$ \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 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 Schild's ladder
Illustration of Schild's ladder

The formula was first introduced by [2], see also [3]. It can be interpreted as follows

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

In the Euclidean space this directly yields the identity.

Matlab Documentation

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% schildsladder(this,x,y,xi) approximates parallel Transport
% 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