Left-Invariant Riemannian Geodesics on Spatial Transformation Groups

Abstract

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.

Horospherical coordinates for optimal Cosserat microrotations

“…First, it exhibits the important role of the reduction to the limiting case ( μ , μ c )=(1,0)—it is more than a routine standardisation; rather, it allows us, by removing the skew‐symmetric part (since μ c =0), to use the much simpler, negative‐curvature geometry and rich structure of the space of positive definite matrices with the Riemannian metric d . In connection with geodesics and exponentials, we should recall that the ordinary matrix exponential from GL( n ) to GL + ( n ) is not surjective and in fact not identical to the Riemannian exponential, , p1535 while for the positive definite matrices, surjectivity holds, as we already noted, and the two exponentials are the same. Second, the preceding computation, which is based on the fact that α I commutes with every other matrix for any α >0, suggests that the “degree of commutativity” of the factor U from the paragraph before , which is related to the sub‐matrices (or subspaces) on which commutativity does hold, should play a major role in the coordinate system that we wish to set up for problem .…”

Transformation diffusion reconstruction of three-dimensional histology volumes from two-dimensional image stacks

“… Alexa (2002 ) proposed that there are no negative eigenvalues if and only if A contains no reflections. However, Zacur et al (2014 ) have shown that a rotation plus an anisometric scaling in fact produces a transformation with a negative eigenvalue, and thus the matrix logarithm is not real. Nonetheless, Alexa's interpolation is still valid for transformations that are not large, such as the ones in typical histological reconstruction applications such as ours.…”

Approximate Joint Diagonalization with Riemannian Optimization on the General Linear Group

“…To conclude this section, a summary of all the necessary tools for Riemannian optimization on the oblique manifold equipped with the left-and right-invariant metrics is given in table 1. The extension to second-order Riemannian optimization methods is straightforward: to define the Riemannian Hessian we need the Levi-Civita connection, which, in the case of the oblique manifold, is obtained by projecting the Levi-Civita connection of GL n (given for instance in [10,35,37]) onto the tangent space; see [2, proposition 5.3.2].…”

“…GL n as a Riemannian manifold. All the results given in this section can be found in [10,27,28,35,37]. The general linear group GL n is an open set in R n×n , thus its tangent space T B GL n at any point B can be identified as R n×n , which is denoted T B GL n R n×n .…”