Tools from topological data analysis have been widely used to represent binary images in many scientific applications. Methods that aim to represent grayscale images (i.e., where pixel intensities instead take on continuous values) have been relatively underdeveloped. In this paper, we introduce the Euler-Radon transform, which generalizes the Euler characteristic transform to grayscale images by using o-minimal structures and Euler integration over definable functions. Coupling the Karhunen-Loeve expansion with our proposed topological representation, we offer hypothesis-testing algorithms based on the chi-squared distribution for detecting significant differences between two groups of grayscale images. We illustrate our framework via extensive numerical experiments and simulations.

翻译：拓扑数据分析工具已被广泛应用于众多科学领域中表示二值图像。然而，旨在表示灰度图像（即像素强度取连续值）的方法相对发展不足。本文引入欧拉-拉东变换，该变换通过使用o-极小结构与可定义函数上的欧拉积分，将欧拉特征变换推广至灰度图像。通过将卡亨南-洛维展开与我们提出的拓扑表示相结合，我们提出基于卡方分布的假设检验算法，用于检测两组灰度图像间的显著差异。我们通过大量数值实验和模拟验证了该框架的有效性。