L. Machado

Riemannian Means as Solutions of Variational Problems

We formulate a variational problem on a Riemannian manifold M whose solutions are piecewise smooth geodesics that best fit a given data set of time labelled points in M . By a limiting process, these solutions converge to a single point in M , that we prove to be the Riemannian mean of the given points for some particular Riemannian manifolds such as Euclidean spaces, connected and compact Lie groups and spheres.

Geodesic regression on spheres from a numerical optimization viewpoint

In the geodesic regression problem it is given a set of data points at known times and the goal is to find a geodesic that best fits the data. This problem corresponds to the generalization of the classical linear regression problem to curved spaces. Here we are interested in the geodesic regression problem on Euclidean spheres. Contrary to the Euclidean situation, the normal equations turn out to be highly nonlinear. To overcome this difficulty, we look at the geodesic regression problem in the unit n-sphere as an optimization problem in the Euclidean space ℝn+1 and use the MATLAB optimization toolbox to solve it numerically.

Interpolation and polynomial fitting in the SPD manifold

Generalizing to Riemannian manifolds classical methods to approximate data (e.g. averaging, interpolation and regularization) has been a theoretical challenge that has also revealed to be computationally very demanding and often unsatisfactory. One particular manifold that shows up in numerous scientific areas that use tensor analysis, including machine learning, medical imaging, and optimization, is the set of symmetric positive definite (SPD) matrices. In this work, we show that when the SPD matrices are endowed with the Log-Euclidean framework, certain optimization problems, such as interpolation and best fitting polynomial problems, can be solved explicitly. This contrasts with what happens in general non-Euclidean spaces. In the Log-Euclidean framework, the SPD manifold has the structure of a commutative Lie group and when equipped with the Log-Euclidean metric it becomes a flat Riemannian manifold. Explicit expressions for polynomial curves in the SPD manifold are therefore obtained easily, and this enables the complete resolution of the proposed problems.

A modified Casteljau algorithm to solve interpolation problems on Stiefel manifolds

The main objective of this paper is to propose a new method to generate smooth interpolating curves on Stiefel manifolds. This method is obtained from a modification of the geometric Casteljau algorithm on manifolds and is based on successive quasi-geodesic interpolation. The quasi-geodesics introduced here for Stiefel manifolds have constant speed, constant covariant acceleration and constant geodesic curvature, and in some particular circumstances they are true geodesics.

Rolling Maps for the Essential Manifold

Computer vision problems typically have geometric constraints. When two cameras view a 3D scene from two distinct positions, or a single camera views a 3D scene from two different locations, there are a number of geometric relations between the 3D points and their projections onto the 2D images. These relations lead to constraints between the image points. In particular, the epipolar constraint encodes the relation between correspondences across two images of the same scene. In a calibrated setting, the epipolar constraint is parameterized by essential matrices, which form the Essential Manifold. The reconstruction of a video from several images of a scene can be formulated as an interpolation problem on this manifold. An approach that simplifies the generation of an interpolating curve consists in projecting the problem to a linear manifold where it can be solved easily, and then projecting back the solution on the nonlinear manifold. The projection is realized by rolling the Essential Manifold, without slip and twist, over an affine tangent space. This gives particular relevance to rolling motions in the context of certain computer vision problems. Having this in mind, we derive the kinematic equations for the rolling motions of the Essential Manifold and present explicit solutions when it rolls along geodesics.

Rolling Symmetric Spaces

Riemannian symmetric spaces play an important role in many areas that are interrelated to information geometry. For instance, in image processing one of the most elementary tasks is image interpolation. Since a set of images may be represented by a point in the Grasmann manifold, image interpolation can be formulated as an interpolation problem on that symmetric space. It turns out that rolling motions, subject to nonholonomic constraints of no-slip and no-twist, provide efficient algorithms to generate interpolating curves on certain Riemannian manifolds, in particular on symmetric spaces. The main goal of this paper is to study rolling motions on symmetric spaces. It is shown that the natural decomposition of the Lie algebra associated to a symmetric space provides the structure of the kinematic equations that describe the rolling motion of that space upon its affine tangent space at a point. This generalizes what can be observed in all the particular cases that are known to the authors. Some of these cases illustrate the general results.

Approximating Sets of Symmetric and Positive-Definite Matrices by Geodesics

We formulate a generalized version of the classical linear regression problem on Riemannian manifolds and derive the counterpart to the normal equations for the manifold of symmetric and positive definite matrices, equipped with the only metric that is invariant under the natural action of the general linear group.

Multi-source domain adaptation using C⁁1-smooth subspaces interpolation

Manifold-based domain adaptation algorithms are receiving increasing attention in computer vision to model distribution shifts between source and target domain. In contrast to early works, that mainly explore intermediate subspaces along geodesics, in this work we propose to interpolate subspaces through C1-smooth curves on the Grassmann manifold. The new methodis based on the geometric Casteljau algorithm that is used to generate smooth interpolating polynomial curves on non-euclidean spaces and can be extended to generate polynomial splines that interpolate a given set of data on the Grassmann manifold. To evaluate the usefulness of the proposed interpolating curves on vision related problems, several experiments were conducted. We show the advantage of using smooth subspaces interpolation in multi-source unsupervised domain adaptation problems and in object recognition problems across datasets.