Added Sphere2 and EssentialMatrix to math.lyx
Frank Dellaert
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@ -1,120 +0,0 @@
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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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\begin_document
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\begin_header
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\textclass article
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\use_default_options true
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\maintain_unincluded_children false
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\language english
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\use_geometry true
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\use_amsmath 1
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\use_esint 1
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\cite_engine basic
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\use_refstyle 1
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\index Index
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\shortcut idx
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\color #008000
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\end_index
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\leftmargin 1in
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\topmargin 1in
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\rightmargin 1in
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\bottommargin 1in
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\secnumdepth 3
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\tocdepth 3
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\quotes_language english
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\end_header
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\begin_body
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\begin_layout Standard
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Derivative of EssentialMatrix epipolar error.
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\end_layout
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\begin_layout Standard
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With respect to orientation:
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\begin_inset Formula
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\[
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e(\omega)=a^{T}[t]_{\times}Re^{\omega}b=a^{T}Ee^{\omega}b
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\]
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\end_inset
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\begin_inset Formula
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\[
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\frac{\partial e(\omega)}{\partial v}=a^{T}E[b]_{\times}
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\]
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\end_inset
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\end_layout
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\begin_layout Standard
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With respect to tangent to sphere:
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\begin_inset Formula
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\[
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e(v)=a^{T}(Bv\times Rb)
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\]
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\end_inset
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\begin_inset Formula
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\[
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\frac{\partial e(v)}{\partial v}=a^{T}\frac{\partial(Bv\times Rb)}{\partial v}=a^{T}[-Rb]_{\times}B=a^{T}R[-b]_{\times}RB
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\]
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\end_inset
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\begin_inset Formula
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\[
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(1*3)(3*3)(3*2)
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\]
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\end_inset
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\end_layout
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\end_body
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\end_document
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@ -3037,10 +3037,10 @@ key "Murray94book"
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\color none
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\begin_inset Formula
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\[
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\exp\left(\left[\begin{array}{c}
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\exp\left(\widehat{\left[\begin{array}{c}
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\omega\\
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v
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\end{array}\right]t\right)=\left[\begin{array}{cc}
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\end{array}\right]}t\right)=\left[\begin{array}{cc}
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e^{\Skew{\omega}t} & (I-e^{\Skew{\omega}t})\left(\omega\times v\right)+\omega\omega^{T}vt\\
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0 & 1
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\end{array}\right]
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621
doc/math.lyx
621
doc/math.lyx
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@ -1594,7 +1594,7 @@ First, the derivative
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\begin_inset Formula $D_{2}f$
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\end_inset
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with respect to in
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with respect to
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\begin_inset Formula $p$
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\end_inset
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@ -1614,11 +1614,11 @@ For the derivative
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\begin_inset Formula $T$
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\end_inset
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, we want
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\end_layout
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\begin_layout Proof
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, we want to find the linear map
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\begin_inset Formula $D_{1}f$
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\end_inset
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such that
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\family roman
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\series medium
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\shape up
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@ -1630,9 +1630,10 @@ For the derivative
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\uwave off
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\noun off
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\color none
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\begin_inset Formula
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\[
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f(Te^{\hat{\xi}},p)=Te^{\hat{\xi}}p\approx Tp+D_{1}f(\xi)
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Tp+D_{1}f(\xi)\approx f(Te^{\hat{\xi}},p)=Te^{\hat{\xi}}p
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\]
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\end_inset
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@ -1661,7 +1662,12 @@ Te^{\hat{\xi}}p\approx T(I+\hat{\xi})p=Tp+T\hat{\xi}p
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\end_inset
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and
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\begin_inset Formula $D_{1}f(\xi)=T\hat{\xi}p$
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\end_inset
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.
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\begin_inset Note Note
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status collapsed
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@ -1679,7 +1685,7 @@ T\hat{\xi}p=\left(T\hat{\xi}T^{-1}\right)Tp=\left(\Ad T\xihat\right)\left(Tp\rig
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\end_inset
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Hence, we need to show that
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Hence, to complete the proof, we need to show that
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\begin_inset Formula
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\begin{equation}
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\xihat p=H(p)\xi\label{eq:Hp}
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@ -4173,9 +4179,9 @@ so
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.
