Statistical Inference on Grayscale Images via the Euler-Radon Transform

TL;DR

This paper introduces the Euler-Radon transform for grayscale images, enabling topological analysis and hypothesis testing, with demonstrated effectiveness through numerical experiments.

Contribution

It develops a novel topological transform for grayscale images using Euler integration, extending previous binary image methods and enabling statistical inference.

Findings

Effective hypothesis-testing algorithms for grayscale images.

Successful numerical experiments demonstrating the method.

Extension of topological tools to continuous-valued images.

Abstract

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.