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

# List of functionals

The second order mid point model functional
1
%  gradientSecondOrderLog(M,f,alpha,beta) the l^2-TV-TV2 mid point model


Computes the second order mid point model

for manifold-valued data $x\in\M^{\vect{n}}$, where $u$ is some given data, that might only be given on a subset of the domain. Then the data term is reduced accordingly.

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The total variation prior
1
% TV(M,x) compute all TV with a p-norm coupling of forward distance terms.


Computes the total variation prior with $p$-norm coupling of a $\M$-valued data set $x\colon\Grid\to\M$, where the pixel grid $\Grid$ is of size ${\vect{n}}$, $\vect{n} = (n_1,\ldots,n_m)\in\mathbb N^m$.

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The second order total variation prior from the mid point prior
1
% proxTV(M,x,lambda) compute the second order TV mid point model value


Computes the total variation of second order from the mid point model. An absolute difference of second order for three points $x,y,z\in\M$ is modeled as the distance of $y$ to the nearest mid point $\geo{x}{z}(\tfrac{1}{2})$ of the geodesics connecting $x$ and $z$.

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