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

A point on the geodesic

The function evaluates a geodesic $\gamma_{x,y}\colon\mathbb R \to\M$ at $t$, i.e. returns $\gamma_{x,y}(t)$ by using the exponential and logarithmic map. When the geodesic is not unique this function still returns a deterministic single value based on the choice of the logarithmic map. This also allows for evaluating the geodesic of an image pixelwise, i.e. if $x,y$ are images, we obtain a pixelwise evaluation.

Matlab Documentation

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% geopoint(x,y,t) - Gives the point \gamma_{x,y}(t)
% placeholder, has many manifolds admit a faster way to compute
% the combination of exp and log, e.g., SymPosDef
%
% INPUT
%   x,y : a point or set of points on the manifold
%   t   : a scalar or set of scalars
%
%
% OUTPUT
%   w : resulting point(s)
% ---
% Manifold-Valued Image Restoration Toolbox 1.0
% J. Persch, R. Bergmann | 2017-03-31 | 2018-02-16