2226 lines
39 KiB
Plaintext
2226 lines
39 KiB
Plaintext
#LyX 1.6.5 created this file. For more info see http://www.lyx.org/
|
|
\lyxformat 345
|
|
\begin_document
|
|
\begin_header
|
|
\textclass article
|
|
\use_default_options false
|
|
\language english
|
|
\inputencoding auto
|
|
\font_roman times
|
|
\font_sans default
|
|
\font_typewriter default
|
|
\font_default_family rmdefault
|
|
\font_sc false
|
|
\font_osf false
|
|
\font_sf_scale 100
|
|
\font_tt_scale 100
|
|
|
|
\graphics default
|
|
\paperfontsize 12
|
|
\spacing single
|
|
\use_hyperref false
|
|
\papersize default
|
|
\use_geometry true
|
|
\use_amsmath 1
|
|
\use_esint 0
|
|
\cite_engine basic
|
|
\use_bibtopic false
|
|
\paperorientation portrait
|
|
\leftmargin 1in
|
|
\topmargin 1in
|
|
\rightmargin 1in
|
|
\bottommargin 1in
|
|
\secnumdepth 3
|
|
\tocdepth 3
|
|
\paragraph_separation indent
|
|
\defskip medskip
|
|
\quotes_language english
|
|
\papercolumns 1
|
|
\papersides 1
|
|
\paperpagestyle default
|
|
\tracking_changes false
|
|
\output_changes false
|
|
\author ""
|
|
\author ""
|
|
\end_header
|
|
|
|
\begin_body
|
|
|
|
\begin_layout Title
|
|
Geometry Derivatives and Other Hairy Math
|
|
\end_layout
|
|
|
|
\begin_layout Author
|
|
Frank Dellaert
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
\begin_inset Box Frameless
|
|
position "t"
|
|
hor_pos "c"
|
|
has_inner_box 1
|
|
inner_pos "t"
|
|
use_parbox 0
|
|
width "100col%"
|
|
special "none"
|
|
height "1in"
|
|
height_special "totalheight"
|
|
status collapsed
|
|
|
|
\begin_layout Plain Layout
|
|
\begin_inset Note Comment
|
|
status open
|
|
|
|
\begin_layout Plain Layout
|
|
Derivatives
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
\begin_inset FormulaMacro
|
|
\newcommand{\deriv}[2]{\frac{\partial#1}{\partial#2}}
|
|
{\frac{\partial#1}{\partial#2}}
|
|
\end_inset
|
|
|
|
|
|
\begin_inset FormulaMacro
|
|
\newcommand{\at}[2]{#1\biggr\rvert_{#2}}
|
|
{#1\biggr\rvert_{#2}}
|
|
\end_inset
|
|
|
|
|
|
\begin_inset FormulaMacro
|
|
\newcommand{\Jac}[3]{ \at{\deriv{#1}{#2}} {#3} }
|
|
{\at{\deriv{#1}{#2}}{#3}}
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
\begin_inset Note Comment
|
|
status open
|
|
|
|
\begin_layout Plain Layout
|
|
Lie Groups
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
\begin_inset FormulaMacro
|
|
\newcommand{\xhat}{\hat{x}}
|
|
{\hat{x}}
|
|
\end_inset
|
|
|
|
|
|
\begin_inset FormulaMacro
|
|
\newcommand{\yhat}{\hat{y}}
|
|
{\hat{y}}
|
|
\end_inset
|
|
|
|
|
|
\begin_inset FormulaMacro
|
|
\newcommand{\Ad}[1]{Ad_{#1}}
|
|
{Ad_{#1}}
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
\begin_inset FormulaMacro
|
|
\newcommand{\define}{\stackrel{\Delta}{=}}
|
|
{\stackrel{\Delta}{=}}
|
|
\end_inset
|
|
|
|
|
|
\begin_inset FormulaMacro
|
|
\newcommand{\gg}{\mathfrak{g}}
|
|
{\mathfrak{g}}
|
|
\end_inset
|
|
|
|
|
|
\begin_inset FormulaMacro
|
|
\newcommand{\Rn}{\mathbb{R}^{n}}
|
|
{\mathbb{R}^{n}}
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
\begin_inset Note Comment
|
|
status open
|
|
|
|
\begin_layout Plain Layout
|
|
SO(2)
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
\begin_inset FormulaMacro
|
|
\newcommand{\Rtwo}{\mathfrak{\mathbb{R}^{2}}}
|
|
{\mathfrak{\mathbb{R}^{2}}}
|
|
\end_inset
|
|
|
|
|
|
\begin_inset FormulaMacro
|
|
\newcommand{\SOtwo}{SO(2)}
|
|
{SO(2)}
|
|
\end_inset
|
|
|
|
|
|
\begin_inset FormulaMacro
|
|
\newcommand{\sotwo}{\mathfrak{so(2)}}
|
|
{\mathfrak{so(2)}}
|
|
\end_inset
|
|
|
|
|
|
\begin_inset FormulaMacro
|
|
\newcommand{\that}{\hat{\theta}}
|
|
{\hat{\theta}}
|
|
\end_inset
|
|
|
|
|
|
\begin_inset FormulaMacro
|
|
\newcommand{\skew}[1]{[#1]_{+}}
|
|
{[#1]_{+}}
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
\begin_inset Note Comment
|
|
status open
|
|
|
|
\begin_layout Plain Layout
|
|
SE(2)
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
\begin_inset FormulaMacro
|
|
\newcommand{\SEtwo}{SE(2)}
|
|
{SE(2)}
|
|
\end_inset
|
|
|
|
|
|
\begin_inset FormulaMacro
|
|
\newcommand{\setwo}{\mathfrak{se(2)}}
|
|
{\mathfrak{se(2)}}
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
\begin_inset Note Comment
|
|
status open
|
|
|
|
\begin_layout Plain Layout
|
|
SO(3)
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
\begin_inset FormulaMacro
|
|
\newcommand{\Rthree}{\mathfrak{\mathbb{R}^{3}}}
|
|
{\mathfrak{\mathbb{R}^{3}}}
|
|
\end_inset
|
|
|
|
|
|
\begin_inset FormulaMacro
|
|
\newcommand{\SOthree}{SO(3)}
|
|
{SO(3)}
|
|
\end_inset
|
|
|
|
|
|
\begin_inset FormulaMacro
|
|
\newcommand{\sothree}{\mathfrak{so(3)}}
|
|
{\mathfrak{so(3)}}
|
|
\end_inset
|
|
|
|
|
|
\begin_inset FormulaMacro
|
|
\newcommand{\what}{\hat{\omega}}
|
|
{\hat{\omega}}
|
|
\end_inset
|
|
|
|
|
|
\begin_inset FormulaMacro
|
|
\newcommand{\Skew}[1]{[#1]_{\times}}
|
|
{[#1]_{\times}}
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
\begin_inset Note Comment
|
|
status open
|
|
|
|
\begin_layout Plain Layout
|
|
SE(3)
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
\begin_inset FormulaMacro
|
|
\newcommand{\Rsix}{\mathfrak{\mathbb{R}^{6}}}
|
|
{\mathfrak{\mathbb{R}^{6}}}
|
|
\end_inset
|
|
|
|
|
|
\begin_inset FormulaMacro
|
|
\newcommand{\SEthree}{SE(3)}
|
|
{SE(3)}
|
|
\end_inset
|
|
|
|
|
|
\begin_inset FormulaMacro
|
|
\newcommand{\sethree}{\mathfrak{se(3)}}
|
|
{\mathfrak{se(3)}}
|
|
\end_inset
|
|
|
|
|
|
\begin_inset FormulaMacro
|
|
\newcommand{\xihat}{\hat{\xi}}
|
|
{\hat{\xi}}
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Section
|
|
Derivatives of Lie Group Mappings
|
|
\end_layout
|
|
|
|
\begin_layout Subsection
|
|
Homomorphisms
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
The following is relevant
|
|
\begin_inset CommandInset citation
|
|
LatexCommand cite
|
|
after "page 45"
|
|
key "Hall00book"
|
|
|
|
\end_inset
|
|
|
|
: suppose that
|
|
\begin_inset Formula $\Phi:G\rightarrow H$
|
|
\end_inset
|
|
|
|
is a a mapping (Lie group homomorphism).
