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

# The inner product for cyclic data

The tangent vectors on $\mathbb S^1$ lie on a line (when looking at the embedded circle), and hence the dot prodcut is the product of the tangent vectors, which are here values.

### Matlab Documentation

1
2
3
4
5
6
7
8
9
10
11
12
13
14
% S1.dot(x,xi,nu)
%     Compute the inner product of two tangent vectors in T_P M
%
% INPUT
%     x  : a point(Set) in S1
%     xi  : a first tangent vector( set) to (each) x
%     nu  : a secod tangent vector( set) to (each) x
%
% OUTPUT
%     ds : the corresponding value(s) of the inner product of (each triple) V,W at X
%
% ---
% Manifold-Valued Image Restoration Toolbox 1.0, J. Persch 2016-12-06
%