Euclidean Equivariant Processing and Machine Learning on Position-Orientation Space

This thesis explores the interplay between Euclidean equivariant processing, position-orientation space, and machine learning. The research is primarily theoretical, with potential use across various domains, including tracking, denoising, enhancement, and optimal transport. However, the primary practical focus and experimentation in this work lie within the domain of machine learning. More precisely, we- Introduce and analyze approximations of the Riemannian distance of a Euclidean equivariant metric on two-dimensional position-orientation. We experimentally investigate if these approximations can improve the accuracy of PDE-G-CNNs when segmenting coronary arteries and performing contour completion.- Rigorously describe an independent and universal collection of Euclidean invariants between pairs of three-dimensional position-orientations. We experimentally investigate if this collection of invariants can improve the accuracy of PONITA when predicting molecular properties.- Classify and parametrize all Euclidean invariant Riemannian metrics on three-dimensional position-orientation space. We describe the _mav distance_ as a computationally efficient alternative to the Riemannian distance. We experimentally investigate if the mav distance can improve the accuracy of PONITA when predicting molecular properties.- Experimentally investigate two geometric adaptations of PDE-G-CNNs on two-dimensional position-orientation space. First, we analyze if a fixed lifting layer using cake wavelets can perform just as well as a trained lifting later. Second, we implement a more general class of Riemannian metrics within the framework. - Generalize scale-space theory on two-dimensional Euclidean space to semifields. Our theory reveals new scale-spaces to use within PDE-CNNS and we experimentally investigate what impact these have on its accuracy when segmenting vessels in retinal images.