$ \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}} $

Pole Ladder – approximate parallel transport

The pole ladder is a way to approximate the parallel transport of a tangent vector to since the original function might be hard to compute. Schild’s ladder is given by the formula

An illustration of the procedure is shown in the following figure adapted from [1].

Illustration of the pole ladder
Illustration of the pole

The formula was first introduced by [2]. It can be interpreted as follows

  1. map onto the manifold as
  2. take the mid point between and ,
  3. reflect at i.e. take the geodesic evaluated at 2,
  4. map into the tangent space of using the negative logaithmic map,

In the Euclidean space this directly yields the identity. It should also be exact on symmetric spaces.

Matlab Documentation

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% poleladder(this,x,y,xi) approximates parallel Transport
% by mappting xi from TxM to TyM using the pole ladder
%
% INPUT
% x,y : two point(set)s on the manifold
% xi   : a tangent vector(Set) on TxM
%
% OUTPUT
%   nu : the resulting vectors in TyM that are approximately
%        parallelTransport(x,y,xi)
%
% ---
% Manifold-valued Image Restoration Toolbox 1.2
% J. Persch, R. Bergmann | 2018-01-04 | 2018-02-19

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. Lorenzi, M and Pennec, X (2014). Efficient parallel transport of deformations in time series of images: from Schild’s to pole ladder. Journal of Mathematical Imaging and Vision. 50 5–17