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

# Pole Ladder – approximate parallel transport

The pole ladder is a way to approximate the parallel transport $P_{x\to y}\xi$ of a tangent vector $\xi\in T_x\M$ to $T_y\M$ 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 .

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

1. map $\xi\in T_x\M$ onto the manifold as $e=exp_x\xi$
2. take the mid point between $x$ and $y$, $c = \gamma(x,y;\tfrac{1}{2})$
3. reflect $e$ at $c$ i.e. take the geodesic evaluated at 2, $p=\gamma(e,c;2)$
4. map $p$ into the tangent space of $y$ using the negative logaithmic map, $\zeta=-\log_yp$

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

### Matlab Documentation

1
2
3
4
5
6
7
8
9
10
11
12
13
14
% 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