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

TL;DR

This paper extends the normalized Radon cumulative distribution transform (R-CDT) to more flexible forms and applies it to multi-dimensional and non-Euclidean data, improving invariance and classification accuracy.

Contribution

It introduces a family of generalized normalizations for R-CDT and explores its application in multi-dimensional and non-Euclidean settings with theoretical guarantees.

Findings

Achieves near-perfect classification accuracy in experiments

Demonstrates invariance under certain transformations

Supports linear separation in feature space

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.