Left-Invariant Riemannian Geodesics on Spatial Transformation Groups

Spatial transformations are mappings between locations of a d-dimensional space and are commonly used in computer vision and image analysis. Many of the spatial transformation sets have a group structure and can be represented by matrix groups. In the computer vision and image analysis fields there is a recent and growing interest in performing analyses on spatial transformations data. Differential and Riemannian geometry have been used as a framework to endow the set of spatial transformations with a metric space structure, allowing the extension of the standard analysis techniques defined on vector spaces. This paper presents a review of the concepts and an overview of approaches to computing Riemannian geodesics on spatial transformation groups. The paper is aimed at providing a bridge for researchers from computer vision and image analysis fields to fill in the gap between differential geometry and computer vision and imaging disciplines. Some application examples are shown to illustrate the use of invariant Riemannian geodesics, such as interpolation of spatial transformations and filtering of matrix-valued images.