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

# Schild's Ladder – approximate parallel transport

Schild’s 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 [1].

The formula was first introduced by [2], see also [3]. 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 of $y$ and $e$, i.e. $c=\gamma(y,e;\frac{1}{2})$
3. reflect $x$ at $c$ i.e. take the geodesic evaluated at 2, $p=\gamma(x,c;2)$
4. map $p$ into the tangent space of $y$ using the logaithmic map, $\zeta=\log_yp$

In the Euclidean space this directly yields the identity.

### Matlab Documentation

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% by mappting xi from TxM to TyM using Schild's ladder
%
% INPUT
% x,y : two point(set)s on the manifold
% v   : 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. Ehlers, J, Pirani, F A E and Schild, A (1972). The geometry of free fall and light propagation. General Relativitiy. Oxford University Press. 63–84 republished 2012
3. Kheyfets, A, Miller, W A and Newton, G A (2000). Schild’s Ladder Parallel Transport Procedure for an Arbitrary Connection. International Journal of Theoretical Physics. 39 2891–8