Abstract
Most datasets encountered in computer vision and medical applications present symmetries that should be taken into account in classification tasks. A typical example is the symmetry by rotation and/or scaling in object detection. A common way to build neural networks that learn the symmetries is to use data augmentation. In order to avoid data augmentation and build more sustainable algorithms, we present an alternative method to mod out symmetries based on the notion of section of a principal fiber bundle. This framework allows the use of simple metrics on the space of objects in order to measure dissimilarities between orbits of objects under the symmetry group. Moreover, the section used can be optimized to maximize separation of classes. We illustrate this methodology on a dataset of contours of objects for the groups of translations, rotations, scalings and reparameterizations. In particular, we present a $2$-parameter family of canonical parameterizations of curves, containing the constant-speed parameterization as a special case, which we believe is interesting in its own right. We hope that this simple application will serve to convey the geometric concepts underlying this method, which have a wide range of possible applications. The code is available at the following link: $\href{this https URL}{this https URL}$. A tutorial notebook showcasing an application of the code to a specific dataset is available at the following link: $\href{this https URL}{this https URL}$
Abstract (translated)
在计算机视觉和医学应用中遇到的大多数数据集都具有应在其分类任务中考虑的对称性。一个典型例子是在对象检测中的旋转和平移及缩放对称性。一种常见的用于构建学习这些对称性的神经网络的方法是使用数据增强技术。为了避免数据增强并开发更可持续的算法，我们提出了一种基于主纤维丛截面的概念来消除对称性的替代方法。这种方法框架允许在对象空间中使用简单的度量来测量在对称性群作用下的对象轨道之间的差异。此外，所使用的截面可以被优化以最大化类别的分离度。 为了说明这一方法的应用，我们使用了一组物体轮廓的数据集，并针对平移、旋转、缩放和再参数化等变换群进行了研究。特别地，我们介绍了一个两参数的曲线规范参数化的家族，其中包含常速参数化作为特殊情况，我们认为这对这项研究本身具有重要意义。 希望这个简单的应用能够帮助传达这种方法背后的基本几何概念，这些概念在广泛的潜在应用场景中都有所体现。 代码可在以下链接获取：[此处提供具体链接]。 一个展示如何将该代码应用于特定数据集的教程笔记本也已在以下链接发布：[此处提供具体链接]。
URL
https://arxiv.org/abs/2509.26070
