
The parallel transport the sphere

This function parallel transports a vector $\xi\in T_x\mathbb S^n$ along a geodesic $\geo{x}{y}$ (where uniqueness is determined) by the logarithmic map implementation) by

$P_{x\to y}(\xi) = \xi - \frac{\langle \log_xy,\xi\rangle_x}{d_{\mathbb S^n}(x,y)} \bigl( \log_xy+\log_yx \bigr)$

This formula is taken from [1,2] and can be interpreted as follows: all components of $\xi$ that share no part with the direction $\log_xy$ are left unchanged. For the remaining part (second term) we have to perform a correction.

Matlab Documentation

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% eta = parallelTransport(x,y,xi) transport xi from TxM parallel to TyM
%
% INPUT
%   x,y : two (sets of) points on the manifold
%   xi  : a (set of) vectors from TxM
%
% OUTPUT
%   eta : the parallel transported vectors in TyM
% ---
% Manifold-valued Image Restoration Toolbox 1.2
% R. Bergmann | 2018-03-01