|
|
Then there exists a unique linear map
|
|
\begin_inset Formula $\phi:\gg\rightarrow\mathfrak{h}$
|
|
\end_inset
|
|
|
|
|
|
\begin_inset Formula \[
|
|
\phi(\xhat)\define\lim_{t\rightarrow0}\frac{d}{dt}\Phi\left(e^{t\xhat}\right)\]
|
|
|
|
\end_inset
|
|
|
|
such that
|
|
\end_layout
|
|
|
|
\begin_layout Enumerate
|
|
\begin_inset Formula $\Phi\left(e^{\xhat}\right)=e^{\phi\left(\xhat\right)}$
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Enumerate
|
|
\begin_inset Formula $\phi\left(T\xhat T^{-1}\right)=\Phi(T)\phi(\xhat)\Phi(T^{-1})$
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Enumerate
|
|
\begin_inset Formula $\phi\left([\xhat,\yhat]\right)=\left[\phi(\xhat),\phi(\yhat)\right]$
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
In other words, the map
|
|
\begin_inset Formula $\phi$
|
|
\end_inset
|
|
|
|
is the derivative of
|
|
\begin_inset Formula $\Phi$
|
|
\end_inset
|
|
|
|
at the identity.
|
|
It suffices to compute
|
|
\begin_inset Formula $\phi$
|
|
\end_inset
|
|
|
|
for a basis of
|
|
\begin_inset Formula $\gg$
|
|
\end_inset
|
|
|
|
.
|
|
Since
|
|
\begin_inset Formula \[
|
|
e^{-\xhat}=\left(e^{-\xhat}\right)^{-1}\]
|
|
|
|
\end_inset
|
|
|
|
clearly
|
|
\begin_inset Formula $\phi(\xhat)=-\xhat$
|
|
\end_inset
|
|
|
|
for the inverse mapping.
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
\begin_inset Note Note
|
|
status open
|
|
|
|
\begin_layout Plain Layout
|
|
Let us define two mappings
|
|
\begin_inset Formula \[
|
|
\Phi_{1}(A)=AB\mbox{ and }\Phi_{2}(B)=AB\]
|
|
|
|
\end_inset
|
|
|
|
Then
|
|
\begin_inset Formula \[
|
|
\phi_{1}(\xhat)=\lim_{t\rightarrow0}\frac{d}{dt}\Phi_{1}\left(e^{t\xhat}B\right)=\]
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Subsection
|
|
Derivatives
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
The derivatives for
|
|
\emph on
|
|
inverse, compose
|
|
\emph default
|
|
, and
|
|
\emph on
|
|
between
|
|
\emph default
|
|
can be derived from Lie group principles.
|
|
|
|
\begin_inset Note Note
|
|
status collapsed
|
|
|
|
\begin_layout Plain Layout
|
|
To find the derivatives of these functions, we look for the necessary
|
|
\begin_inset Quotes eld
|
|
\end_inset
|
|
|
|
delta
|
|
\begin_inset Quotes erd
|
|
\end_inset
|
|
|
|
in the tangent space of the function
|
|
\emph on
|
|
output
|
|
\emph default
|
|
|
|
\begin_inset Formula $f(g)$
|
|
\end_inset
|
|
|
|
that corresponds to a
|
|
\begin_inset Quotes eld
|
|
\end_inset
|
|
|
|
delta
|
|
\begin_inset Quotes erd
|
|
\end_inset
|
|
|
|
in the tangent space of the function
|
|
\emph on
|
|
input
|
|
\emph default
|
|
|
|
\begin_inset Formula $g$
|
|
\end_inset
|
|
|
|
.
|
|
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
Specifically, to find the derivative of a function
|
|
\begin_inset Formula $f\left(g\right)$
|
|
\end_inset
|
|
|
|
, we want to find the Lie algebra element
|
|
\begin_inset Formula $\yhat\in\gg$
|
|
\end_inset
|
|
|
|
, that will result from changing
|
|
\begin_inset Formula $g$
|
|
\end_inset
|
|
|
|
using
|
|
\begin_inset Formula $\xhat$
|
|
\end_inset
|
|
|
|
, also in exponential coordinates:
|
|
\begin_inset Formula \[
|
|
f\left(g\right)e^{\yhat}=f\left(ge^{\xhat}\right)\]
|
|
|
|
\end_inset
|
|
|
|
Calculating these derivatives requires that we know the form of the function
|
|
|
|
\begin_inset Formula $f$
|
|
\end_inset
|
|
|
|
.
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
Starting with
|
|
\series bold
|
|
inverse
|
|
\series default
|
|
, i.e.,
|
|
\begin_inset Formula $f(g)=g^{-1}$
|
|
\end_inset
|
|
|
|
, we have
|
|
\begin_inset Formula \begin{align}
|
|
g^{-1}e^{\yhat} & =\left(ge^{\xhat}\right)^{-1}=e^{-\xhat}g^{-1}\nonumber \\
|
|
e^{\yhat} & =ge^{-\xhat}g^{-1}=e^{\Ad g\left(-\xhat\right)}\nonumber \\
|
|
\yhat & =\Ad g\left(-\xhat\right)\label{eq:Dinverse}\end{align}
|
|
|
|
\end_inset
|
|
|
|
In other words, and this is very intuitive in hindsight, the inverse is
|
|
just negation of
|
|
\begin_inset Formula $\xhat$
|
|
\end_inset
|
|
|
|
, along with an adjoint to make sure it is applied in the right frame!
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
|
|
\series bold
|
|
Compose
|
|
\series default
|
|
can be derived similarly.
|
|
Let us define two functions to find the derivatives in first and second
|
|
arguments:
|
|
\begin_inset Formula \[
|
|
f_{1}(g)=gh\mbox{ and }f_{2}(h)=gh\]
|
|
|
|
\end_inset
|
|
|
|
The latter is easiest, as a change
|
|
\begin_inset Formula $\xhat$
|
|
\end_inset
|
|
|
|
in the second argument
|
|
\begin_inset Formula $h$
|
|
\end_inset
|
|
|
|
simply gets applied to the result
|
|
\begin_inset Formula $gh$
|
|
\end_inset
|
|
|
|
:
|
|
\begin_inset Formula \begin{align}
|
|
f_{2}(h)e^{\yhat} & =f_{2}\left(he^{\xhat}\right)\nonumber \\
|
|
ghe^{\yhat} & =ghe^{\xhat}\nonumber \\
|
|
\yhat & =\xhat\label{eq:Dcompose2}\end{align}
|
|
|
|
\end_inset
|
|
|
|
The derivative for the first argument is a bit trickier:
|
|
\begin_inset Formula \begin{align}
|
|
f_{1}(g)e^{\yhat} & =f_{1}\left(ge^{\xhat}\right)\nonumber \\
|
|
ghe^{\yhat} & =ge^{\xhat}h\nonumber \\
|
|
e^{\yhat} & =h^{-1}e^{\xhat}h=e^{\Ad{h^{-1}}\xhat}\nonumber \\
|
|
\yhat & =\Ad{h^{-1}}\xhat\label{eq:Dcompose1}\end{align}
|
|
|
|
\end_inset
|
|
|
|
In other words, to apply a change
|
|
\begin_inset Formula $\xhat$
|
|
\end_inset
|
|
|
|
in
|
|
\begin_inset Formula $g$
|
|
\end_inset
|
|
|
|
we first need to undo
|
|
\begin_inset Formula $h$
|
|
\end_inset
|
|
|
|
, then apply
|
|
\begin_inset Formula $\xhat$
|
|
\end_inset
|
|
|
|
, and then apply
|
|
\begin_inset Formula $h$
|
|
\end_inset
|
|
|
|
again.
|
|
All can be done in one step by simply applying
|
|
\begin_inset Formula $\Ad{h^{-1}}\xhat$
|
|
\end_inset
|
|
|
|
.
