🤖 AI Summary
Computer vision and medical image classification often suffer from symmetry-induced ambiguities—such as rotation, scaling, and reparameterization—whose invariance is typically enforced via data augmentation, lacking geometric interpretability and sustainability. This paper proposes a geometry-aware learning framework that eliminates the need for data augmentation. Grounded in the theory of principal fiber bundles, we construct a two-parameter family of canonical parametrizations for planar curves—including constant-speed parametrization as a special case—and learn discriminative representations by optimizing bundle sections to maximize inter-class separation. Coupled with an orbit-space metric design, our approach achieves strict invariance to translation, rotation, scaling, and reparameterization. Empirical evaluation on object contour datasets demonstrates that the learned shape representations are symmetry-invariant and yield significant classification improvements. The implementation, including code and tutorials, is publicly available to support reproducible research and practical deployment.
📝 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: $href{https://github.com/GiLonga/Geometric-Learning}{https://github.com/GiLonga/Geometric-Learning}$. A tutorial notebook showcasing an application of the code to a specific dataset is available at the following link: $href{https://github.com/ioanaciuclea/geometric-learning-notebook}{https://github.com/ioanaciuclea/geometric-learning-notebook}$