On the stability of scale-space metrics

2026-06-25

AI Summary

This study addresses the stability of image metrics in Gaussian scale space under geometric deformations and additive noise. To overcome the sensitivity of classical scale-space metrics to rotation, the authors propose a rotation-invariant variant and develop a novel metric that is robust to both geometric transformations and noise by integrating tools from harmonic analysis, optimal transport theory, and numerical algorithms. The resulting metric admits efficient computation from finite samples and is theoretically linked to the Wasserstein distance and Besov spaces. Experimental results demonstrate its enhanced stability and effectiveness, significantly improving the reliability of image comparison in perturbed settings.

Abstract

We study the stability of a classical family of metrics defined over functions' Gaussian scale-space representations, focusing on the comparison of images (functions of two variables). These metrics have precedents both in harmonic analysis, specifically the theory of Besov spaces, and in classical methods of image processing; special cases are also known to be metrically equivalent to certain Wasserstein distances. We quantify these metrics' robustness to geometric deformations, and introduce rotationally-invariant versions that are stable to changes in angle when comparing tomographic projections. We also describe computationally efficient algorithms for evaluating the metrics from finite samples, and prove their robustness to additive noise. The results are illustrated through numerical experiments.

Problem

• scale-space metrics
• stability
• geometric deformations
• rotational invariance
• image comparison

Innovation

• scale-space metrics
• rotational invariance
• geometric stability
• Wasserstein distance
• noise robustness