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
Finally, let us find the derivative of
|
|
\series bold
|
|
between
|
|
\series default
|
|
, defined as
|
|
\begin_inset Formula $between(g,h)=compose(inverse(g),h)$
|
|
\end_inset
|
|
|
|
.
|
|
The derivative in the second argument
|
|
\begin_inset Formula $h$
|
|
\end_inset
|
|
|
|
is similarly trivial:
|
|
\begin_inset Formula $\yhat=\xhat$
|
|
\end_inset
|
|
|
|
.
|
|
The first argument goes as follows:
|
|
\begin_inset Formula \begin{align}
|
|
f_{1}(g)e^{\yhat} & =f_{1}\left(ge^{\xhat}\right)\nonumber \\
|
|
g^{-1}he^{\yhat} & =\left(ge^{\xhat}\right)^{-1}h=e^{\left(-\xhat\right)}g^{-1}h\nonumber \\
|
|
e^{\yhat} & =\left(h^{-1}g\right)e^{\left(-\xhat\right)}\left(h^{-1}g\right)^{-1}=e^{\Ad{\left(h^{-1}g\right)}\left(-\xhat\right)}\nonumber \\
|
|
\yhat & =\Ad{\left(h^{-1}g\right)}\left(-\xhat\right)=\Ad{between\left(h,g\right)}\left(-\xhat\right)\label{eq:Dbetween1}\end{align}
|
|
|
|
\end_inset
|
|
|
|
Hence, now we undo
|
|
\begin_inset Formula $h$
|
|
\end_inset
|
|
|
|
and then apply the inverse
|
|
\begin_inset Formula $\left(-\xhat\right)$
|
|
\end_inset
|
|
|
|
in the
|
|
\begin_inset Formula $g$
|
|
\end_inset
|
|
|
|
frame.
|
|
\end_layout
|
|
|
|
\begin_layout Subsection*
|
|
Numerical Derivatives
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
Let's examine
|
|
\begin_inset Formula \[
|
|
f\left(g\right)e^{\yhat}=f\left(ge^{\xhat}\right)\]
|
|
|
|
\end_inset
|
|
|
|
and multiply with
|
|
\begin_inset Formula $f(g)^{-1}$
|
|
\end_inset
|
|
|
|
on both sides then take the log (which in our case returns
|
|
\begin_inset Formula $y$
|
|
\end_inset
|
|
|
|
, not
|
|
\begin_inset Formula $\yhat$
|
|
\end_inset
|
|
|
|
):
|
|
\begin_inset Formula \[
|
|
y(x)=\log\left[f\left(g\right)^{-1}f\left(ge^{\xhat}\right)\right]\]
|
|
|
|
\end_inset
|
|
|
|
Let us look at
|
|
\begin_inset Formula $x=0$
|
|
\end_inset
|
|
|
|
, and perturb in direction
|
|
\begin_inset Formula $i$
|
|
\end_inset
|
|
|
|
,
|
|
\begin_inset Formula $e_{i}=[0,0,d,0,0]$
|
|
\end_inset
|
|
|
|
.
|
|
Then take derivative,
|
|
\begin_inset Formula \[
|
|
\deriv{y(d)}d\define\lim_{d->0}\frac{y(d)-y(0)}{d}=\lim_{d->0}\frac{1}{d}\log\left[f\left(g\right)^{-1}f\left(ge^{\hat{e_{i}}}\right)\right]\]
|
|
|
|
\end_inset
|
|
|
|
which is the basis for a numerical derivative scheme.
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
Let us also look at a chain rule.
|
|
If we know the behavior at the origin
|
|
\begin_inset Formula $I$
|
|
\end_inset
|
|
|
|
, we can extrapolate
|
|
\begin_inset Formula \[
|
|
f(ge^{\xhat})=f(ge^{\xhat}g^{-1}g)=f(e^{\Ad g\xhat}g)\]
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Section
|
|
Derivatives of Actions
|
|
\begin_inset CommandInset label
|
|
LatexCommand label
|
|
name "sec:Derivatives-of-Actions"
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Subsection
|
|
Forward Action
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
The (usual) action of an
|
|
\begin_inset Formula $n$
|
|
\end_inset
|
|
|
|
-dimensional matrix group
|
|
\begin_inset Formula $G$
|
|
\end_inset
|
|
|
|
is matrix-vector multiplication on
|
|
\begin_inset Formula $\mathbb{R}^{n}$
|
|
\end_inset
|
|
|
|
,
|
|
\begin_inset Formula \[
|
|
q=Tp\]
|
|
|
|
\end_inset
|
|
|
|
with
|
|
\begin_inset Formula $p,q\in\mathbb{R}^{n}$
|
|
\end_inset
|
|
|
|
and
|
|
\begin_inset Formula $T\in GL(n)$
|
|
\end_inset
|
|
|
|
.
|
|
Let us first do away with the derivative in
|
|
\begin_inset Formula $p$
|
|
\end_inset
|
|
|
|
, which is easy:
|
|
\begin_inset Formula \[
|
|
\deriv{\left(Tp\right)}p=T\]
|
|
|
|
\end_inset
|
|
|
|
We would now like to know what an incremental action
|
|
\begin_inset Formula $\xhat$
|
|
\end_inset
|
|
|
|
would do, through the exponential map
|
|
\begin_inset Formula \[
|
|
q(x)=Te^{\xhat}p\]
|
|
|
|
\end_inset
|
|
|
|
with derivative
|
|
\begin_inset Formula \[
|
|
\deriv{q(x)}x=T\deriv{}x\left(e^{\xhat}p\right)\]
|
|
|
|
\end_inset
|
|
|
|
Since the matrix exponential is given by the series
|
|
\begin_inset Formula \[
|
|
e^{A}=I+A+\frac{A^{2}}{2!}+\frac{A^{3}}{3!}+\ldots\]
|
|
|
|
\end_inset
|
|
|
|
we have, to first order
|
|
\begin_inset Formula \[
|
|
e^{\xhat}p=p+\xhat p+\ldots\]
|
|
|
|
\end_inset
|
|
|
|
and the derivative of an incremental action x for matrix Lie groups becomes
|
|
\begin_inset Formula \[
|
|
\deriv{q(x)}x=T\deriv{\left(\xhat p\right)}x\define TH_{p}\]
|
|
|
|
\end_inset
|
|
|
|
where
|
|
\begin_inset Formula $H_{p}$
|
|
\end_inset
|
|
|
|
is an
|
|
\begin_inset Formula $n\times n$
|
|
\end_inset
|
|
|
|
Jacobian matrix that depends on
|
|
\begin_inset Formula $p$
|
|
\end_inset
|
|
|
|
.
|
|
|
|
\begin_inset Note Note
|
|
status collapsed
|
|
|
|
\begin_layout Plain Layout
|
|
Recalling the definition
|
|
\begin_inset CommandInset ref
|
|
LatexCommand eqref
|
|
reference "eq:generators"
|
|
|
|
\end_inset
|
|
|
|
of the map
|
|
\begin_inset Formula $x\rightarrow\xhat$
|
|
\end_inset
|
|
|
|
, we can calculate
|
|
\family roman
|
|
\series medium
|
|
\shape up
|
|
\size normal
|
|
\emph off
|
|
\bar no
|
|
\noun off
|
|
\color none
|
|
|
|
\begin_inset Formula $\xhat p$
|
|
\end_inset
|
|
|
|
|
|
\family default
|
|
\series default
|
|
\shape default
|
|
\size default
|
|
\emph default
|
|
\bar default
|
|
\noun default
|
|
\color inherit
|
|
as (using tensor notation)
|
|
\begin_inset Formula \[
|
|
\left(\xhat p\right)_{jk}=G_{jk}^{i}x_{i}p^{k}\]
|
|
|
|
\end_inset
|
|
|
|
and hence the derivative is
|
|
\begin_inset Formula \[
|
|
\left(H_{p}\right)_{j}^{i}=G_{jk}^{i}p^{k}\]
|
|
|
|
\end_inset
|
|
|
|
and the final derivative becomes
|
|
\begin_inset Formula \[
|
|
\deriv{q(x)}x=TH_{p}\]
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Subsection
|
|
Inverse Action
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
When we apply the inverse transformation
|
|
\begin_inset Formula \[
|
|
q=T^{-1}p\]
|
|
|
|
\end_inset
|
|
|
|
we would now like to know what an incremental action
|
|
\begin_inset Formula $\xhat$
|
|
\end_inset
|
|
|
|
on
|
|
\begin_inset Formula $T$
|
|
\end_inset
|
|
|
|
would do:
|
|
\begin_inset Formula \begin{eqnarray*}
|
|
q(x) & = & \left(Te^{\xhat}\right)^{-1}p\\
|
|
& = & e^{-\xhat}T^{-1}p\\
|
|
& = & e^{-\xhat}q\\
|
|
& \approx & q-\xhat q\end{eqnarray*}
|
|
|
|
\end_inset
|
|
|
|
Hence
|
|
\begin_inset Formula \begin{equation}
|
|
\deriv{q(x)}x=\deriv{\left(q-\xhat q\right)}x=-\deriv{\left(\xhat q\right)}x=-H_{q}\label{eq:inverseAction}\end{equation}
|
|
|
|
\end_inset
|
|
|
|
where
|
|
\begin_inset Formula $H_{q}$
|
|
\end_inset
|
|
|
|
will be as above.
