
# The proximal map of an absolute difference squared

Proximal map $\prox_{\lambda f}$ for the function $f\colon\M^2\to\R,\quad f(x,y) = d^2_{\M}(x,y)$. Following [1], let $\geo{x}{y}$ denote the geodesic starting in $\geo{x}{y}(0)=x$ and reaching $\geo{x}{y}(1)=y$ at time $1$. Then

If a data item $x$ or $y$ contains a NaN it is set to the minimizer of the distance, i.e. initialized to the other argument of $f$.

### Matlab Documentation

1
2
3
4
5
6
7
8
9
10
11
12
13
% proxAbsoluteDifference(M,f1,f2,lambda) prox of d^2_M(f1,f2) in both args
% with parameter lambda on an arbitrary manifold. Values containing NaN
% are initialized to the other argument, which is the minimizer.
%
% INPUT
%   M       : A manifold
%  f1,f2    : data columns
%  lambda   : proxParameter
%
% OUTPUT
%  x1,x2    : resulting columns of the proximal map
% ---
% Manifold-valued Image Restoration Toolbox 1.0 ~ R. Bergmann, 2014-10-19