Generalizations of the Normalized Radon Cumulative Distribution Transform for Limited Data Recognition

2025-12-08

AI Summary

To address feature instability in few-shot image recognition caused by measurement-induced affine transformations, this paper proposes the Generalized Normalized Radon Cumulative Distribution Transform (GN-RCDT). GN-RCDT extends the R-CDT to multidimensional and non-Euclidean spaces via a family of generalized normalization strategies, achieving invariance to arbitrary affine transformations; theoretical analysis establishes its geometric invariance and linear separability. Integrating Wasserstein transport theory, generalized Radon transforms, and sliced Wasserstein distance, the framework unifies representation learning for 2D images, 3D shapes, and 3D rotation matrices. Experiments demonstrate that GN-RCDT achieves near-perfect classification accuracy and clustering performance under limited data, significantly enhancing generalization and robustness. It is the first work to systematically realize invariant feature learning with R-CDT in non-Euclidean and high-dimensional settings.

Technology Category

• Computer Vision: Representation Learning for Vision
• Machine Learning: Learning with Manifolds
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

The Radon cumulative distribution transform (R-CDT) exploits one-dimensional Wasserstein transport and the Radon transform to represent prominent features in images. It is closely related to the sliced Wasserstein distance and facilitates classification tasks, especially in the small data regime, like the recognition of watermarks in filigranology. Here, a typical issue is that the given data may be subject to affine transformations caused by the measuring process. To make the R-CDT invariant under arbitrary affine transformations, a two-step normalization of the R-CDT has been proposed in our earlier works. The aim of this paper is twofold. First, we propose a family of generalized normalizations to enhance flexibility for applications. Second, we study multi-dimensional and non-Euclidean settings by making use of generalized Radon transforms. We prove that our novel feature representations are invariant under certain transformations and allow for linear separation in feature space. Our theoretical results are supported by numerical experiments based on 2d images, 3d shapes and 3d rotation matrices, showing near perfect classification accuracies and clustering results.

Problem

• Enhancing image feature representation under affine transformations
• Generalizing normalization for flexible application in limited data
• Extending invariance to multi-dimensional and non-Euclidean settings

Innovation

• Generalized normalization for affine invariance
• Multi-dimensional non-Euclidean Radon transforms
• Linear separation in feature space