$ \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 inner product on symmetric positive definite matrices

The distance on the symmetric positive definite matrices is given by

where denotes the matrix logarithm and is the Frobenius norm of the matrix.

Matlab Documentation

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% d = dist(x,y) compute the distance between x,y from P(n).
%
% INPUT
%   y,x : two points (matrices) or sets of points (matrices)
%         on P(m)
%
% OUTPUT
%     d : resulting distances of each pair of points of p,q.
% ---
% Manifold-Valued Image Restoration Toolbox 1.0
% R. Bergmann ~ 2014-10-19 | 2015-04-11

See also