$ \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 second order mixed difference

Proximal map for the function

Following [1] we can compute the gradient of . Let and denote the mid points of geodesics, such that their distance is minimal among all geodesics connecting and , respectively.

Then, using the adjoint Differential of the start point of a geodesic denoted by the gradint is given by

The proximal map is computed approximately using a subgradient descent, where both the step size as well as the stopping criterion are optional parameters of this function.

Matlab Documentation

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% proxAbsoluteSecondOrderMixedDifference(M,f1,f2,f3,f4,lambda)
% Compute the second order mixed difference based on the mid point model
% employing a sub gradient descent
%
% INPUT
%    M          : a manifold
%   f1,f2,f3,f4 : Data items from a 2x2 matrix each
%   lambda      : parameter of the proximal map
%
% OUTPUT
%  x1,x2,x3,x4 : result of the proximal map
% ---
% MVIRT | R. Bergmann, 2018-01-22

See also

used in

References

  1. Bačák, M, Bergmann, R, Steidl, G and Weinmann, A (2016). A second order non-smooth variational model for restoring manifold-valued images. SIAM Journal on Scientific Computing. 38 A567–A597