Proximal map for the function , where relaxes the nondifferential point at . The function is given as in [1] by
For small we obtain a parabola and for large s a linear increase with ascent . From both the proximal map of an absolute difference and its squared. The closed form of the proximal map also results from these two as
If a data item or contains a NaN
it is set to the minimizer of
the distance, i.e. initialized to the other argument of .
Matlab Documentation
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% proxAbsoluteDifferenceHuber(M,f1,f2,lambda,tau,omega) prox d(x,y) relaxed
% Given the Huber function
% h(t) = tau^2t^2 for t< omega/(sqrt(2)tau), omega*sqrt(2)*tau*t  omega^2/2
% i.e. for small t a suqared function (steered by tau) and for large t a
% linear part with ascent omega.
% lambda is the prox parameter, as for the absolute difference, this
% function inpaints NaN values and works on arbitrary manifolds
%
% INPUT
% M : A manifold
% f1,f2 : data columns
% lambda : parameter within the proximal map
% tau, omega : parameters of the Huber function
%
% OUTPUT
% x1,x2 : resulting columns of the proximal map
% 
% Manifoldvalued Image Restoration Toolbox 1.0 ~ R. Bergmann, 20141019
See also
 The Cyclic Proximal Point Algorithm
 The proximal map of an absolute difference
 The proximal map of an absolute difference
used in
References

Weinmann, A, Demaret, L and Storath, M (2014). Total variation regularization for manifoldvalued data. SIAM Journal on Imaging Sciences. 7 2226â€“57
We consider total variation (TV) minimization for manifoldvalued data. We propose a cyclic proximal point algorithm and a parallel proximal point algorithm to minimize TV functionals with \(\ell^p\)type data terms in the manifold case. These algorithms are based on iterative geodesic averaging which makes them easily applicable to a large class of data manifolds. As an application, we consider denoising images which take their values in a manifold. We apply our algorithms to diffusion tensor images and interferometric SAR images as well as sphere and cylindervalued images. For the class of Cartanâ€“Hadamard manifolds (which includes the data space in diffusion tensor imaging) we show the convergence of the proposed TV minimizing algorithms to a global minimizer.@article{WDS14, author = {Weinmann, Andreas and Demaret, Laurent and Storath, Martin}, title = {Total variation regularization for manifoldvalued data}, journal = {SIAM Journal on Imaging Sciences}, volume = {7}, number = {4}, pages = {22262257}, year = {2014}, doi = {10.1137/130951075}, eprint = {1312.7710}, eprinttype = {arXiv} }