Added Sim3
dellaert
parent
a056086ea4
commit
6e1879a60d
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@ -1,5 +1,5 @@
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#LyX 2.0 created this file. For more info see http://www.lyx.org/
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\lyxformat 413
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#LyX 2.1 created this file. For more info see http://www.lyx.org/
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\lyxformat 474
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\begin_document
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\begin_header
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\textclass article
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@ -15,13 +15,13 @@ theorems-std
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@ -32,15 +32,24 @@ theorems-std
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\use_hyperref false
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\papersize default
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\use_geometry true
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\use_amsmath 1
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\use_mhchem 1
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\use_package amsmath 1
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\use_package amssymb 1
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\use_package mathtools 1
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\index Index
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\shortcut idx
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@ -175,7 +184,7 @@ status collapsed
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\end_inset
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\begin_inset Caption
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\begin_inset Caption Standard
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\begin_layout Plain Layout
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Robot moving along a circular trajectory.
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@ -255,7 +264,7 @@ status open
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\end_inset
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\begin_inset Caption
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\begin_inset Caption Standard
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\begin_layout Plain Layout
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\begin_inset CommandInset label
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@ -2944,6 +2953,218 @@ p\\
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\end_inset
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\end_layout
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\begin_layout Section
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3D Similarity Transformations
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\end_layout
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\begin_layout Standard
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The group of 3D similarity transformations
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\begin_inset Formula $Sim(3)$
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\end_inset
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is the set of
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\begin_inset Formula $4\times4$
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\end_inset
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invertible matrices of the form
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\begin_inset Formula
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\[
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T\define\left[\begin{array}{cc}
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R & t\\
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0 & s^{-1}
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\end{array}\right]
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\]
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\end_inset
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where
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\begin_inset Formula $s$
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\end_inset
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is a scalar.
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There are several different conventions in use for the Lie algebra generators,
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but we use
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\begin_inset Formula
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\[
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G^{1}=\left(\begin{array}{cccc}
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0 & 0 & 0 & 0\\
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0 & 0 & -1 & 0\\
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0 & 1 & 0 & 0\\
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0 & 0 & 0 & 0
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\end{array}\right)\mbox{}G^{2}=\left(\begin{array}{cccc}
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0 & 0 & 1 & 0\\
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0 & 0 & 0 & 0\\
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-1 & 0 & 0 & 0\\
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0 & 0 & 0 & 0
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\end{array}\right)\mbox{ }G^{3}=\left(\begin{array}{cccc}
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0 & -1 & 0 & 0\\
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1 & 0 & 0 & 0\\
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0 & 0 & 0 & 0\\
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0 & 0 & 0 & 0
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\end{array}\right)
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\]
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\end_inset
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\begin_inset Formula
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\[
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G^{4}=\left(\begin{array}{cccc}
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0 & 0 & 0 & 1\\
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0 & 0 & 0 & 0\\
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0 & 0 & 0 & 0\\
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0 & 0 & 0 & 0
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\end{array}\right)\mbox{}G^{5}=\left(\begin{array}{cccc}
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0 & 0 & 0 & 0\\
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0 & 0 & 0 & 1\\
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0 & 0 & 0 & 0\\
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0 & 0 & 0 & 0
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\end{array}\right)\mbox{ }G^{6}=\left(\begin{array}{cccc}
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0 & 0 & 0 & 0\\
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0 & 0 & 0 & 0\\
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0 & 0 & 0 & 1\\
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0 & 0 & 0 & 0
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\end{array}\right)\mbox{ }G^{7}=\left(\begin{array}{cccc}
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0 & 0 & 0 & 0\\
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0 & 0 & 0 & 0\\
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0 & 0 & 0 & 0\\
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0 & 0 & 0 & -1
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\end{array}\right)
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\]
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\end_inset
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\end_layout
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\begin_layout Subsection
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Actions
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\end_layout
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\begin_layout Standard
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The action of
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\begin_inset Formula $\SEthree$
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\end_inset
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on 3D points is done by embedding the points in
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\begin_inset Formula $\mathbb{R}^{4}$
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\end_inset
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by using homogeneous coordinates
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\begin_inset Formula
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\[
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\hat{q}=\left[\begin{array}{c}
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q\\
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s^{-1}
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\end{array}\right]=\left[\begin{array}{c}
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Rp+t\\
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s^{-1}
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\end{array}\right]=\left[\begin{array}{cc}
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R & t\\
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0 & s^{-1}
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\end{array}\right]\left[\begin{array}{c}
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p\\
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1
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\end{array}\right]=T\hat{p}
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\]
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\end_inset
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The derivative
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\begin_inset Formula $D_{1}f(\xi)$
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\end_inset
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in an incremental change
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\begin_inset Formula $\xi$
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\end_inset
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to
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\begin_inset Formula $T$
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\end_inset
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is given by
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\begin_inset Formula $TH(p)$
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\end_inset
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where
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\begin_inset Formula
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\[
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H(p)=G_{jk}^{i}p^{j}=\left(\begin{array}{ccccccc}
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0 & z & -y & 1 & 0 & 0 & 0\\
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-z & 0 & x & 0 & 1 & 0 & 0\\
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y & -x & 0 & 0 & 0 & 1 & 0\\
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0 & 0 & 0 & 0 & 0 & 0 & -1
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\end{array}\right)
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\]
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\end_inset
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In other words
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\begin_inset Formula
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\[
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D_{1}f(\xi)=\left[\begin{array}{cc}
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R & t\\
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0 & s^{-1}
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\end{array}\right]\left[\begin{array}{ccc}
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-\left[p\right]_{x} & I_{3} & 0\\
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0 & 0 & -1
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\end{array}\right]=\left[\begin{array}{ccc}
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-R\left[p\right]_{x} & R & -t\\
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0 & 0 & -s^{-1}
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\end{array}\right]
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\]
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\end_inset
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This is the derivative for the action on homogeneous coordinates.
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Switching back to non-homogeneous coordinates is done by
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\begin_inset Formula
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\[
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\left[\begin{array}{c}
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q\\
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a
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\end{array}\right]\rightarrow q/a
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\]
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\end_inset
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with derivative
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\begin_inset Formula
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\[
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\left[\begin{array}{cc}
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a^{-1}I_{3} & -qa^{-2}\end{array}\right]
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\]
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\end_inset
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For
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\begin_inset Formula $a=s^{-1}$
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\end_inset
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we obtain
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\begin_inset Formula
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\[
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D_{1}f(\xi)=\left[\begin{array}{cc}
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sI_{3} & -qs^{2}\end{array}\right]\left[\begin{array}{ccc}
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-R\left[p\right]_{x} & R & -t\\
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0 & 0 & -s^{-1}
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\end{array}\right]=\left[\begin{array}{ccc}
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-sR\left[p\right]_{x} & sR & -st+qs\end{array}\right]=\left[\begin{array}{ccc}
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-sR\left[p\right]_{x} & sR & sRp\end{array}\right]
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\]
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\end_inset
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\end_layout
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\begin_layout Standard
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\begin_inset Newpage pagebreak
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\end_inset
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\end_layout
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\begin_layout Section
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