Geometric Learning of Canonical Parameterizations of $2D$-curves

TL;DR

This work develops a geometry-inspired framework to normalize 2D contours with respect to shape-preserving symmetries by employing sections of the principal fiber bundle of parameterized curves. It introduces a two-parameter family of canonical parameterizations, the curvature-weighted clock parameterizations, and defines a simple L2-based distance restricted to a chosen section, enabling metric learning to maximize class separation. Through extensive experiments on leaf contour datasets, including Swedish leaves and Flavia, the authors demonstrate that optimally chosen sections substantially improve clustering and classification with minimal parameterization. The approach also provides automatic contour registration and deformations, and the authors show that the method can serve as an efficient pre-processing step for more complex classifiers, with potential extensions to higher-dimensional shape spaces. Code and tutorials are provided to facilitate adoption and replication.

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: $\href{https://github.com/ioanaciuclea/geometric-learning-notebook}{https://github.com/ioanaciuclea/geometric-learning-notebook}$

Figures (20)

• Figure 1: (a) Left: The pair of leaves from the Swedish dataset that maximizes the intraclass distance is extracted from the training set, and the interpolation of their optimal parameterizations for the Dunn index is displayed for the parameters $(n = 3, \lambda = 2000)$. Right: The pair of leaves from the Swedish dataset that minimizes the interclass distance is extracted from the training set, and the interpolation of their optimal parameterizations for the Dunn index is displayed for $(n = 3, \lambda = 2000)$. (b) Left: The pair of leaves from the Swedish dataset that maximizes the intraclass distance is extracted from the training set, and the interpolation of their optimal parameterizations for the Davies Bouldin index is displayed for the parameters $(n = 5, \lambda = +\infty)$. Right: The pair of leaves from the Swedish dataset that minimizes the interclass distance is extracted from the training set, and the interpolation of their optimal parameterizations is displayed for $(n = 5, \lambda = +\infty)$. We can see that the same pair of leaves maximizes the intraclass distance both for the Dunn index and the Davies Bouldin index, and the same pair of leaves minimizes the interclass distance for both indices.
• Figure 2: Emmy Noether and the moving frame associated with her profile. The signed curvature $\kappa$ is defined as the rate of turning angle of the moving frame associated with a parameterized plane curve. The maximum and the minimum of the signed curvature correspond to two points where the curvature is extremal.
• Figure 3: Illustration of a fiber bundle $\pi: \mathcal{P}\rightarrow \mathcal{P}/\mathcal{G}$ with three different sections $S_i: \mathcal{P}/\mathcal{G}\rightarrow \mathcal{P}$.
• Figure 4: A one-parameter family of canonical parameterizations: Each contour of Emmy Noether is parameterized in a unique way using Equation \ref{['ulambda']} for a given parameter $\lambda$. The sample points are the images of a uniform sampling of the interval $[0;1]$. The leftmost contour is parameterized proportionally to the curvature-length with parameter space $\mathbb{R}/\mathbb{Z} = \{t \in [0,1], 0\sim 1\}$ and corresponds to $\lambda = 0$. For this parameterization, sample points are concentrated on high-curvature portions of the curve, whereas flat pieces contain no sample points. The rightmost contour is parameterized proportionally to arc-length with parameter space $\mathbb{R}/\mathbb{Z} = \{t \in [0,1], 0\sim 1\}$ and corresponds to $\lambda = +\infty$. In this case, sample points are uniformly distributed along the contour. In between, from left to right, the following parameters are used $\lambda = 0.3$, $\lambda = 1$, $\lambda = 2$ (see Equation \ref{['ulambda']}).
• Figure 5: Dataset of Swedish leaves from the Linköpling University datasethttps://www.cvl.isy.liu.se/en/research/datasets/swedish-leaf/ (accessed on 8 September 2025). (a) A sample image from each class of leaves is depicted (the classes are ordered from left to right and top to bottom) (b) corresponding classes (c) extracted contours using Matlab's function bwboundaries on binarized images.

Theorems & Definitions (13)

• Definition 2.1
• Remark 2.2
• Proposition 2.3
• Definition 2.4
• Definition 2.5
• Remark 2.6
• Proposition 2.7
• proof : Proof of Proposition \ref{['distance_non_zero']}
• Proposition 2.8
• proof