From cf219c3a1be06191a9a2b510be979391834f4d4e Mon Sep 17 00:00:00 2001 From: Frank Dellaert Date: Tue, 17 Dec 2013 15:18:58 +0000 Subject: [PATCH] Added Sphere2 and EssentialMatrix to math.lyx --- doc/EssentialMatrix.lyx | 120 -------- doc/LieGroups.lyx | 4 +- doc/math.lyx | 621 +++++++++++++++++++++++++++++++++++++++- doc/math.pdf | Bin 173687 -> 140339 bytes doc/sphere.lyx | 475 ------------------------------ doc/sphere.pdf | Bin 44587 -> 0 bytes 6 files changed, 613 insertions(+), 607 deletions(-) delete mode 100644 doc/EssentialMatrix.lyx delete mode 100644 doc/sphere.lyx delete mode 100644 doc/sphere.pdf diff --git a/doc/EssentialMatrix.lyx b/doc/EssentialMatrix.lyx deleted file mode 100644 index 77855f6c2..000000000 --- a/doc/EssentialMatrix.lyx +++ /dev/null @@ -1,120 +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 1in -\topmargin 1in -\rightmargin 1in -\bottommargin 1in -\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 Standard -Derivative of EssentialMatrix epipolar error. -\end_layout - -\begin_layout Standard -With respect to orientation: -\begin_inset Formula -\[ -e(\omega)=a^{T}[t]_{\times}Re^{\omega}b=a^{T}Ee^{\omega}b -\] - -\end_inset - - -\begin_inset Formula -\[ -\frac{\partial e(\omega)}{\partial v}=a^{T}E[b]_{\times} -\] - -\end_inset - - -\end_layout - -\begin_layout Standard -With respect to tangent to sphere: -\begin_inset Formula -\[ -e(v)=a^{T}(Bv\times Rb) -\] - -\end_inset - - -\begin_inset Formula -\[ -\frac{\partial e(v)}{\partial v}=a^{T}\frac{\partial(Bv\times Rb)}{\partial v}=a^{T}[-Rb]_{\times}B=a^{T}R[-b]_{\times}RB -\] - -\end_inset - - -\begin_inset Formula -\[ -(1*3)(3*3)(3*2) -\] - -\end_inset - - -\end_layout - -\end_body -\end_document diff --git a/doc/LieGroups.lyx b/doc/LieGroups.lyx index 4af039252..adf62314c 100644 --- a/doc/LieGroups.lyx +++ b/doc/LieGroups.lyx @@ -3037,10 +3037,10 @@ key "Murray94book" \color none \begin_inset Formula \[ -\exp\left(\left[\begin{array}{c} +\exp\left(\widehat{\left[\begin{array}{c} \omega\\ v -\end{array}\right]t\right)=\left[\begin{array}{cc} +\end{array}\right]}t\right)=\left[\begin{array}{cc} e^{\Skew{\omega}t} & (I-e^{\Skew{\omega}t})\left(\omega\times v\right)+\omega\omega^{T}vt\\ 0 & 1 \end{array}\right] diff --git a/doc/math.lyx b/doc/math.lyx index 8d56963cb..98b1f0b15 100644 --- a/doc/math.lyx +++ b/doc/math.lyx @@ -1594,7 +1594,7 @@ First, the derivative \begin_inset Formula $D_{2}f$ \end_inset - with respect to in + with respect to \begin_inset Formula $p$ \end_inset @@ -1614,11 +1614,11 @@ For the derivative \begin_inset Formula $T$ \end_inset -, we want -\end_layout - -\begin_layout Proof +, we want to find the linear map +\begin_inset Formula $D_{1}f$ +\end_inset + such that \family roman \series medium \shape up @@ -1630,9 +1630,10 @@ For the derivative \uwave off \noun off \color none + \begin_inset Formula \[ -f(Te^{\hat{\xi}},p)=Te^{\hat{\xi}}p\approx Tp+D_{1}f(\xi) +Tp+D_{1}f(\xi)\approx f(Te^{\hat{\xi}},p)=Te^{\hat{\xi}}p \] \end_inset @@ -1661,7 +1662,12 @@ Te^{\hat{\xi}}p\approx T(I+\hat{\xi})p=Tp+T\hat{\xi}p \end_inset +and +\begin_inset Formula $D_{1}f(\xi)=T\hat{\xi}p$ +\end_inset +. + \begin_inset Note Note status collapsed @@ -1679,7 +1685,7 @@ T\hat{\xi}p=\left(T\hat{\xi}T^{-1}\right)Tp=\left(\Ad T\xihat\right)\left(Tp\rig \end_inset -Hence, we need to show that +Hence, to complete the proof, we need to show that \begin_inset Formula \begin{equation} \xihat p=H(p)\xi\label{eq:Hp} @@ -4173,9 +4179,9 @@ so . Hence, the final derivative of