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Hence, the final derivative of an action in its first argument is
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\begin_inset Formula
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\[
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\deriv{\left(Rp\right)}{\omega}=RH(p)=-R\Skew p
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\]
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\begin{equation}
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\deriv{\left(Rp\right)}{\omega}=RH(p)=-R\Skew p\label{eq:Rot3action}
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\end{equation}
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\end_inset
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@ -5027,6 +5033,601 @@ Re^{\Skew{\omega}} & t+R\left[v+\left(\omega\times v\right)/2\right]\\
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\end_inset
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\end_layout
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\begin_layout Section
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The Sphere
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\begin_inset Formula $S^{2}$
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\end_inset
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\end_layout
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\begin_layout Subsection
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Definitions
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\end_layout
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\begin_layout Standard
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The sphere
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\begin_inset Formula $S^{2}$
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\end_inset
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is the set of all unit vectors in
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\begin_inset Formula $\Rthree$
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\end_inset
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, i.e., all directions in three-space:
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\begin_inset Formula
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\[
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S^{2}=\{p\in\Rthree|\left\Vert p\right\Vert =1\}
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\]
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\end_inset
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The tangent space
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\begin_inset Formula $T_{p}S^{2}$
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\end_inset
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at a point
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\begin_inset Formula $p$
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\end_inset
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consists of three-vectors
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\begin_inset Formula $\xihat$
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\end_inset
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such that
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\begin_inset Formula $\xihat$
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\end_inset
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is tangent to
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\begin_inset Formula $S^{2}$
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\end_inset
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at
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\begin_inset Formula $p$
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\end_inset
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, i.e.,
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\begin_inset Formula
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\[
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T_{p}S^{2}\define\left\{ \xihat\in\Rthree|p^{T}\xihat=0\right\}
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\]
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\end_inset
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While not a Lie group, we can define an exponential map, which is given
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in Ma et.
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al
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\begin_inset CommandInset citation
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LatexCommand cite
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key "Ma01ijcv"
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\end_inset
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, as well as in this CVPR tutorial by Anuj Srivastava:
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\begin_inset CommandInset href
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LatexCommand href
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name "http://stat.fsu.edu/~anuj/CVPR_Tutorial/Part2.pdf"
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\end_inset
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.
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\begin_inset Formula
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\[
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\exp_{p}\xihat=\cos\left(\left\Vert \xihat\right\Vert \right)p+\sin\left(\left\Vert \xihat\right\Vert \right)\frac{\xihat}{\left\Vert \xihat\right\Vert }
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\]
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\end_inset
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The latter also gives the inverse, i.e., get the tangent vector
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\begin_inset Formula $z$
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\end_inset
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to go from
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\begin_inset Formula $p$
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\end_inset
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to
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\begin_inset Formula $q$
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\end_inset
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:
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\begin_inset Formula
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\[
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z=\log_{p}q=\frac{\theta}{\sin\theta}\left(q-p\cos\theta\right)p
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\]
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\end_inset
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with
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\begin_inset Formula $\theta=\cos^{-1}\left(p^{T}q\right)$
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\end_inset
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.
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\end_layout
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\begin_layout Subsection
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Local Coordinates
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\end_layout
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\begin_layout Standard
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We can find a basis
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\begin_inset Formula $B_{p}$
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\end_inset
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for the tangent space
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\begin_inset Formula $T_{p}S^{2}$
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\end_inset
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, with
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\begin_inset Formula $B_{p}=\left[b_{1}|b_{2}\right]$
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\end_inset
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a
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\begin_inset Formula $3\times2$
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\end_inset
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matrix, by either
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\end_layout
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\begin_layout Enumerate
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Decompose
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\begin_inset Formula $p=QR$
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\end_inset
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, with
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\begin_inset Formula $Q$
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\end_inset
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orthonormal and
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\begin_inset Formula $R$
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\end_inset
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of the form
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\begin_inset Formula $[1\,0\,0]^{T}$
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\end_inset
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, and hence
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\begin_inset Formula $p=Q_{1}$
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\end_inset
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.
