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].
The formula was first introduced by [2]. It can be interpreted as follows
 map onto the manifold as
 take the mid point between and ,
 reflect at i.e. take the geodesic evaluated at 2,
 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
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)
%
% 
% Manifoldvalued Image Restoration Toolbox 1.2
% J. Persch, R. Bergmann  20180104  20180219
References

Bergmann, R, Fitschen, J H, Persch, J and Steidl, G (2017). Priors with coupled first and second order differences for manifoldvalued image processing
We generalize discrete variational models involving the infimal convolution (IC) of first and second order differences and the total generalized variation (TGV) to manifoldvalued images. We propose both extrinsic and intrinsic approaches. The extrinsic models are based on embedding the manifold into an Euclidean space of higher dimension with manifold constraints. An alternating direction methods of multipliers can be employed for finding the minimizers. However, the components within the extrinsic IC or TGV decompositions live in the embedding space which makes their interpretation difficult. Therefore we investigate two intrinsic approaches: for Lie groups, we employ the group action within the models; for more general manifolds our IC model is based on recently developed absolute second order differences on manifolds, while our TGV approach uses an approximation of the parallel transport by the pole ladder. For computing the minimizers of the intrinsic models we apply gradient descent algorithms. Numerical examples demonstrate that our approaches work well for certain manifolds.@online{BFPS17, author = {Bergmann, Ronny and Fitschen, Jan Henrik and Persch, Johannes and Steidl, Gabriele}, title = {Priors with coupled first and second order differences for manifoldvalued image processing}, year = {2017}, eprint = {1709.01343}, eprinttype = {arXiv} }

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
@article{LP14, author = {Lorenzi, M. and Pennec, X.}, title = {Efficient parallel transport of deformations in time series of images: from {S}child's to pole ladder}, journal = {Journal of Mathematical Imaging and Vision}, year = {2014}, volume = {50}, number = {1}, pages = {517}, doi = {10.1007/s1085101304703} }