$\DeclareMathOperator{\arccosh}{arccosh} \DeclareMathOperator*{\argmin}{arg\,min} \DeclareMathOperator{\Exp}{Exp} \newcommand{\geo}[2]{\gamma_{\overset{\frown}{#1,#2}}} \newcommand{\geoS}{\gamma} \newcommand{\geoD}[2]{\gamma_} \newcommand{\geoL}[2]{\gamma(#2; #1)} \newcommand{\gradM}{\nabla_{\M}} \newcommand{\gradMComp}[1]{\nabla_{\M,#1}} \newcommand{\Grid}{\mathcal G} \DeclareMathOperator{\Log}{Log} \newcommand{\M}{\mathcal M} \newcommand{\N}{\mathcal N} \newcommand{\mat}[1]{\mathbf{#1}} \DeclareMathOperator{\prox}{prox} \newcommand{\PT}[3]{\mathrm{PT}_{#1\to#2}#3} \newcommand{\R}{\mathbb R} \newcommand{\SPD}[1]{\mathcal{P}(#1)} \DeclareMathOperator{\Tr}{Tr} \newcommand{\tT}{\mathrm{T}} \newcommand{\vect}[1]{\mathbf{#1}}$

# The exponential map on symmetric positive definite matrices

The exponential map on the sphere $\SPD{n}$ is given for $x\in\SPD{n}$, $\xi\in T_x\SPD{n}$ by the formula

where $\Exp$ denotes the matrix exponential.

### Matlab Documentation

 1 2 3 4 5 6 7 8 9 10 11 12 13 14 % y = exp(x,xi) exponential map at x from P(n) towards xi in TxP(n) % % INPUT % x : a point or set of points on the manifold P(n) % xi : a point or set of point in the tangential spaces TxM % % OPTIONAL % t : shorten vectors V by factor t % (given as one value or an array of same length as number of Vs) % % OUTPUT % Y : resulting point(s) on P(m) % --- % Manifold-Valued Image Restoration Toolbox 1.0, R. Bergmann ~ 2015-01-20 | 2015-04-10