Bi-invariant Means in Lie Groups. Application to Left-invariant Polyaffine Transformations
Résumé
In this work, we present a general framework to define rigorously a novel type of mean in Lie groups, called the bi-invariant mean. This mean enjoys many desirable invariance properties, which generalize to the non-linear case the properties of the arithmetic mean: it is invariant with respect to left- and right-multiplication, as well as inversion. Previously, this type of mean was only defined in Lie groups endowed with a bi-invariant Riemannian metric, like compact Lie groups such as the group of rotations. But Riemannian bi-invariant metrics do not always exist. In particular, we prove in this work that such metrics do not exist in any dimension for rigid transformations, which form but the most simple Lie group involved in bio-medical image registration. To overcome the lack of existence of bi-invariant Riemannian metrics for many Lie groups, we propose in this article to define bi-invariant means in any finite-dimensional real Lie group via a general barycentric equation, whose solution is by definition the bi-invariant mean. We show the existence and uniqueness of this novel type of mean, provided the dispersion of the data is small enough, and the convergence of an efficient iterative algorithm for computing this mean has also been shown. The intuition of the existence of such a mean was first given by R.P.Woods (without any precise definition), along with an efficient algorithm for computing it (without proof of convergence), in the case of matrix groups. In the case of rigid transformations, we give a simple criterion for the general existence and uniqueness of the bi-invariant mean, which happens to be the same as for rotations. We also give closed forms for the bi-invariant mean in a number of simple but instructive cases, including 2D rigid transformations. Interestingly, for general linear transformations, we show that similarly to the Log-Euclidean mean, which we proposed in recent work, the bi-invariant mean is a generalization of the (scalar) geometric mean, since the determinant of the bi-invariant mean is exactly equal to the geometric mean of the determinants of the data. Last but not least, we use this new type of mean to define a novel class of polyaffine transformations, called left-invariant polyaffine, which allows to fuse local rigid or affine components arbitrarily far away from the identity, contrary to Log-Euclidean polyaffine fusion, which we have recently introduced.
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