Krzysztof A. Krakowski

An algorithm based on rolling to generate smooth interpolating curves on ellipsoids

We present an algorithm to generate a smooth curve interpolating a set of data on an

A modified Casteljau algorithm to solve interpolation problems on Stiefel manifolds

n

Geometry of the rolling ellipsoid

-dimensional ellipsoid, which is given in closed form. This is inspired by an algorithm based on a rolling and wrapping technique, described in [11] for data on a general manifold embedded in Euclidean space. Since the ellipsoid can be embedded in an Euclidean space, this algorithm can be implemented, at least theoretically. However, one of the basic steps of that algorithm consists in rolling the ellipsoid, over its affine tangent space at a point, along a curve. This would allow to project data from the ellipsoid to a space where interpolation problems can be easily solved. However, even if one chooses to roll along a geodesic, the fact that explicit forms for Euclidean geodesics on the ellipsoid are not known, would be a major obstacle to implement the rolling part of the algorithm. To overcome this problem and achieve our goal, we embed the ellipsoid and its affine tangent space in

Rolling Symmetric Spaces

\mathbb{R}^{n+1}

Wheels for Staircases

equipped with an appropriate Riemannian metric, so that geodesics are given in explicit form and, consequently, the kinematics of the rolling motion are easy to solve. By doing so, we can rewrite the algorithm to generate a smooth interpolating curve on the ellipsoid which is given in closed form.

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

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.

Covariant differentiation under rolling maps

We study rolling maps of the Euclidean ellipsoid, rolling upon its affine tangent space at a point. Driven by the geometry of rolling maps, we find a simple formula for the angular velocity of the rolling ellipsoid along any piecewise smooth curve in terms of the Gauss map. This result is then generalised to rolling any smooth hyper-surface. On the way, we derive a formula for the Gaussian curvature of an ellipsoid which has an elementary proof and has been previously known only for dimension two.

WHY CONTROLLABILITY OF ROLLING MOTIONS MAY FAIL

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.

Exploring quasi-geodesics on Stiefel manifolds in order to smooth interpolate between domains

Our objective is to design innovative robot wheels capable of rolling on staircases, without sliding and without bouncing. This is the first step to reach the ultimate goal of building wheelchairs capable to overcome the obstacles imposed by staircases on people with limited mobility. We show that, given a staircase with equal steps, there is an infinite number of wheels that roll over it, with the constraints of no-sliding and no-bouncing. Some of these wheels appear to be very interesting for real applications. We also present an algorithm for the construction of a wheel, which depends only on the measures of the tread (the part of the staircase that is stepped on) and the riser (the vertical portion between each tread) of the step.

THE SIGNIFICANCE OF LIE THEORY IN THE GEOMETRY OF ROLLING MOTIONS

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.