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.

翻译：Radon累积分布变换（R-CDT）通过结合一维Wasserstein传输与Radon变换，实现了图像显著特征的表征。该方法与切片Wasserstein距离密切相关，并在小数据场景（如防伪水印识别）的分类任务中表现出优越性能。此类应用常面临测量过程引起的仿射变换干扰数据的问题。为实现R-CDT在任意仿射变换下的不变性，我们前期研究提出了两步归一化方法。本文目标有二：首先，我们提出一系列广义归一化方法以增强应用灵活性；其次，通过引入广义Radon变换，我们拓展了该方法至多维与非欧几里得空间。我们证明了新特征表示在特定变换下具有不变性，并允许在特征空间中进行线性分离。基于二维图像、三维形状及三维旋转矩阵的数值实验验证了理论结果，显示出近乎完美的分类准确度与聚类效果。