$ \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}} $

gradient of the TV prior

For data on a -dimensional grid with this function comutes the gradient of the TV prior

with respect to all entries of , i.e. .

In order to compute the gradient , one has to take into account all summands, where appears, which are besides the summands over the neighborhood all summands, where is a forward neighbor, i.e. the following [1] we denote an inner sum by for . Then the gradient reads by [2] and the chain rule

for and

for , where we set a summand to the zero vector whenever the denominator is zero and obtain a subgradient in this case.

Optional Parameters

epsilon
relax the TV (especially for by replacing any distance by and for each inner sum (over ) is relaxed to avoid subgradients in constant areas.
p
denotes the outer coupling of the differences
weights for each item of
introduces a weight that is muliplied to any term, i.e. if weights is zero this the corresponding data item is ignored.

Matlab Documentation

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% gradTV(M,x) compute the gradient of the manifold total variation
%
% INPUT
%   M      :  a manifold
%   x      : data (size [manDims,dataDims])
%
% 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
%  'weights'  : (ones(dataDims)) exclude certain data points from all
%               gradient terms
%
% OUTPUT
%   eta : the gradient
% ---
% MVIRT, R. Bergmann, 2017-12-08

See also

References

  1. Bergmann, R, Fitschen, J H, Persch, J and Steidl, G (2017). Priors with coupled first and second order differences for manifold-valued image processing
  2. Afsari, B (2011). Riemannian \(L^p\)center of mass: Existence, uniqueness, and convexity. Proceedings of the American Mathematical Society. 139 655–73