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

# The CPP algorithm for Huber relaxed TV

This algorithm computes the cyclic or randomized proximal point algorithm on a manifold $\M$ for the first order differences relaxed by a Huber functional. It consits of proxDistanceSquared, proxTV, with a modified/relaxed inner difference, the Huber function relaxed absolute difference. The functional to minimize reads for given (noisy or lossy) data $u\in\M^{\vect{n}}$$as where the data term is reduced to known data from $u\in\M^{\vect{n}}$ for the lossy case, and the first order differences are relaxed to a quadratic term for small distances. The weighghts $\alpha>0$ can be also given as vectors for the dimensions with different weights, i.e.$\alpha_1\$ as a weight for the differences in $x_1$-direction Real values are interpreted as constant vectors.

When given as matrices, the upper triangular part of $\alpha$ weights the diagonal differences in both positive directions (to $\vect{k}+\vect{e}_i+\vect{e}_j$ direction, while the lower triangular denotes weights for $\vect{k}+\vect{e}_i-\vect{e}_j$.

The parameters lambda, lambdaIterate, and stoppingCriterion are explained in the optional parameters section of the CyclicProximalPoint this function internally uses.

The FixedMask sets pixel in $u$ to the value initially provided by $x$ and leaves them unchanged. the UnknownMask specifies unknown pixel, which are intialized and regularized during the algorithm.

### Matlab Documentation

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%   CPP_HuberTV(M,f,alpha,stoppincCriterion)
% Compute the CPP of the first order TV with Huber relaxation
% can also be called with a struct with the corresponding fields, i.e. as
%   CPP_HuberTV(problem).
%
% INPUT
%   M                 : a manifold.
%   f                 : an image from M^{m_1,m_2,...,m_n} of
%                           manifold-valued data
%   alpha             : weight(s) for first order absolute difference terms
%                       either a number or one for each dimension.
%   stoppingCriterion : a functional @(x,xold,lambda,iter) as a stopping
%                       criterion based on x, the last iterate xold, lambda
%
%   OUTPUT:
%      x   : result of the cyclic proximal point algorithm
%  recData : recorded data array of the by the record-function
%
%   OPTIONAL PARAMETERS
%
%  'UnknownMask'   : ([]) Specify a mask the same size as the data, i.e.,
%                        m_1xm_2x...xm_nw, which are labeled as unknown
%                        they are initialized in the cycles, when possible.
%                       values that are fixed. They are fixed on a 'poor
%                       mans method', i.e. reset after each step
%  'lambdaIterate' : (@ (iter)) a functional @(iter) computing the
%                        iterates value of lambda
%  'lambda'        : if no iterate (see above) is given, this value is used
%                        to set the iterate to @(iter) lambda/iter
%   Debug          : ([]) a function @ (x,xold,iter) producing debug
%                       output
%   Record         : (@(x,iter)) a function returning a column vector, if
%                       there's a second return value and this function is
%                       given, data is recorded in an array and returned
% ---
% MVIRT | R. Bergmann | 2018-01-22