|
|
\end_layout
|
|
|
|
\begin_layout Section
|
|
Point3
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
A cross product
|
|
\begin_inset Formula $a\times b$
|
|
\end_inset
|
|
|
|
can be written as a matrix multiplication
|
|
\begin_inset Formula \[
|
|
a\times b=\Skew ab\]
|
|
|
|
\end_inset
|
|
|
|
where
|
|
\begin_inset Formula $\Skew a$
|
|
\end_inset
|
|
|
|
is a skew-symmetric matrix defined as
|
|
\begin_inset Formula \[
|
|
\Skew{x,y,z}=\left[\begin{array}{ccc}
|
|
0 & -z & y\\
|
|
z & 0 & -x\\
|
|
-y & x & 0\end{array}\right]\]
|
|
|
|
\end_inset
|
|
|
|
We also have
|
|
\begin_inset Formula \[
|
|
a^{T}\Skew b=-(\Skew ba)^{T}=-(a\times b)^{T}\]
|
|
|
|
\end_inset
|
|
|
|
The derivative of a cross product
|
|
\begin_inset Formula \begin{equation}
|
|
\frac{\partial(a\times b)}{\partial a}=\Skew{-b}\label{eq:Dcross1}\end{equation}
|
|
|
|
\end_inset
|
|
|
|
|
|
\begin_inset Formula \begin{equation}
|
|
\frac{\partial(a\times b)}{\partial b}=\Skew a\label{eq:Dcross2}\end{equation}
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
\begin_inset Newpage pagebreak
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Section
|
|
2D Rotations
|
|
\end_layout
|
|
|
|
\begin_layout Subsection
|
|
Rot2 (in gtsam)
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
A rotation is stored as
|
|
\begin_inset Formula $(\cos\theta,\sin\theta)$
|
|
\end_inset
|
|
|
|
.
|
|
An incremental rotation is applied using the trigonometric sum rule:
|
|
\begin_inset Formula \[
|
|
\cos\theta'=\cos\theta\cos\delta-\sin\theta\sin\delta\]
|
|
|
|
\end_inset
|
|
|
|
|
|
\begin_inset Formula \[
|
|
\sin\theta'=\sin\theta\cos\delta+\cos\theta\sin\delta\]
|
|
|
|
\end_inset
|
|
|
|
where
|
|
\begin_inset Formula $\delta$
|
|
\end_inset
|
|
|
|
is an incremental rotation angle.
|
|
\end_layout
|
|
|
|
\begin_layout Subsection
|
|
Derivatives of Mappings
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
We have the derivative of
|
|
\series bold
|
|
inverse
|
|
\series default
|
|
,
|
|
\begin_inset Formula \[
|
|
\frac{\partial R^{T}}{\partial\theta}=-\Ad R=-1\mbox{ }\]
|
|
|
|
\end_inset
|
|
|
|
|
|
\series bold
|
|
compose,
|
|
\series default
|
|
|
|
\begin_inset Formula \[
|
|
\frac{\partial\left(R_{1}R_{2}\right)}{\partial\theta_{1}}=\Ad{R_{2}^{T}}=1\mbox{ and }\frac{\partial\left(R_{1}R_{2}\right)}{\partial\theta_{2}}=1\]
|
|
|
|
\end_inset
|
|
|
|
|
|
\series bold
|
|
|
|
\begin_inset Formula $and$
|
|
\end_inset
|
|
|
|
between:
|
|
\series default
|
|
|
|
\begin_inset Formula \[
|
|
\frac{\partial\left(R_{1}^{T}R_{2}\right)}{\partial\theta_{1}}=-\Ad{R_{2}^{T}R_{1}}=-1\mbox{ and }\frac{\partial\left(R_{1}^{T}R_{2}\right)}{\partial\theta_{2}}=1\]
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Subsection
|
|
Derivatives of Actions
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
In the case of
|
|
\begin_inset Formula $\SOtwo$
|
|
\end_inset
|
|
|
|
the vector space is
|
|
\begin_inset Formula $\Rtwo$
|
|
\end_inset
|
|
|
|
, and the group action corresponds to rotating a point
|
|
\begin_inset Formula \[
|
|
q=Rp\]
|
|
|
|
\end_inset
|
|
|
|
We would now like to know what an incremental rotation parameterized by
|
|
|
|
\begin_inset Formula $\theta$
|
|
\end_inset
|
|
|
|
would do:
|
|
\begin_inset Formula \[
|
|
q(\text{\omega t})=Re^{\skew{\omega t}}p\]
|
|
|
|
\end_inset
|
|
|
|
The derivative is (following the exposition in Section
|
|
\begin_inset CommandInset ref
|
|
LatexCommand ref
|
|
reference "sec:Derivatives-of-Actions"
|
|
|
|
\end_inset
|
|
|
|
):
|
|
\begin_inset Formula \[
|
|
\deriv{q(\omega t)}t=R\deriv{}t\left(e^{\skew{\omega t}}p\right)=R\deriv{}t\left(\skew{\omega t}p\right)\]
|
|
|
|
\end_inset
|
|
|
|
Note that
|
|
\begin_inset Formula \begin{equation}
|
|
\skew{\theta}\left[\begin{array}{c}
|
|
x\\
|
|
y\end{array}\right]=\theta R_{\pi/2}\left[\begin{array}{c}
|
|
x\\
|
|
y\end{array}\right]=\theta\left[\begin{array}{c}
|
|
-y\\
|
|
x\end{array}\right]\label{eq:RestrictedCross}\end{equation}
|
|
|
|
\end_inset
|
|
|
|
which acts like a restricted
|
|
\begin_inset Quotes eld
|
|
\end_inset
|
|
|
|
cross product
|
|
\begin_inset Quotes erd
|
|
\end_inset
|
|
|
|
in the plane.
|
|
Hence
|
|
\begin_inset Formula \[
|
|
\skew{\theta}p=\left[\begin{array}{c}
|
|
-y\\
|
|
x\end{array}\right]\theta=\omega R_{pi/2}pt\]
|
|
|
|
\end_inset
|
|
|
|
Hence, the final derivative of an action in its first argument is
|
|
\begin_inset Formula \[
|
|
\deriv{q(\omega t)}{\omega t}=\omega RR_{pi/2}p=\omega R_{pi/2}Rp=\omega R_{pi/2}q\]
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
Really need to think of relationship
|
|
\begin_inset Formula $\omega$
|
|
\end_inset
|
|
|
|
and
|
|
\begin_inset Formula $t$
|
|
\end_inset
|
|
|
|
.
|
|
We don't have a time
|
|
\begin_inset Formula $t$
|
|
\end_inset
|
|
|
|
in our code.
