Exponential map

release/4.3a0
Frank Dellaert 2013-12-15 21:52:03 +00:00
parent bad1c38fe6
commit 6f537b2b44
2 changed files with 57 additions and 4 deletions

View File

@ -65,7 +65,11 @@
\begin_body
\begin_layout Title
Retraction on a Sphere
Manifold Geometry of the Sphere
\begin_inset Formula $S^{2}$
\end_inset
\end_layout
\begin_layout Author
@ -74,8 +78,8 @@ Frank, Can, and Manohar
\begin_layout Standard
\begin_inset FormulaMacro
\newcommand{\xihat}{\hat{\xi}}
{\hat{\xi}}
\newcommand{\xihat}{z}
{z}
\end_inset
@ -147,7 +151,7 @@ retraction
\begin_inset Formula
\[
q=R_{p}(\xihat)=\frac{p+\xihat}{\left|p+\xihat\right|}=\frac{p+\xihat}{\alpha}
q=R_{p}(\xihat)=\frac{p+\xihat}{\left\Vert p+z\right\Vert }=\frac{p+\xihat}{\alpha}
\]
\end_inset
@ -407,6 +411,55 @@ and because
\end_layout
\begin_layout Subsubsection*
Exponential Map
\end_layout
\begin_layout Standard
The exponential map itself is not so difficult, and is given in Ma01ijcv,
as well as in this CVPR tutorial by Anuj Srivastava:
\begin_inset CommandInset href
LatexCommand href
name "http://stat.fsu.edu/~anuj/CVPR_Tutorial/Part2.pdf"
\end_inset
.
\begin_inset Formula
\[
\exp_{p}\xihat=\cos\left(\left\Vert \xihat\right\Vert \right)p+\sin\left(\left\Vert \xihat\right\Vert \right)\frac{\xihat}{\left\Vert \xihat\right\Vert }
\]
\end_inset
The latter also gives the inverse, i.e., get the tangent vector
\begin_inset Formula $z$
\end_inset
to go from
\begin_inset Formula $p$
\end_inset
to
\begin_inset Formula $q$
\end_inset
:
\begin_inset Formula
\[
z=\log_{p}q=\frac{\theta}{\sin\theta}\left(q-p\cos\theta\right)p
\]
\end_inset
with
\begin_inset Formula $\theta=\cos^{-1}\left(p^{T}q\right)$
\end_inset
.
\end_layout
\begin_layout Standard
\begin_inset CommandInset bibtex
LatexCommand bibtex

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doc/sphere.pdf Normal file

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