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The basis
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\begin_inset Formula $B_{p}=\left[Q_{2}|Q_{3}\right]$
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\end_inset
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, i.e., the last two columns of
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\begin_inset Formula $Q$
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\end_inset
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.
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\end_layout
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\begin_layout Enumerate
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Form
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\begin_inset Formula $b_{1}=p\times a$
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\end_inset
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, with
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\begin_inset Formula $a$
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\end_inset
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(consistently) chosen to be non-parallel to
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\begin_inset Formula $p$
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\end_inset
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, and
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\begin_inset Formula $b_{2}=p\times b_{1}$
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\end_inset
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.
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\begin_inset Note Note
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status collapsed
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\begin_layout Plain Layout
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To choose
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\begin_inset Formula $a$
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\end_inset
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, one way is to divide the sphere into regions, e.g., pick the axis
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\begin_inset Formula $e_{i}$
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\end_inset
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such that
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\begin_inset Formula $e_{i}^{T}p$
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\end_inset
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is smallest.
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However, that leads to discontinuous boundaries.
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Since
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\begin_inset Formula $0\leq\left|e_{i}^{T}p\right|\leq1$
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\end_inset
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for all
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\begin_inset Formula $p\in S^{2}$
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\end_inset
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, a better idea might be to use a mixture, e.g.,
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\begin_inset Formula
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\[
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a=\frac{1}{2(x^{2}+y^{2}+z^{2})}\left[\begin{array}{c}
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y^{2}+z^{2}\\
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x^{2}+z^{2}\\
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x^{2}+y^{2}
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\end{array}\right]
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\]
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\end_inset
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\end_layout
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\end_inset
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\end_layout
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\begin_layout Standard
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Now we can write
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\begin_inset Formula $\xihat=B_{p}\xi$
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\end_inset
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with
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\begin_inset Formula $\xi\in R^{2}$
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\end_inset
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the 2D coordinate in the tangent plane basis
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\begin_inset Formula $B_{p}$
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\end_inset
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.
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\end_layout
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\begin_layout Subsection
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Retraction
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\end_layout
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\begin_layout Standard
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The exponential map uses
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\begin_inset Formula $\cos$
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\end_inset
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and
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\begin_inset Formula $\sin$
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\end_inset
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, and is more than we need for optimization.
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Suppose we have a point
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\begin_inset Formula $p\in S^{2}$
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\end_inset
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and a 3-vector
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\begin_inset Formula $\xihat\in T_{p}S^{2}$
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\end_inset
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, Absil
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\begin_inset CommandInset citation
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LatexCommand cite
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key "Absil07book"
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\end_inset
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tells us we can simply add
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\begin_inset Formula $\xihat$
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\end_inset
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to
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\begin_inset Formula $p$
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\end_inset
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and renormalize to get a new point
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\begin_inset Formula $q$
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\end_inset
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on the sphere.
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This is what he calls a
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\series bold
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retraction
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\family roman
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\series medium
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\shape up
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\size normal
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\emph off
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\bar no
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\strikeout off
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\uuline off
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\uwave off
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\noun off
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\color none
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\begin_inset Formula $\retract_{p}(\xihat)$
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\end_inset
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,
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\family default
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\series default
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\shape default
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\size default
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\emph default
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\bar default
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\strikeout default
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\uuline default
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\uwave default
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\noun default
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\color inherit
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\begin_inset Formula
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\[
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q=\retract_{p}(\xihat)=\frac{p+\xihat}{\left\Vert p+z\right\Vert }=\frac{p+\xihat}{\alpha}
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\]
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\end_inset
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with
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\begin_inset Formula $\alpha$
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\end_inset
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the norm of
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\begin_inset Formula $p+\xihat$
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\end_inset
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.