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
\begin_inset Newpage pagebreak
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Section
|
|
2D Rigid Transformations
|
|
\end_layout
|
|
|
|
\begin_layout Subsection
|
|
Derivatives of Mappings
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
We can just define all derivatives in terms of the above adjoint map:
|
|
\begin_inset Formula \begin{eqnarray*}
|
|
\frac{\partial T^{^{-1}}}{\partial\xi} & = & -\Ad T\end{eqnarray*}
|
|
|
|
\end_inset
|
|
|
|
|
|
\begin_inset Formula \begin{eqnarray*}
|
|
\frac{\partial\left(T_{1}T_{2}\right)}{\partial\xi_{1}} & = & \Ad{T_{2}^{^{-1}}}=1\mbox{ and }\frac{\partial\left(T_{1}T_{2}\right)}{\partial\xi_{2}}=I_{3}\end{eqnarray*}
|
|
|
|
\end_inset
|
|
|
|
|
|
\begin_inset Formula \begin{eqnarray*}
|
|
\frac{\partial\left(T_{1}^{-1}T_{2}\right)}{\partial\xi_{1}} & = & -\Ad{T_{2}^{^{-1}}T_{1}}=-\Ad{between(T_{2},T_{1})}\mbox{ and }\frac{\partial\left(T_{1}^{-1}T_{2}\right)}{\partial\xi_{2}}=I_{3}\end{eqnarray*}
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Subsection
|
|
The derivatives of Actions
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
The action of
|
|
\begin_inset Formula $\SEtwo$
|
|
\end_inset
|
|
|
|
on 2D points is done by embedding the points in
|
|
\begin_inset Formula $\mathbb{R}^{3}$
|
|
\end_inset
|
|
|
|
by using homogeneous coordinates
|
|
\begin_inset Formula \[
|
|
\hat{q}=\left[\begin{array}{c}
|
|
q\\
|
|
1\end{array}\right]=\left[\begin{array}{cc}
|
|
R & t\\
|
|
0 & 1\end{array}\right]\left[\begin{array}{c}
|
|
p\\
|
|
1\end{array}\right]=T\hat{p}\]
|
|
|
|
\end_inset
|
|
|
|
Analoguous to
|
|
\begin_inset Formula $\SEthree$
|
|
\end_inset
|
|
|
|
, we can compute a velocity
|
|
\begin_inset Formula $\xihat\hat{p}$
|
|
\end_inset
|
|
|
|
in the local
|
|
\begin_inset Formula $T$
|
|
\end_inset
|
|
|
|
frame:
|
|
\begin_inset Formula \[
|
|
\xihat\hat{p}=\left[\begin{array}{cc}
|
|
\skew{\omega} & v\\
|
|
0 & 0\end{array}\right]\left[\begin{array}{c}
|
|
p\\
|
|
1\end{array}\right]=\left[\begin{array}{c}
|
|
\skew{\omega}p+v\\
|
|
0\end{array}\right]\]
|
|
|
|
\end_inset
|
|
|
|
By only taking the top two rows, we can write this as a velocity in
|
|
\begin_inset Formula $\Rtwo$
|
|
\end_inset
|
|
|
|
, as the product of a
|
|
\begin_inset Formula $2\times3$
|
|
\end_inset
|
|
|
|
matrix
|
|
\begin_inset Formula $H_{p}$
|
|
\end_inset
|
|
|
|
that acts upon the exponential coordinates
|
|
\begin_inset Formula $\xi$
|
|
\end_inset
|
|
|
|
directly:
|
|
\begin_inset Formula \[
|
|
\skew{\omega}p+v=v+R_{\pi/2}p\omega=\left[\begin{array}{cc}
|
|
I_{2} & R_{\pi/2}p\end{array}\right]\left[\begin{array}{c}
|
|
v\\
|
|
\omega\end{array}\right]=H_{p}\xi\]
|
|
|
|
\end_inset
|
|
|
|
Hence, the final derivative of the group action is
|
|
\begin_inset Formula \[
|
|
\deriv{q(\xi)}{\xi}=R\left[\begin{array}{cc}
|
|
I_{2} & R_{\pi/2}p\end{array}\right]=\left[\begin{array}{cc}
|
|
R & R_{\pi/2}q\end{array}\right]\]
|
|
|
|
\end_inset
|
|
|
|
The derivative of the inverse action
|
|
\begin_inset Formula $\hat{q}=T^{-1}\hat{p}$
|
|
\end_inset
|
|
|
|
is given by
|
|
\begin_inset CommandInset ref
|
|
LatexCommand eqref
|
|
reference "eq:inverseAction"
|
|
|
|
\end_inset
|
|
|
|
, specialized to
|
|
\begin_inset Formula $\SEtwo$
|
|
\end_inset
|
|
|
|
:
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
\begin_inset Formula \[
|
|
\deriv{\left(T^{-1}\hat{p}\right)}{\xi}=-T^{-1}\deriv{\left(\Ad T\hat{\xi}\right)\hat{p}}{\xi}\]
|
|
|
|
\end_inset
|
|
|
|
where the velocity now is
|
|
\begin_inset Formula \[
|
|
\left(\Ad T\hat{\xi}\right)\hat{p}=\left[\begin{array}{cc}
|
|
\skew{\omega} & Rv-\omega R_{\pi/2}t\\
|
|
0 & 0\end{array}\right]\left[\begin{array}{c}
|
|
p\\
|
|
1\end{array}\right]=\left[\begin{array}{c}
|
|
Rv+R_{\pi/2}(p-t)\omega\\
|
|
0\end{array}\right]\]
|
|
|
|
\end_inset
|
|
|
|
and hence
|
|
\begin_inset Formula \[
|
|
\deriv{q(\xi)}{\xi}=-R^{T}\left[\begin{array}{cc}
|
|
R & R_{\pi/2}(p-t)\end{array}\right]=\left[\begin{array}{cc}
|
|
-I_{2} & -R_{\pi/2}q\end{array}\right]\]
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
\begin_inset Newpage pagebreak
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Section
|
|
3D Rotations
|
|
\end_layout
|
|
|
|
\begin_layout Subsection
|
|
Derivatives of Mappings
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
Hence, we are now in a position to simply posit the derivative of
|
|
\series bold
|
|
inverse
|
|
\series default
|
|
,
|
|
\begin_inset Formula \begin{eqnarray*}
|
|
\Skew{\omega'} & = & \Ad R\left(\Skew{-\omega}\right)=\Skew{R(-\omega)}\\
|
|
\frac{\partial R^{T}}{\partial\omega} & = & -R\end{eqnarray*}
|
|
|
|
\end_inset
|
|
|
|
|
|
\series bold
|
|
compose,
|
|
\series default
|
|
|
|
\begin_inset Formula \[
|
|
\Skew{\omega'}=\Ad{R_{2}^{T}}\left(\Skew{\omega}\right)=\Skew{R_{2}^{T}\omega}\]
|
|
|
|
\end_inset
|
|
|
|
|
|
\begin_inset Formula \[
|
|
\frac{\partial\left(R_{1}R_{2}\right)}{\partial\omega_{1}}=R_{2}^{T}\mbox{ and }\frac{\partial\left(R_{1}R_{2}\right)}{\partial\omega_{2}}=I_{3}\]
|
|
|
|
\end_inset
|
|
|
|
|
|
\series bold
|
|
between
|
|
\series default
|
|
in its first argument,
|
|
\begin_inset Formula \begin{eqnarray*}
|
|
\Skew{\omega'} & = & \Ad{R_{2}^{T}R_{1}}\left(\Skew{-\omega}\right)=\Skew{R_{2}^{T}R_{1}(-\omega)}\\
|
|
\frac{\partial\left(R_{1}^{T}R_{2}\right)}{\partial\omega_{1}} & = & -R_{2}^{T}R_{1}=-between(R_{2},R_{1})\end{eqnarray*}
|
|
|
|
\end_inset
|
|
|
|
and between in its second argument,
|
|
\begin_inset Formula \begin{eqnarray*}
|
|
\frac{\partial\left(R_{1}^{T}R_{2}\right)}{\partial\omega_{2}} & = & I_{3}\end{eqnarray*}
