$ \DeclareMathOperator{\arccosh}{arccosh} \DeclareMathOperator*{\argmin}{arg\,min} \DeclareMathOperator{\Exp}{Exp} \newcommand{\geo}[2]{\gamma_{\overset{\frown}{#1,#2}}} \newcommand{\geoS}{\gamma} \newcommand{\geoD}[2]{\gamma_} \newcommand{\geoL}[2]{\gamma(#2; #1)} \newcommand{\gradM}{\nabla_{\M}} \newcommand{\gradMComp}[1]{\nabla_{\M,#1}} \newcommand{\Grid}{\mathcal G} \DeclareMathOperator{\Log}{Log} \newcommand{\M}{\mathcal M} \newcommand{\N}{\mathcal N} \newcommand{\mat}[1]{\mathbf{#1}} \DeclareMathOperator{\prox}{prox} \newcommand{\PT}[3]{\mathrm{PT}_{#1\to#2}#3} \newcommand{\R}{\mathbb R} \newcommand{\SPD}[1]{\mathcal{P}(#1)} \DeclareMathOperator{\Tr}{Tr} \newcommand{\tT}{\mathrm{T}} \newcommand{\vect}[1]{\mathbf{#1}} $

The parallel transport the sphere

This function parallel transports a vector along a geodesic (where uniqueness is determined) by the logarithmic map implementation) by

This formula is taken from [1,2] and can be interpreted as follows: all components of that share no part with the direction 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

See also

References

  1. Bergmann, R, Fitschen, J H, Persch, J and Steidl, G (2017). Priors with coupled first and second order differences for manifold-valued image processing
  2. Hosseini, S and Uschmajew, A (2017). A Riemannian gradient sampling algorithm for nonsmooth optimization on manifolds. SIAM Journal on Optimization. 27 173–89