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\end_layout
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\begin_layout Standard
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We can also define a retraction from local coordinates
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\begin_inset Formula $\xi\in\Rtwo$
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\end_inset
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:
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\begin_inset Formula
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\[
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q=\retract_{p}(\xi)=\frac{p+B_{p}\xi}{\left\Vert p+B_{p}\xi\right\Vert }=\frac{p+B_{p}\xi}{\alpha}
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\]
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\end_inset
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\end_layout
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\begin_layout Standard
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Since
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\begin_inset Formula $\xihat$
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\end_inset
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is in the tangent space
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\begin_inset Formula $T_{p}S^{2}$
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\end_inset
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at
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\begin_inset Formula $p$
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\end_inset
|
||||
|
||||
, we have
|
||||
\begin_inset Formula $p^{T}\xihat=0$
|
||||
\end_inset
|
||||
|
||||
.
|
||||
|
||||
\end_layout
|
||||
|
||||
\begin_layout Subsubsection*
|
||||
Inverse Retraction
|
||||
\end_layout
|
||||
|
||||
\begin_layout Standard
|
||||
If
|
||||
\begin_inset Formula $\xihat=B_{p}\xi$
|
||||
\end_inset
|
||||
|
||||
with
|
||||
\begin_inset Formula $\xi\in R^{2}$
|
||||
\end_inset
|
||||
|
||||
the 2D coordinate in the tangent plane basis
|
||||
\begin_inset Formula $B_{p}$
|
||||
\end_inset
|
||||
|
||||
, we have
|
||||
\begin_inset Formula
|
||||
\[
|
||||
\xi=\frac{B^{T}q}{p^{T}q}
|
||||
\]
|
||||
|
||||
\end_inset
|
||||
|
||||
|
||||
\end_layout
|
||||
|
||||
\begin_layout Proof
|
||||
We seek
|
||||
\begin_inset Formula
|
||||
\[
|
||||
\alpha q=p+B_{p}\xi
|
||||
\]
|
||||
|
||||
\end_inset
|
||||
|
||||
If we multiply both sides with
|
||||
\begin_inset Formula $B_{p}^{T}$
|
||||
\end_inset
|
||||
|
||||
(project on the basis
|
||||
\begin_inset Formula $B_{p}$
|
||||
\end_inset
|
||||
|
||||
) we obtain
|
||||
\begin_inset Formula
|
||||
\[
|
||||
\alpha B_{p}^{T}q=B_{p}^{T}p+B_{p}^{T}B_{p}\xi
|
||||
\]
|
||||
|
||||
\end_inset
|
||||
|
||||
and because
|
||||
\begin_inset Formula $B_{p}^{T}p=0$
|
||||
\end_inset
|
||||
|
||||
and
|
||||
\begin_inset Formula $B_{p}^{T}B_{p}=I$
|
||||
\end_inset
|
||||
|
||||
we trivially obtain
|
||||
\begin_inset Formula $\xi$
|
||||
\end_inset
|
||||
|
||||
as the scaled projection
|
||||
\begin_inset Formula $B^{T}q$
|
||||
\end_inset
|
||||
|
||||
:
|
||||
\begin_inset Formula
|
||||
\[
|
||||
\xi=\alpha B^{T}q
|
||||
\]
|
||||
|
||||
\end_inset
|
||||
|
||||
To recover the scale factor
|
||||
\begin_inset Formula $\alpha$
|
||||
\end_inset
|
||||
|
||||
we multiply with
|
||||
\begin_inset Formula $p^{T}$
|
||||
\end_inset
|
||||
|
||||
on both sides we have
|
||||
\begin_inset Formula
|
||||
\[
|
||||
\alpha p^{T}q=p^{T}p+p^{T}B_{p}\xi
|
||||
\]
|
||||
|
||||
\end_inset
|
||||
|
||||
and (since
|
||||
\begin_inset Formula $p^{T}p=1$
|
||||
\end_inset
|
||||
|
||||
and
|
||||
\begin_inset Formula $p^{T}B_{p}\xi=0$
|
||||
\end_inset
|
||||
|
||||
) we have
|
||||
\begin_inset Formula $\alpha=1/(p^{T}q)$
|
||||
\end_inset
|
||||
|
||||
.
|
||||
\end_layout
|
||||
|
||||
\begin_layout Section
|
||||
The Essential Matrix Manifold
|
||||
\end_layout
|
||||
|
||||
\begin_layout Standard
|
||||
We parameterize essential matrices as a pair
|
||||
\begin_inset Formula $(R,t)$
|
||||
\end_inset
|
||||
|
||||
, where
|
||||
\begin_inset Formula $R\in\SOthree$
|
||||
\end_inset
|
||||
|
||||
and
|
||||
\begin_inset Formula $t\in S^{2}$
|
||||
\end_inset
|
||||
|
||||
, the unit sphere.
