Proximal map for the function , where the geodesic is the one closest to . Following [1] we can compute the gradient of . Let denote the mid point of a geodesic, such that the distance is minimal among all geodesics connecting and . Then, using the adjoint Differential of the start point of a geodesic denoted by the gradint is given by
An illustration is given by the following figure.
The proximal map is computed approximately using a subgradient descent, where both the step size as well as the stopping criterion are optional parameters of this function.
If one of the three values is NaN
it can be set to the minimizer of
using the geodesic of the remaining ones, e.g. or
.
Matlab Documentation
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% proxAbsoluteSecondOrderDifference(M,f1,f2,f3,lambda) prox d2(f1,f2,f3)
% Compute the proximal map of a second order difference (mid point model)
% by a subgradient descent. This formula was derived in
%
% R. Bergmann, M. Bacak, G. Steidl, A. Weinmann,
% A second order nonsmooth variational model for restoring
% manifoldvalued images,
% SIAM Journal on Scientific Computing, 38, (1), A567?A597, 2016.
%
% INPUT
% M : a manifold
% f1,f2,f3 : three point(sets)
% lambda : prox paramater
%
% OPTIONAL
% stepSize : (@(x,eta,iter,old_step) 1/iter) a functional determining
% the step size of the subgradient algorithm.
% stopCrit : (stopCritMaxIterEpsilonCreator(M,3,10^(9))) a stopping
% criterion functional ( @(x,xold,lambda,iter) )
% 
% Manifoldvalued Image Restoration Toolbox 1.0
% R. Bergmann ~ 20150422  20180308
% see LICENSE.txt
See also
used in
References

Bačák, M, Bergmann, R, Steidl, G and Weinmann, A (2016). A second order nonsmooth variational model for restoring manifoldvalued images. SIAM Journal on Scientific Computing. 38 A567–A597
We introduce a new nonsmooth variational model for the restoration of manifoldvalued data which includes second order differences in the regularization term. While such models were successfully applied for realvalued images, we introduce the second order difference and the corresponding variational models for manifold data, which up to now only existed for cyclic data. The approach requires a combination of techniques from numerical analysis, convex optimization and differential geometry. First, we establish a suitable definition of absolute second order differences for signals and images with values in a manifold. Employing this definition, we introduce a variational denoising model based on first and second order differences in the manifold setup. In order to minimize the corresponding functional, we develop an algorithm using an inexact cyclic proximal point algorithm. We propose an efficient strategy for the computation of the corresponding proximal mappings in symmetric spaces utilizing the machinery of Jacobi fields. For the \(n\)sphere and the manifold of symmetric positive definite matrices, we demonstrate the performance of our algorithm in practice. We prove the convergence of the proposed exact and inexact variant of the cyclic proximal point algorithm in Hadamard spaces. These results which are of interest on its own include, e.g., the manifold of symmetric positive definite matrices.@article{BBSW16, author = {Ba{\v{c}}{\'a}k, Miroslav and Bergmann, Ronny and Steidl, Gabriele and Weinmann, Andreas}, title = {A second order nonsmooth variational model for restoring manifoldvalued images}, journal = {SIAM Journal on Scientific Computing}, year = {2016}, volume = {38}, number = {1}, pages = {A567A597}, doi = {10.1137/15M101988X}, eprint = {1506.02409}, eprinttype = {arXiv} }