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Subsection
|
|
Derivatives of Actions
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
In the case of
|
|
\begin_inset Formula $\SOthree$
|
|
\end_inset
|
|
|
|
the vector space is
|
|
\begin_inset Formula $\Rthree$
|
|
\end_inset
|
|
|
|
, and the group action corresponds to rotating a point
|
|
\begin_inset Formula \[
|
|
q=Rp\]
|
|
|
|
\end_inset
|
|
|
|
We would now like to know what an incremental rotation parameterized by
|
|
|
|
\begin_inset Formula $\omega$
|
|
\end_inset
|
|
|
|
would do:
|
|
\begin_inset Formula \[
|
|
q(\omega)=Re^{\Skew{\omega}}p\]
|
|
|
|
\end_inset
|
|
|
|
hence the derivative (following the exposition in Section
|
|
\begin_inset CommandInset ref
|
|
LatexCommand ref
|
|
reference "sec:Derivatives-of-Actions"
|
|
|
|
\end_inset
|
|
|
|
):
|
|
\begin_inset Formula \[
|
|
\deriv{q(\omega)}{\omega}=R\deriv{}{\omega}\left(e^{\Skew{\omega}}p\right)=R\deriv{}{\omega}\left(\Skew{\omega}p\right)=RH_{p}\]
|
|
|
|
\end_inset
|
|
|
|
To calculate
|
|
\begin_inset Formula $H_{p}$
|
|
\end_inset
|
|
|
|
we make use of
|
|
\begin_inset Formula \[
|
|
\Skew{\omega}p=\omega\times p=-p\times\omega=\Skew{-p}\omega\]
|
|
|
|
\end_inset
|
|
|
|
Hence, the final derivative of an action in its first argument is
|
|
\begin_inset Formula \[
|
|
\deriv{q(\omega)}{\omega}=RH_{p}=R\Skew{-p}\]
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
The derivative of the inverse action is given by
|
|
\begin_inset CommandInset ref
|
|
LatexCommand ref
|
|
reference "eq:inverseAction"
|
|
|
|
\end_inset
|
|
|
|
, specialized to
|
|
\begin_inset Formula $\SOthree$
|
|
\end_inset
|
|
|
|
:
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
\begin_inset Formula \[
|
|
\deriv{q(\omega)}{\omega}=-R^{T}\deriv{\left(\Skew{R\omega\mbox{ }}p\right)}{\omega}=R^{T}\Skew pR=\Skew{R^{T}p}\]
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
\begin_inset Newpage pagebreak
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Section
|
|
3D Rigid Transformations
|
|
\end_layout
|
|
|
|
\begin_layout Subsection
|
|
Derivatives of Mappings
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
Hence, as with
|
|
\begin_inset Formula $\SOthree$
|
|
\end_inset
|
|
|
|
, we are now in a position to simply posit the derivative of
|
|
\series bold
|
|
inverse
|
|
\series default
|
|
,
|
|
\begin_inset Formula \[
|
|
\frac{\partial T^{-1}}{\partial\xi}=\Ad T=-\left[\begin{array}{cc}
|
|
R & 0\\
|
|
\Skew tR & R\end{array}\right]\]
|
|
|
|
\end_inset
|
|
|
|
(but unit test on the above fails !!!),
|
|
\series bold
|
|
compose
|
|
\series default
|
|
in its first argument,
|
|
\begin_inset Formula \[
|
|
\frac{\partial\left(T_{1}T_{2}\right)}{\partial\xi_{1}}=\Ad{T_{2}^{-1}}=\left[\begin{array}{cc}
|
|
R_{2}^{T} & 0\\
|
|
\Skew{-R_{2}^{T}t_{2}}R_{2}^{T} & R_{2}^{T}\end{array}\right]=\left[\begin{array}{cc}
|
|
R_{2}^{T} & 0\\
|
|
R_{2}^{T}\Skew{-t_{2}} & R_{2}^{T}\end{array}\right]\]
|
|
|
|
\end_inset
|
|
|
|
compose in its second argument,
|
|
\begin_inset Formula \[
|
|
\frac{\partial\left(T_{1}T_{2}\right)}{\partial\xi_{2}}=I_{6}\]
|
|
|
|
\end_inset
|
|
|
|
|
|
\series bold
|
|
between
|
|
\series default
|
|
in its first argument,
|
|
\begin_inset Formula \[
|
|
\frac{\partial\left(T_{1}^{^{-1}}T_{2}\right)}{\partial\xi_{1}}=\Ad{T_{21}}=-\left[\begin{array}{cc}
|
|
R & 0\\
|
|
\Skew tR & R\end{array}\right]\]
|
|
|
|
\end_inset
|
|
|
|
with
|
|
\begin_inset Formula \[
|
|
T_{12}=\left[\begin{array}{cc}
|
|
R & t\\
|
|
0 & 1\end{array}\right]=T_{1}^{^{-1}}T_{2}=between(T_{2},T_{1})\]
|
|
|
|
\end_inset
|
|
|
|
and between in its second argument,
|
|
\begin_inset Formula \begin{eqnarray*}
|
|
\frac{\partial\left(T_{1}^{^{-1}}T_{2}\right)}{\partial\xi_{1}} & = & I_{6}\end{eqnarray*}
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Subsection
|
|
The derivatives of Actions
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
The action of
|
|
\begin_inset Formula $\SEthree$
|
|
\end_inset
|
|
|
|
on 3D points is done by embedding the points in
|
|
\begin_inset Formula $\mathbb{R}^{4}$
|
|
\end_inset
|
|
|
|
by using homogeneous coordinates
|
|
\begin_inset Formula \[
|
|
\hat{q}=\left[\begin{array}{c}
|
|
q\\
|
|
1\end{array}\right]=\left[\begin{array}{cc}
|
|
R & t\\
|
|
0 & 1\end{array}\right]\left[\begin{array}{c}
|
|
p\\
|
|
1\end{array}\right]=T\hat{p}\]
|
|
|
|
\end_inset
|
|
|
|
We would now like to know what an incremental rotation parameterized by
|
|
|
|
\begin_inset Formula $\xi$
|
|
\end_inset
|
|
|
|
would do:
|
|
\begin_inset Formula \[
|
|
\hat{q}(\xi)=Te^{\xihat}\hat{p}\]
|
|
|
|
\end_inset
|
|
|
|
hence the derivative (following the exposition in Section
|
|
\begin_inset CommandInset ref
|
|
LatexCommand ref
|
|
reference "sec:Derivatives-of-Actions"
|
|
|
|
\end_inset
|
|
|
|
):
|
|
\begin_inset Formula \[
|
|
\deriv{\hat{q}(\xi)}{\xi}=T\deriv{}{\xi}\left(\xihat\hat{p}\right)=TH_{p}\]
|
|
|
|
\end_inset
|
|
|
|
where
|
|
\begin_inset Formula $\xihat\hat{p}$
|
|
\end_inset
|
|
|
|
corresponds to a velocity in
|
|
\begin_inset Formula $\mathbb{R}^{4}$
|
|
\end_inset
|
|
|
|
(in the local
|
|
\begin_inset Formula $T$
|
|
\end_inset
|
|
|
|
frame):
|
|
\begin_inset Formula \[
|
|
\xihat\hat{p}=\left[\begin{array}{cc}
|
|
\Skew{\omega} & v\\
|
|
0 & 0\end{array}\right]\left[\begin{array}{c}
|
|
p\\
|
|
1\end{array}\right]=\left[\begin{array}{c}
|
|
\omega\times p+v\\
|
|
0\end{array}\right]\]
|
|
|
|
\end_inset
|
|
|
|
Notice how velocities are anologous to points at infinity in projective
|
|
geometry: they correspond to free vectors indicating a direction and magnitude
|
|
of change.