|
||||
The epipolar matrix is then given by
|
||||
\begin_inset Formula
|
||||
\[
|
||||
E=\Skew tR
|
||||
\]
|
||||
|
||||
\end_inset
|
||||
|
||||
and the epipolar error given two corresponding points
|
||||
\begin_inset Formula $a$
|
||||
\end_inset
|
||||
|
||||
and
|
||||
\begin_inset Formula $b$
|
||||
\end_inset
|
||||
|
||||
is
|
||||
\begin_inset Formula
|
||||
\[
|
||||
e(R,t;a,b)=a^{T}Eb
|
||||
\]
|
||||
|
||||
\end_inset
|
||||
|
||||
We are of course interested in the derivative with respect to orientation
|
||||
(using
|
||||
\begin_inset CommandInset ref
|
||||
LatexCommand ref
|
||||
reference "eq:Rot3action"
|
||||
|
||||
\end_inset
|
||||
|
||||
)
|
||||
\begin_inset Formula
|
||||
\[
|
||||
\frac{\partial(a^{T}[t]_{\times}Rb)}{\partial\omega}=a^{T}[t]_{\times}\frac{\partial(Rb)}{\partial\omega}=-a^{T}[t]_{\times}R\Skew b=-a^{T}E[b]_{\times}
|
||||
\]
|
||||
|
||||
\end_inset
|
||||
|
||||
and with respect to change in the direction
|
||||
\begin_inset Formula $t$
|
||||
\end_inset
|
||||
|
||||
|
||||
\begin_inset Formula
|
||||
\[
|
||||
\frac{\partial e(a^{T}[t]_{\times}Rb)}{\partial\xi}=a^{T}\frac{\partial(B\xi\times Rb)}{\partial v}=-a^{T}[Rb]_{\times}B
|
||||
\]
|
||||
|
||||
\end_inset
|
||||
|
||||
where we made use of the fact that the retraction can be written as
|
||||
\begin_inset Formula $t+B\xi$
|
||||
\end_inset
|
||||
|
||||
, with
|
||||
\begin_inset Formula $B$
|
||||
\end_inset
|
||||
|
||||
a local basis, and we made use of
|
||||
\begin_inset CommandInset ref
|
||||
LatexCommand eqref
|
||||
reference "eq:Dcross1"
|
||||
|
||||
\end_inset
|
||||
|
||||
:
|
||||
\family roman
|
||||
\series medium
|
||||
\shape up
|
||||
\size normal
|
||||
\emph off
|
||||
\bar no
|
||||
\strikeout off
|
||||
\uuline off
|
||||
\uwave off
|
||||
\noun off
|
||||
\color none
|
||||
|
||||
\begin_inset Formula
|
||||
\[
|
||||
\frac{\partial(a\times b)}{\partial a}=\Skew{-b}
|
||||
\]
|
||||
|
||||
\end_inset
|
||||
|
||||
|
||||
\end_layout
|
||||
|
||||
\begin_layout Section
|
||||
|
|
|
|||
BIN
doc/math.pdf
BIN
doc/math.pdf
Binary file not shown.
