  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.