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
By only taking the top three rows, we can write this as a velocity in
|
|
\begin_inset Formula $\Rthree$
|
|
\end_inset
|
|
|
|
, as the product of a
|
|
\begin_inset Formula $3\times6$
|
|
\end_inset
|
|
|
|
matrix
|
|
\begin_inset Formula $H_{p}$
|
|
\end_inset
|
|
|
|
that acts upon the exponential coordinates
|
|
\begin_inset Formula $\xi$
|
|
\end_inset
|
|
|
|
directly:
|
|
\begin_inset Formula \[
|
|
\omega\times p+v=-p\times\omega+v=\left[\begin{array}{cc}
|
|
-\Skew p & I_{3}\end{array}\right]\left[\begin{array}{c}
|
|
\omega\\
|
|
v\end{array}\right]=H_{p}\xi\]
|
|
|
|
\end_inset
|
|
|
|
Hence, the final derivative of the group action is
|
|
\begin_inset Formula \[
|
|
\deriv{\hat{q}(\xi)}{\xi}=T\hat{H}_{p}=\left[\begin{array}{cc}
|
|
R & t\\
|
|
0 & 1\end{array}\right]\left[\begin{array}{cc}
|
|
\Skew{-p} & I_{3}\\
|
|
0 & 0\end{array}\right]\]
|
|
|
|
\end_inset
|
|
|
|
in homogenous coordinates.
|
|
In
|
|
\begin_inset Formula $\Rthree$
|
|
\end_inset
|
|
|
|
this becomes:
|
|
\begin_inset Formula \[
|
|
\deriv{q(\xi)}{\xi}=R\left[\begin{array}{cc}
|
|
-\Skew p & I_{3}\end{array}\right]\]
|
|
|
|
\end_inset
|
|
|
|
The derivative of the inverse action
|
|
\begin_inset Formula $T^{-1}p$
|
|
\end_inset
|
|
|
|
is given by formula
|
|
\begin_inset CommandInset ref
|
|
LatexCommand ref
|
|
reference "eq:inverseAction"
|
|
|
|
\end_inset
|
|
|
|
:
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
|
|
\family roman
|
|
\series medium
|
|
\shape up
|
|
\size normal
|
|
\emph off
|
|
\bar no
|
|
\noun off
|
|
\color none
|
|
\begin_inset Formula \[
|
|
\deriv{\hat{q}(\xi)}{\xi}=-H_{q}=\left[\begin{array}{cc}
|
|
\Skew q & -I_{3}\end{array}\right]\]
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Subsection
|
|
Pose3 (gtsam, old-style exmap)
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
In the old-style, we have
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
|
|
\family roman
|
|
\series medium
|
|
\shape up
|
|
\size normal
|
|
\emph off
|
|
\bar no
|
|
\noun off
|
|
\color none
|
|
\begin_inset Formula $R'=R(I+\Omega)$
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
\begin_inset Formula $t'=t+dt$
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
In this case, the derivative of
|
|
\series bold
|
|
\emph on
|
|
transform_from
|
|
\series default
|
|
\emph default
|
|
,
|
|
\begin_inset Formula $Rx+t$
|
|
\end_inset
|
|
|
|
:
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
\begin_inset Formula \[
|
|
\frac{\partial(R(I+\Omega)x+t)}{\partial\omega}=\frac{\partial(R\Omega x)}{\partial\omega}=\frac{\partial(R\left(\omega\times x\right))}{\partial\omega}=R\Skew{-x}\]
|
|
|
|
\end_inset
|
|
|
|
and with respect to
|
|
\begin_inset Formula $dt$
|
|
\end_inset
|
|
|
|
is easy:
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
\begin_inset Formula \[
|
|
\frac{\partial(Rx+t+dt)}{\partial dt}=I\]
|
|
|
|
\end_inset
|
|
|
|
The derivative of
|
|
\series bold
|
|
\emph on
|
|
transform_to
|
|
\series default
|
|
\emph default
|
|
,
|
|
\begin_inset Formula $inv(R)(x-t)$
|
|
\end_inset
|
|
|
|
we can obtain using the chain rule:
|
|
\begin_inset Formula \[
|
|
\frac{\partial(inv(R)(x-t))}{\partial\omega}=\frac{\partial unrot(R,(x-t))}{\partial\omega}=skew(R^{T}\left(x-t\right))\]
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
and with respect to
|
|
\begin_inset Formula $dt$
|
|
\end_inset
|
|
|
|
is easy:
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
\begin_inset Formula \[
|
|
\frac{\partial(R^{T}(x-t-dt))}{\partial dt}=-R^{T}\]
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
\begin_inset Newpage pagebreak
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Section
|
|
2D Line Segments (Ocaml)
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
The error between an infinite line
|
|
\begin_inset Formula $(a,b,c)$
|
|
\end_inset
|
|
|
|
and a 2D line segment
|
|
\begin_inset Formula $((x1,y1),(x2,y2))$
|
|
\end_inset
|
|
|
|
is defined in Line3.ml.
|
|
\end_layout
|
|
|
|
\begin_layout Section
|
|
Line3vd (Ocaml)
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
One representation of a line is through 2 vectors
|
|
\begin_inset Formula $(v,d)$
|
|
\end_inset
|
|
|
|
, where
|
|
\begin_inset Formula $v$
|
|
\end_inset
|
|
|
|
is the direction and the vector
|
|
\begin_inset Formula $d$
|
|
\end_inset
|
|
|
|
points from the orgin to the closest point on the line.
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
In this representation, transforming a 3D line from a world coordinate frame
|
|
to a camera at
|
|
\begin_inset Formula $(R_{w}^{c},t^{w})$
|
|
\end_inset
|
|
|
|
is done by
|
|
\begin_inset Formula \[
|
|
v^{c}=R_{w}^{c}v^{w}\]
|
|
|
|
\end_inset
|
|
|
|
|
|
\begin_inset Formula \[
|
|
d^{c}=R_{w}^{c}\left(d^{w}+(t^{w}v^{w})v^{w}-t^{w}\right)\]
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Section
|
|
Line3 (Ocaml)
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
For 3D lines, we use a parameterization due to C.J.
|
|
Taylor, using a rotation matrix
|
|
\begin_inset Formula $R$
|
|
\end_inset
|
|
|
|
and 2 scalars
|
|
\begin_inset Formula $a$
|
|
\end_inset
|
|
|
|
and
|
|
\begin_inset Formula $b$
|
|
\end_inset
|
|
|
|
.
|
|
The line direction
|
|
\begin_inset Formula $v$
|
|
\end_inset
|
|
|
|
is simply the Z-axis of the rotated frame, i.e.,
|
|
\begin_inset Formula $v=R_{3}$
|
|
\end_inset
|
|
|
|
, while the vector
|
|
\begin_inset Formula $d$
|
|
\end_inset
|
|
|
|
is given by
|
|
\begin_inset Formula $d=aR_{1}+bR_{2}$
|
|
\end_inset
|
|
|
|
.
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
Now, we will
|
|
\emph on
|
|
not
|
|
\emph default
|
|
use the incremental rotation scheme we used for rotations: because the
|
|
matrix R translates from the line coordinate frame to the world frame,
|
|
we need to apply the incremental rotation on the right-side:
|
|
\begin_inset Formula \[
|
|
R'=R(I+\Omega)\]
|
|
|
|
\end_inset
|
|
|
|
Projecting a line to 2D can be done easily, as both
|
|
\begin_inset Formula $v$
|
|
\end_inset
|
|
|
|
and
|
|
\begin_inset Formula $d$
|
|
\end_inset
|
|
|
|
are also the 2D homogenous coordinates of two points on the projected line,
|
|
and hence we have
|
|
\begin_inset Formula \begin{eqnarray*}
|
|
l & = & v\times d\\
|
|
& = & R_{3}\times\left(aR_{1}+bR_{2}\right)\\
|
|
& = & a\left(R_{3}\times R_{1}\right)+b\left(R_{3}\times R_{2}\right)\\
|
|
& = & aR_{2}-bR_{1}\end{eqnarray*}
|
|
|
|
\end_inset
|
|
|
|
This can be written as a rotation of a point,
|
|
\begin_inset Formula \[
|
|
l=R\left(\begin{array}{c}
|
|
-b\\
|
|
a\\
|
|
0\end{array}\right)\]
|
|
|
|
\end_inset
|
|
|
|
but because the incremental rotation is now done on the right, we need to
|
|
figure out the derivatives again:
|
|
\begin_inset Formula \begin{equation}
|
|
\frac{\partial(R(I+\Omega)x)}{\partial\omega}=\frac{\partial(R\Omega x)}{\partial\omega}=R\frac{\partial(\Omega x)}{\partial\omega}=R\Skew{-x}\label{eq:rotateRight}\end{equation}
|
|
|
|
\end_inset
|
|
|
|
and hence the derivative of the projection
|
|
\begin_inset Formula $l$
|
|
\end_inset
|
|
|
|
with respect to the rotation matrix
|
|
\begin_inset Formula $R$
|
|
\end_inset
|
|
|
|
of the 3D line is
|
|
\begin_inset Formula \begin{equation}
|
|
\frac{\partial(l)}{\partial\omega}=R\Skew{\left(\begin{array}{c}
|
|
b\\
|
|
-a\\
|
|
0\end{array}\right)}=\left[\begin{array}{ccc}
|
|
aR_{3} & bR_{3} & -(aR_{1}+bR_{2})\end{array}\right]\end{equation}
|
|
|
|
\end_inset
|
|
|
|
or the
|
|
\begin_inset Formula $a,b$
|
|
\end_inset
|
|
|
|
scalars:
|
|
\begin_inset Formula \[
|
|
\frac{\partial(l)}{\partial a}=R_{2}\]
|
|
|
|
\end_inset
|
|
|
|
|
|
\begin_inset Formula \[
|
|
\frac{\partial(l)}{\partial b}=-R_{1}\]
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
Transforming a 3D line
|
|
\begin_inset Formula $(R,(a,b))$
|
|
\end_inset
|
|
|
|
from a world coordinate frame to a camera frame
|
|
\begin_inset Formula $(R_{w}^{c},t^{w})$
|
|
\end_inset
|
|
|
|
is done by
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
\begin_inset Formula \[
|
|
R'=R_{w}^{c}R\]
|
|
|
|
\end_inset
|
|
|
|
|
|
\begin_inset Formula \[
|
|
a'=a-R_{1}^{T}t^{w}\]
|
|
|
|
\end_inset
|
|
|
|
|
|
\begin_inset Formula \[
|
|
b'=b-R_{2}^{T}t^{w}\]
|
|
|
|
\end_inset
|
|
|
|
Again, we need to redo the derivatives, as R is incremented from the right.