475
doc/sphere.lyx
475
doc/sphere.lyx
|
|
@ -1,475 +0,0 @@
|
|||
#LyX 2.0 created this file. For more info see http://www.lyx.org/
|
||||
\lyxformat 413
|
||||
\begin_document
|
||||
\begin_header
|
||||
\textclass article
|
||||
\use_default_options true
|
||||
\maintain_unincluded_children false
|
||||
\language english
|
||||
\language_package default
|
||||
\inputencoding auto
|
||||
\fontencoding global
|
||||
\font_roman default
|
||||
\font_sans default
|
||||
\font_typewriter default
|
||||
\font_default_family default
|
||||
\use_non_tex_fonts false
|
||||
\font_sc false
|
||||
\font_osf false
|
||||
\font_sf_scale 100
|
||||
\font_tt_scale 100
|
||||
|
||||
\graphics default
|
||||
\default_output_format default
|
||||
\output_sync 0
|
||||
\bibtex_command default
|
||||
\index_command default
|
||||
\paperfontsize 11
|
||||
\spacing single
|
||||
\use_hyperref false
|
||||
\papersize default
|
||||
\use_geometry true
|
||||
\use_amsmath 1
|
||||
\use_esint 1
|
||||
\use_mhchem 1
|
||||
\use_mathdots 1
|
||||
\cite_engine basic
|
||||
\use_bibtopic false
|
||||
\use_indices false
|
||||
\paperorientation portrait
|
||||
\suppress_date false
|
||||
\use_refstyle 1
|
||||
\index Index
|
||||
\shortcut idx
|
||||
\color #008000
|
||||
\end_index
|
||||
\leftmargin 3cm
|
||||
\topmargin 3cm
|
||||
\rightmargin 3cm
|
||||
\bottommargin 3cm
|
||||
\secnumdepth 3
|
||||
\tocdepth 3
|
||||
\paragraph_separation indent
|
||||
\paragraph_indentation default
|
||||
\quotes_language english
|
||||
\papercolumns 1
|
||||
\papersides 1
|
||||
\paperpagestyle default
|
||||
\tracking_changes false
|
||||
\output_changes false
|
||||
\html_math_output 0
|
||||
\html_css_as_file 0
|
||||
\html_be_strict false
|
||||
\end_header
|
||||
|
||||
\begin_body
|
||||
|
||||
\begin_layout Title
|
||||
Manifold Geometry of the Sphere
|
||||
\begin_inset Formula $S^{2}$
|
||||
\end_inset
|
||||
|
||||
|
||||
\end_layout
|
||||
|
||||
\begin_layout Author
|
||||
Frank, Can, and Manohar
|
||||
\end_layout
|
||||
|
||||
\begin_layout Standard
|
||||
\begin_inset FormulaMacro
|
||||
\newcommand{\xihat}{z}
|
||||
{z}
|
||||
\end_inset
|
||||
|
||||
|
||||
\end_layout
|
||||
|
||||
\begin_layout Subsubsection*
|
||||
Retraction
|
||||
\end_layout
|
||||
|
||||
\begin_layout Standard
|
||||
Suppose we have a point
|
||||
\begin_inset Formula $p\in S^{2}$
|
||||
\end_inset
|
||||
|
||||
and a 3-vector
|
||||
\begin_inset Formula $\xihat$
|
||||
\end_inset
|
||||
|
||||
, Absil
|
||||
\begin_inset CommandInset citation
|
||||
LatexCommand cite
|
||||
key "Absil07book"
|
||||
|
||||
\end_inset
|
||||
|
||||
tells us we can simply add
|
||||
\begin_inset Formula $\xihat$
|
||||
\end_inset
|
||||
|
||||
to
|
||||
\begin_inset Formula $p$
|
||||
\end_inset
|
||||
|
||||
and renormalize to get a new point
|
||||
\begin_inset Formula $q$
|
||||
\end_inset
|
||||
|
||||
on the sphere.
|
||||
This is what he calls a
|
||||
\series bold
|
||||
retraction
|
||||
\family roman
|
||||
\series medium
|
||||
\shape up
|
||||
\size normal
|
||||
\emph off
|
||||
\bar no
|
||||
\strikeout off
|
||||
\uuline off
|
||||
\uwave off
|
||||
\noun off
|
||||
\color none
|
||||
|
||||
\begin_inset Formula $R_{p}(\xihat)$
|
||||
\end_inset
|
||||
|
||||
,
|
||||
\family default
|
||||
\series default
|
||||
\shape default
|
||||
\size default
|
||||
\emph default
|
||||
\bar default
|
||||
\strikeout default
|
||||
\uuline default
|
||||
\uwave default
|
||||
\noun default
|
||||
\color inherit
|
||||
|
||||
\begin_inset Formula
|
||||
\[
|
||||
q=R_{p}(\xihat)=\frac{p+\xihat}{\left\Vert p+z\right\Vert }=\frac{p+\xihat}{\alpha}
|
||||
\]
|
||||
|
||||
\end_inset
|
||||
|
||||
with
|
||||
\begin_inset Formula $\alpha$
|
||||
\end_inset
|
||||
|
||||
the norm of
|
||||
\begin_inset Formula $p+\xihat$
|
||||
\end_inset
|
||||
|
||||
.
