the assistance of ChatGPT. Our strategy
outlines a productive process wherein a mathematician trains ChatGPT on pure
mathematical concepts, steers ChatGPT towards generating computational topology
code, and subsequently validates the generated code using established examples.
Our specific case studies encompass the computation of Betti numbers, Laplacian
matrices, and Dirac matrices for simplicial complexes, as well as the
persistence of various homologies and Laplacians. Furthermore, we explore the
application of ChatGPT in computing recently developed topological theories for
hypergraphs and digraphs. This work serves as an initial step towards
effectively transforming pure mathematical theories into practical
computational tools, with the ultimate goal of enabling real applications
across diverse fields.

本研究以 ChatGPT 协助实现了从纯数学理论到实际计算拓扑工具的转变，桥接了理论拓扑概念与其在计算拓扑中的实际应用，展示了在纯数学概念缺乏计算经验和编码技能的情况下，通过数学家训练 ChatGPT 并利用其生成的计算拓扑代码，验证在计算拓扑中的运用性，如计算 Betti 数、拉普拉斯矩阵、Dirac 矩阵、各种同调和拉普拉斯的持续性，并探索 ChatGPT 在超图和有向图的拓扑理论计算中的应用。

ChatGPT 计算拓扑

ChatGPT for Computational Topology

This work introduces a novel framework for quantifying the presence and
strength of recurrent dynamics in video data. Specifically, we provide
continuous measures of periodicity (perfect repetition) and quasiperiodicity
(superposition of periodic modes with non-commensurate periods), in a way which
does not require segmentation, training, object tracking or 1-dimensional
surrogate signals. Our methodology operates directly on video data. The
approach combines ideas from nonlinear time series analysis (delay embeddings)
and computational topology (persistent homology), by translating the problem of
finding recurrent dynamics in video data, into the problem of determining the
circularity or toroidality of an associated geometric space. Through extensive
testing, we show the robustness of our scores with respect to several noise
models/levels, we show that our periodicity score is superior to other methods
when compared to human-generated periodicity rankings, and furthermore, we show
that our quasiperiodicity score clearly indicates the presence of biphonation
in videos of vibrating vocal folds, which has never before been accomplished
end to end quantitatively.

本文提出了一种新的框架来量化视频数据中经常性动态的存在和强度，并通过将非线性时间序列分析的思想与计算拓扑的思想相结合，将在视频数据中找到经常性动态的问题转化为确定相关几何空间的圆形性或环形性来提供周期性（完美重复）和准周期性（具有非可比周期的周期模式的叠加）的连续度量。经过广泛的测试，我们证明了我们的得分对于几个噪声模型 / 水平的鲁棒性，并且我们的周期性得分在与人类生成的周期性排名相比时优于其他方法，而我们的准周期性得分清楚地表明了振动声门的视频中存在双音发声，这在以前从未 quantitatvely 完成。