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

# The proximal map of an absolute difference

Proximal map $\prox_{\lambda f}$ for the function $f\colon\M^2\to\R,\quad f(x,y) = d_{\M}(x,y)$. Following , let $\geo{x}{y}$ denote the geodesic starting in $\geo{x}{y}(0)=x$ and reaching $\geo{x}{y}(1)=y$ at time $1$. Then

If a data item $x$ or $y$ contains a NaN it is set to the minimizer of the distance, i.e. initialized to the other argument of $f$.

### Matlab Documentation

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% proxAbsoluteDifference(M,f1,f2,lambda) prox of d_M(f1,f2) in both args
% with parameter lambda on an arbitrary manifold. Values containing NaN
% are initialized to the other argument, which is the minimizer.
%
%
% INPUT
%   M         : A manifold
%  f1,f2   : data columns
%  lambda     : parameter within the proximal map
%
% OUTPUT
%  x1,x2   : resulting columns of the proximal map
% ---
% Manifold-valued Image Restoration Toolbox 1.0 ~ R. Bergmann, 2014-10-19