|
||||
The only restriction on
|
||||
\begin_inset Formula $\xihat$
|
||||
\end_inset
|
||||
|
||||
is that it is in the tangent space
|
||||
\begin_inset Formula $T_{p}S^{2}$
|
||||
\end_inset
|
||||
|
||||
at
|
||||
\begin_inset Formula $p$
|
||||
\end_inset
|
||||
|
||||
, i.e.,
|
||||
\begin_inset Formula $p^{T}\xihat=0$
|
||||
\end_inset
|
||||
|
||||
.
|
||||
Multiplying with
|
||||
\begin_inset Formula $p^{T}$
|
||||
\end_inset
|
||||
|
||||
on both sides we have
|
||||
\begin_inset Formula
|
||||
\[
|
||||
\alpha p^{T}q=p^{T}p+p^{T}\xihat
|
||||
\]
|
||||
|
||||
\end_inset
|
||||
|
||||
and (since
|
||||
\begin_inset Formula $p^{T}p=1$
|
||||
\end_inset
|
||||
|
||||
and
|
||||
\begin_inset Formula $p^{T}\xihat=0$
|
||||
\end_inset
|
||||
|
||||
) we have
|
||||
\begin_inset Formula $\alpha=1/(p^{T}q)$
|
||||
\end_inset
|
||||
|
||||
.
|
||||
\end_layout
|
||||
|
||||
\begin_layout Subsubsection*
|
||||
Inverse
|
||||
\end_layout
|
||||
|
||||
\begin_layout Standard
|
||||
Suppose we are given points
|
||||
\begin_inset Formula $p$
|
||||
\end_inset
|
||||
|
||||
and
|
||||
\begin_inset Formula $q$
|
||||
\end_inset
|
||||
|
||||
on the sphere, what is the tangent vector
|
||||
\begin_inset Formula $\xihat$
|
||||
\end_inset
|
||||
|
||||
that takes
|
||||
\begin_inset Formula $p$
|
||||
\end_inset
|
||||
|
||||
to
|
||||
\begin_inset Formula $q$
|
||||
\end_inset
|
||||
|
||||
? We can find a basis
|
||||
\begin_inset Formula $B$
|
||||
\end_inset
|
||||
|
||||
for the tangent space, with
|
||||
\begin_inset Formula $B=\left[b_{1}|b_{2}\right]$
|
||||
\end_inset
|
||||
|
||||
a
|
||||
\begin_inset Formula $3\times2$
|
||||
\end_inset
|
||||
|
||||
matrix, by either
|
||||
\end_layout
|
||||
|
||||
\begin_layout Enumerate
|
||||
Decompose
|
||||
\begin_inset Formula $p=QR$
|
||||
\end_inset
|
||||
|
||||
, with
|
||||
\begin_inset Formula $Q$
|
||||
\end_inset
|
||||
|
||||
orthonormal and
|
||||
\begin_inset Formula $R$
|
||||
\end_inset
|
||||
|
||||
of the form
|
||||
\begin_inset Formula $[1\,0\,0]^{T}$
|
||||
\end_inset
|
||||
|
||||
, and hence
|
||||
\begin_inset Formula $p=Q_{1}$
|
||||
\end_inset
|
||||
|
||||
.
|
||||
The basis
|
||||
\begin_inset Formula $B=\left[Q_{2}|Q_{3}\right]$
|
||||
\end_inset
|
||||
|
||||
, i.e., the last two columns of
|
||||
\begin_inset Formula $Q$
|
||||
\end_inset
|
||||
|
||||
.
