$ \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}} $

The proximal map of an absolute difference squared

Proximal map for the function . Following [1], let denote the geodesic starting in and reaching at time . Then

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

Matlab Documentation

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% proxAbsoluteDifference(M,f1,f2,lambda) prox of d^2_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   : proxParameter
%
% OUTPUT
%  x1,x2    : resulting columns of the proximal map
% ---
% Manifold-valued Image Restoration Toolbox 1.0 ~ R. Bergmann, 2014-10-19

See also

References

  1. Weinmann, A, Demaret, L and Storath, M (2014). Total variation regularization for manifold-valued data. SIAM Journal on Imaging Sciences. 7 2226–57