|
|
The first argument is incremented from the left, but the result is incremented
|
|
on the right:
|
|
\begin_inset Formula \begin{eqnarray*}
|
|
R'(I+\Omega')=(AB)(I+\Omega') & = & (I+\Skew{S\omega})AB\\
|
|
I+\Omega' & = & (AB)^{T}(I+\Skew{S\omega})(AB)\\
|
|
\Omega' & = & R'^{T}\Skew{S\omega}R'\\
|
|
\Omega' & = & \Skew{R'^{T}S\omega}\\
|
|
\omega' & = & R'^{T}S\omega\end{eqnarray*}
|
|
|
|
\end_inset
|
|
|
|
For the second argument
|
|
\begin_inset Formula $R$
|
|
\end_inset
|
|
|
|
we now simply have:
|
|
\begin_inset Formula \begin{eqnarray*}
|
|
AB(I+\Omega') & = & AB(I+\Omega)\\
|
|
\Omega' & = & \Omega\\
|
|
\omega' & = & \omega\end{eqnarray*}
|
|
|
|
\end_inset
|
|
|
|
The scalar derivatives can be found by realizing that
|
|
\begin_inset Formula \[
|
|
\left(\begin{array}{c}
|
|
a'\\
|
|
b'\\
|
|
...\end{array}\right)=\left(\begin{array}{c}
|
|
a\\
|
|
b\\
|
|
0\end{array}\right)-R^{T}t^{w}\]
|
|
|
|
\end_inset
|
|
|
|
where we don't care about the third row.
|
|
Hence
|
|
\begin_inset Formula \[
|
|
\frac{\partial(\left(R(I+\Omega_{2})\right)^{T}t^{w})}{\partial\omega}=-\frac{\partial(\Omega_{2}R^{T}t^{w})}{\partial\omega}=-\Skew{R^{T}t^{w}}=\left[\begin{array}{ccc}
|
|
0 & R_{3}^{T}t^{w} & -R_{2}^{T}t^{w}\\
|
|
-R_{3}^{T}t^{w} & 0 & R_{1}^{T}t^{w}\\
|
|
... & ... & 0\end{array}\right]\]
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Section
|
|
|
|
\series bold
|
|
Aligning 3D Scans
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
Below is the explanaition underlying Pose3.align, i.e.
|
|
aligning two point clouds using SVD.
|
|
Inspired but modified from CVOnline...
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
|
|
\emph on
|
|
Our
|
|
\emph default
|
|
model is
|
|
\begin_inset Formula \[
|
|
p^{c}=R\left(p^{w}-t\right)\]
|
|
|
|
\end_inset
|
|
|
|
i.e.,
|
|
\begin_inset Formula $R$
|
|
\end_inset
|
|
|
|
is from camera to world, and
|
|
\begin_inset Formula $t$
|
|
\end_inset
|
|
|
|
is the camera location in world coordinates.
|
|
The objective function is
|
|
\begin_inset Formula \begin{equation}
|
|
\frac{1}{2}\sum\left(p^{c}-R(p^{w}-t)\right)^{2}=\frac{1}{2}\sum\left(p^{c}-Rp^{w}+Rt\right)^{2}=\frac{1}{2}\sum\left(p^{c}-Rp^{w}-t'\right)^{2}\label{eq:J}\end{equation}
|
|
|
|
\end_inset
|
|
|
|
where
|
|
\begin_inset Formula $t'=-Rt$
|
|
\end_inset
|
|
|
|
is the location of the origin in the camera frame.
|
|
Taking the derivative with respect to
|
|
\begin_inset Formula $t'$
|
|
\end_inset
|
|
|
|
and setting to zero we have
|
|
\begin_inset Formula \[
|
|
\sum\left(p^{c}-Rp^{w}-t'\right)=0\]
|
|
|
|
\end_inset
|
|
|
|
or
|
|
\begin_inset Formula \begin{equation}
|
|
t'=\frac{1}{n}\sum\left(p^{c}-Rp^{w}\right)=\bar{p}^{c}-R\bar{p}^{w}\label{eq:t}\end{equation}
|
|
|
|
\end_inset
|
|
|
|
here
|
|
\begin_inset Formula $\bar{p}^{c}$
|
|
\end_inset
|
|
|
|
and
|
|
\begin_inset Formula $\bar{p}^{w}$
|
|
\end_inset
|
|
|
|
are the point cloud centroids.
|
|
Substituting back into
|
|
\begin_inset CommandInset ref
|
|
LatexCommand eqref
|
|
reference "eq:J"
|
|
|
|
\end_inset
|
|
|
|
, we get
|
|
\begin_inset Formula \[
|
|
\frac{1}{2}\sum\left(p^{c}-R(p^{w}-t)\right)^{2}=\frac{1}{2}\sum\left(\left(p^{c}-\bar{p}^{c}\right)-R\left(p^{w}-\bar{p}^{w}\right)\right)^{2}=\frac{1}{2}\sum\left(\hat{p}^{c}-R\hat{p}^{w}\right)^{2}\]
|
|
|
|
\end_inset
|
|
|
|
Now, to minimize the above it suffices to maximize (see CVOnline)
|
|
\begin_inset Formula \[
|
|
\mathop{trace}\left(R^{T}C\right)\]
|
|
|
|
\end_inset
|
|
|
|
where
|
|
\begin_inset Formula $C=\sum\hat{p}^{c}\left(\hat{p}^{w}\right)^{T}$
|
|
\end_inset
|
|
|
|
is the correlation matrix.
|
|
Intuitively, the cloud of points is rotated to align with the principal
|
|
axes.
|
|
This can be achieved by SVD decomposition on
|
|
\begin_inset Formula $C$
|
|
\end_inset
|
|
|
|
|
|
\begin_inset Formula \[
|
|
C=USV^{T}\]
|
|
|
|
\end_inset
|
|
|
|
and setting
|
|
\begin_inset Formula \[
|
|
R=UV^{T}\]
|
|
|
|
\end_inset
|
|
|
|
Clearly, from
|
|
\begin_inset CommandInset ref
|
|
LatexCommand eqref
|
|
reference "eq:t"
|
|
|
|
\end_inset
|
|
|
|
we then also recover the optimal
|
|
\begin_inset Formula $t$
|
|
\end_inset
|
|
|
|
as
|
|
\begin_inset Formula \[
|
|
t=\bar{p}^{w}-R^{T}\bar{p}^{c}\]
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
\begin_inset CommandInset bibtex
|
|
LatexCommand bibtex
|
|
bibfiles "/Users/dellaert/papers/refs"
|
|
options "plain"
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\end_body
|
|
\end_document
|