par_trans {GeodRegr}

Parallel transport

Description

Performs \Gamma_{p_1 \rightarrow p_2}(v), parallel transport along the unique minimizing geodesic connecting p_1 and p_2, if it exists, on the given manifold.

Usage

par_trans(manifold, p1, p2, v)

Arguments

manifold

Type of manifold ('euclidean', 'sphere', 'hyperbolic', or 'kendall').

p1

A vector (or column matrix) representing a point on the manifold.

p2

A vector (or column matrix) representing a point on the manifold.

v

A vector (or column matrix) tangent to p1.

Details

On the sphere, there is no unique minimizing geodesic connecting p_1 and -p_1.

Value

A vector tangent to p2.

Author(s)

Ha-Young Shin

References

Fletcher, P. T. (2013). Geodesic regression and the theory of least squares on Riemannian manifolds. International Journal of Computer Vision, 105, 171-185.

Cornea, E., Zhu, H., Kim, P. and Ibrahim, J. G. (2017). Regression models on Riemannian symmetric spaces. Journal of the Royal Statistical Society: Series B, 79, 463-482.

Calinon, S. (2020). Gaussians on Riemannian manifolds: Applications for robot learning and adaptive control. IEEE Robotics & Automation Magazine, 27, 33-45.

Shin, H.-Y. and Oh H.-S. (2020). Robust Geodesic Regression. <arXiv:2007.04518>

Examples

p1 <- matrix(rnorm(10), ncol = 2)
p1 <- p1[, 1] + (1i) * p1[, 2]
p1 <- (p1 - mean(p1)) / norm(p1 - mean(p1), type = '2') # project onto pre-shape space
p2 <- matrix(rnorm(10), ncol = 2)
p2 <- p2[, 1] + (1i) * p2[, 2]
p2 <- (p2 - mean(p2)) / norm(p2 - mean(p2), type = '2') # project onto pre-shape space
p3 <- matrix(rnorm(10), ncol = 2)
p3 <- p3[, 1] + (1i) * p3[, 2]
p3 <- (p3 - mean(p3)) / norm(p3 - mean(p3), type = '2') # project onto pre-shape space
v <- log_map('kendall', p1, p3)
par_trans('kendall', p1, p2, v)