|
||||
\end_layout
|
||||
|
||||
\begin_layout Enumerate
|
||||
Form
|
||||
\begin_inset Formula $b_{1}=p\times a$
|
||||
\end_inset
|
||||
|
||||
, with
|
||||
\begin_inset Formula $a$
|
||||
\end_inset
|
||||
|
||||
(consistently) chosen to be non-parallel to
|
||||
\begin_inset Formula $p$
|
||||
\end_inset
|
||||
|
||||
, and
|
||||
\begin_inset Formula $b_{2}=p\times b_{1}$
|
||||
\end_inset
|
||||
|
||||
.
|
||||
|
||||
\begin_inset Note Note
|
||||
status collapsed
|
||||
|
||||
\begin_layout Plain Layout
|
||||
To choose
|
||||
\begin_inset Formula $a$
|
||||
\end_inset
|
||||
|
||||
, one way is to divide the sphere into regions, e.g., pick the axis
|
||||
\begin_inset Formula $e_{i}$
|
||||
\end_inset
|
||||
|
||||
such that
|
||||
\begin_inset Formula $e_{i}^{T}p$
|
||||
\end_inset
|
||||
|
||||
is smallest.
|
||||
However, that leads to discontinuous boundaries.
|
||||
Since
|
||||
\begin_inset Formula $0\leq\left|e_{i}^{T}p\right|\leq1$
|
||||
\end_inset
|
||||
|
||||
for all
|
||||
\begin_inset Formula $p\in S^{2}$
|
||||
\end_inset
|
||||
|
||||
, a better idea might be to use a mixture, e.g.,
|
||||
\begin_inset Formula
|
||||
\[
|
||||
a=\frac{1}{2(x^{2}+y^{2}+z^{2})}\left[\begin{array}{c}
|
||||
y^{2}+z^{2}\\
|
||||
x^{2}+z^{2}\\
|
||||
x^{2}+y^{2}
|
||||
\end{array}\right]
|
||||
\]
|
||||
|
||||
\end_inset
|
||||
|
||||
|
||||
\end_layout
|
||||
|
||||
\end_inset
|
||||
|
||||
|
||||
\end_layout
|
||||
|
||||
\begin_layout Standard
|
||||
Now, if
|
||||
\begin_inset Formula $\xihat=B\xi$
|
||||
\end_inset
|
||||
|
||||
with
|
||||
\begin_inset Formula $\xi\in R^{2}$
|
||||
\end_inset
|
||||
|
||||
the 2D coordinate in the tangent plane basis
|
||||
\begin_inset Formula $B$
|
||||
\end_inset
|
||||
|
||||
, we have
|
||||
\begin_inset Formula
|
||||
\[
|
||||
\alpha q=p+\xihat=p+B\xi
|
||||
\]
|
||||
|
||||
\end_inset
|
||||
|
||||
If we multiply both sides with
|
||||
\begin_inset Formula $B^{T}$
|
||||
\end_inset
|
||||
|
||||
(project on the basis
|
||||
\begin_inset Formula $B$
|
||||
\end_inset
|
||||
|
||||
) we obtain
|
||||
\begin_inset Formula
|
||||
\[
|
||||
\alpha B^{T}q=B^{T}p+B^{T}B\xi
|
||||
\]
|
||||
|
||||
\end_inset
|
||||
|
||||
and because
|
||||
\begin_inset Formula $B^{T}p=0$
|
||||
\end_inset
|
||||
|
||||
and
|
||||
\begin_inset Formula $B^{T}B=I$
|
||||
\end_inset
|
||||
|
||||
we trivially obtain
|
||||
\begin_inset Formula $\xi$
|
||||
\end_inset
|
||||
|
||||
as the scaled projection
|
||||
\begin_inset Formula $B^{T}q$
|
||||
\end_inset
|
||||
|
||||
:
|
||||
\begin_inset Formula
|
||||
\[
|
||||
\xi=\alpha B^{T}q=\frac{B^{T}q}{p^{T}q}
|
||||
\]
|
||||
|
||||
\end_inset
|
||||
|
||||
|
||||
\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
|
||||
bibfiles "../../../papers/refs"
|
||||
options "plain"
|
||||
|
||||
\end_inset
|
||||
|
||||
|
||||
\end_layout
|
||||
|
||||
\end_body
|
||||
\end_document
|
||||
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Reference in New Issue