Roto-Translation Invariant Metrics on Position-Orientation Space

Riemannian metrics on the position-orientation space M(3) that are roto-translation group SE(3) invariant play a key role in image analysis tasks like enhancement, denoising, and segmentation. These metrics enable roto-translation equivariant algorithms, with the associated Riemannian distance often used in implementation. However, computing the Riemannian distance is costly, which makes it unsuitable in situations where constant recomputation is needed. We propose the mav (minimal angular velocity) distance, defined as the Riemannian length of a geometrically meaningful curve, as a practical alternative. We see an application of the mav distance in geometric deep learning. Namely, neural networks architectures such as PONITA, relies on geometric invariants to create their roto-translation equivariant model. The mav distance offers a trainable invariant, with the parameters that determine the Riemannian metric acting as learnable weights. In this paper we: 1) classify and parametrize all SE(3) invariant metrics on M(3), 2) describes how to efficiently calculate the mav distance, and 3) investigate if including the mav distance within PONITA can positively impact its accuracy in predicting molecular properties.