论文标题
封闭曲线签名的指标和独特性标准
Metrics and Uniqueness Criteria on the Signatures of Closed Curves
论文作者
论文摘要
本文探讨了Calabi等人在1996年引入的差异特征的范例。这种方法在诸如计算机视觉之类的领域具有巨大的影响，这些技术可以使用这些技术来验证人的笔迹与先前的文档或医学成像一致，以举一些示例。由希克曼（Hickman）在2011年提供的示例以及2009年的Musso和Nicolodi在此不变式中提供的示例，我们为曲线及其签名之间的对应关系提供了新标准，可以在一般环境中独一无二。为了显示此结果，我们介绍了有关签名的新方法，尤其是通过微分方程的镜头，以及签名的扩展，以包括有关曲线函数的高阶衍生物的信息，与曲线相对应和所需的组动作。我们还显示了有关签名的鲁棒性的结果，表明，如果两个签名足够接近，则在$ \ mathbb {r}^n $子集中的一个合适的度量标准下，鉴于这些签名的某些条件，它们也对应于它们对应的相应等价曲线类别。
This paper explores the paradigm of the differential signature introduced in 1996 by Calabi et al. This methodology has vast implications in fields such as computer vision, where these techniques can potentially be used to verify a person's handwriting is consistent with prior documents, or in medical imaging, to name a few examples. Motivated by examples provided by Hickman in 2011 and Musso and Nicolodi in 2009 regarding key failures in this invariant, we provide new criteria for the correspondence between a curve and its signature to be unique in a general setting. To show this result, we introduce new methods regarding the signature, particularly through the lens of differential equations, and the extension of the signature to include information on higher order derivatives of the curvature function corresponding to the curve and desired group action. We additionally show results regarding the robustness of the signature, showing that under a suitable metric on the space of subsets of $\mathbb{R}^n$, if two signatures are sufficiently close then so too will the corresponding equivalence classes of curves they correspond to, given certain conditions on these signatures.