an action in its first argument is \begin_inset Formula -\[ -\deriv{\left(Rp\right)}{\omega}=RH(p)=-R\Skew p -\] +\begin{equation} +\deriv{\left(Rp\right)}{\omega}=RH(p)=-R\Skew p\label{eq:Rot3action} +\end{equation} \end_inset @@ -5027,6 +5033,601 @@ Re^{\Skew{\omega}} & t+R\left[v+\left(\omega\times v\right)/2\right]\\ \end_inset +\end_layout + +\begin_layout Section +The Sphere +\begin_inset Formula $S^{2}$ +\end_inset + + +\end_layout + +\begin_layout Subsection +Definitions +\end_layout + +\begin_layout Standard +The sphere +\begin_inset Formula $S^{2}$ +\end_inset + + is the set of all unit vectors in +\begin_inset Formula $\Rthree$ +\end_inset + +, i.e., all directions in three-space: +\begin_inset Formula +\[ +S^{2}=\{p\in\Rthree|\left\Vert p\right\Vert =1\} +\] + +\end_inset + +The tangent space +\begin_inset Formula $T_{p}S^{2}$ +\end_inset + + at a point +\begin_inset Formula $p$ +\end_inset + + consists of three-vectors +\begin_inset Formula $\xihat$ +\end_inset + + such that +\begin_inset Formula $\xihat$ +\end_inset + + is tangent to +\begin_inset Formula $S^{2}$ +\end_inset + + at +\begin_inset Formula $p$ +\end_inset + +, i.e., +\begin_inset Formula +\[ +T_{p}S^{2}\define\left\{ \xihat\in\Rthree|p^{T}\xihat=0\right\} +\] + +\end_inset + +While not a Lie group, we can define an exponential map, which is given + in Ma et. + al +\begin_inset CommandInset citation +LatexCommand cite +key "Ma01ijcv" + +\end_inset + +, 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 Subsection +Local Coordinates +\end_layout + +\begin_layout Standard +We can find a basis +\begin_inset Formula $B_{p}$ +\end_inset + + for the tangent space +\begin_inset Formula $T_{p}S^{2}$ +\end_inset + +, with +\begin_inset Formula $B_{p}=\left[b_{1}|b_{2}\right]$ +\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 $B_{p}=\left[Q_{2}|Q_{3}\right]$ +\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 we can write +\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 + +. +\end_layout + +\begin_layout Subsection +Retraction +\end_layout + +\begin_layout Standard +The exponential map uses +\begin_inset Formula $\cos$ +\end_inset + + and +\begin_inset Formula $\sin$ +\end_inset + +, and is more than we need for optimization. + Suppose we have a point +\begin_inset Formula $p\in S^{2}$ +\end_inset + + and a 3-vector +\begin_inset Formula $\xihat\in T_{p}S^{2}$ +\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 $\retract_{p}(\xihat)$ +\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 +\[ +q=\retract_{p}(\xihat)=\frac{p+\xihat}{\left\Vert p+z\right\Vert }=\frac{p+\xihat}{\alpha} +\] + +\end_inset + +with +\begin_inset Formula $\alpha$ +\end_inset + + the norm of +\begin_inset Formula $p+\xihat$ +\end_inset + +. +\end_layout + +\begin_layout Standard +We can also define a retraction from local coordinates +\begin_inset Formula $\xi\in\Rtwo$ +\end_inset + +: +\begin_inset Formula +\[ +q=\retract_{p}(\xi)=\frac{p+B_{p}\xi}{\left\Vert p+B_{p}\xi\right\Vert }=\frac{p+B_{p}\xi}{\alpha} +\] + +\end_inset + + +\end_layout + +\begin_layout Standard +Since +\begin_inset Formula $\xihat$ +\end_inset + + is in the tangent space +\begin_inset Formula $T_{p}S^{2}$ +\end_inset + + at +\begin_inset Formula $p$ +\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 diff --git a/doc/math.pdf b/doc/math.pdf index 9b1046f68c4dccaa64b02e2792dbae722fc233a2..6589af06a5a71b0c1551933c8dfb428f4f6ea8eb 100644 GIT binary patch literal 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