$ \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 second order mid point model functional

Computes the second order mid point model

for manifold-valued data , where is some given data, that might only be given on a subset of the domain. Then the data term is reduced accordingly.For the prior both the total variation and the second order mid point model are employed. For more details, see [1].

Matlab Documentation

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%  gradientSecondOrderLog(M,f,alpha,beta) the l^2-TV-TV2 mid point model
%
% INPUT
%    M : a manifold
%    f : given (original) data
%    x : value to take the gradient at
%    alpha : weight of TV
%    beta : weight of TV2
%
% OPTIONAL
%   'p'      : (p=1) compute TV with p-norm coupling in the dimensions of
%              the data, i.e. anisotropic TV for p=1 and isotropic for p=2
%  'epsilon' : compute the gradient of the epsilon-relaxed TV and TV2
% ---
% MVIRT - R. Bergmann, 2017-12-07

See also

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