归一化Radon累积分布变换在有限数据识别中的推广
Generalizations of the Normalized Radon Cumulative Distribution Transform for Limited Data Recognition
AI总结：
本文提出了一种归一化Radon累积分布变换的推广方法，用于在有限数据下实现图像和形状的鲁棒分类。
摘要：
Radon累积分布变换（R-CDT）利用一维Wasserstein传输和Radon变换来表示图像中的显著特征。它与切片Wasserstein距离密切相关，有助于分类任务，尤其是在小数据情况下，例如在丝网印刷中识别水印。这里，一个典型的问题是给定的数据可能由于测量过程而受到仿射变换的影响。为了使R-CDT对任意仿射变换具有不变性，我们之前的工作中提出了一种两步归一化方法。本文的目标是双方面的。首先，我们提出了一种通用的归一化家族以增强应用的灵活性。其次，我们通过使用通用的Radon变换研究多维和非欧几里得设置。我们证明了我们的新特征表示在某些变换下具有不变性，并允许在特征空间中进行线性分离。我们的理论结果通过基于2D图像、3D形状和3D旋转矩阵的数值实验得到支持，显示出接近完美的分类准确率和聚类结果。
英文摘要：
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.
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