From 39b4090b21532166c5979ebc7c21967a85aa8a67 Mon Sep 17 00:00:00 2001
From: Richard Roberts <richard.jw.roberts@gmail.com>
Date: Fri, 21 Feb 2014 15:47:51 -0500
Subject: [PATCH] Moved coordinate frames document into GTSAM

---
 doc/gtsam-coordinate-frames.lyx | 2527 +++++++++++++++++++++++++++++++
 doc/gtsam-coordinate-frames.pdf |  Bin 0 -> 80806 bytes
 2 files changed, 2527 insertions(+)
 create mode 100644 doc/gtsam-coordinate-frames.lyx
 create mode 100644 doc/gtsam-coordinate-frames.pdf

diff --git a/doc/gtsam-coordinate-frames.lyx b/doc/gtsam-coordinate-frames.lyx
new file mode 100644
index 000000000..33d0dd977
--- /dev/null
+++ b/doc/gtsam-coordinate-frames.lyx
@@ -0,0 +1,2527 @@
+#LyX 2.0 created this file. For more info see http://www.lyx.org/
+\lyxformat 413
+\begin_document
+\begin_header
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+
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+
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+{\mathbb{SE}\left(#1\right)}
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+
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+\begin_inset FormulaMacro
+\newcommand{\se}[1]{\mathfrak{se}\left(#1\right)}
+{\mathfrak{se}\left(#1\right)}
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+
+\end_layout
+
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+\begin_inset FormulaMacro
+\newcommand{\R}[1]{\mathbb{R}^{#1}}
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+
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+
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+\begin_inset FormulaMacro
+\newcommand{\norm}[1]{\left\Vert #1\right\Vert }
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+\begin_inset FormulaMacro
+\newcommand{\lin}[1]{\overset{{\scriptscriptstyle \circ}}{#1}}
+{\overset{{\scriptscriptstyle \circ}}{#1}}
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Introduction
+\end_layout
+
+\begin_layout Standard
+This document describes the coordinate frame conventions in which GTSAM
+ inputs and represents states and uncertainties.
+ When specifying initial conditions, measurements and their uncertainties,
+ and interpreting estimated uncertainties and the results of geometry operations
+, the coordinate frame convention comes into play.
+\end_layout
+
+\begin_layout Standard
+GTSAM as consistently as possible represents all states and uncertainties
+ in the body frame.
+ In cases where several frames are used simultaneously, a good rule of thumb
+ is that measurements and uncertainties will be represented in the 
+\begin_inset Quotes eld
+\end_inset
+
+last
+\begin_inset Quotes erd
+\end_inset
+
+ frame of the series.
+\end_layout
+
+\begin_layout Section
+Frame Conventions in Geometry, Lie Group, and Manifold Operations
+\end_layout
+
+\begin_layout Standard
+\begin_inset Float table
+wide false
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+
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+
+\begin_inset Newline newline
+\end_inset
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+
+\begin_inset Formula $a=\exp\hat{\delta}$
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+
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+\begin_inset Formula $a.\mathbf{logmap}()==\delta$
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+\begin_inset Text
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+\begin_layout Plain Layout
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+\begin_inset Formula $q=a.\mathbf{transform\_to}\left(p\right)$
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+<cell alignment="left" valignment="top" topline="true" usebox="none">
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+\size footnotesize
+\begin_inset Formula $a$
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+
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+
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+<cell alignment="left" valignment="top" topline="true" usebox="none">
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+
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+
+\end_layout
+
+\begin_layout Plain Layout
+\begin_inset Caption
+
+\begin_layout Plain Layout
+\begin_inset CommandInset label
+LatexCommand label
+name "tab:Coordinate-frame-transformations"
+
+\end_inset
+
+Coordinate frame transformations performed by GTSAM geometry operations.
+ Here, 
+\begin_inset Formula $a$
+\end_inset
+
+, 
+\begin_inset Formula $b$
+\end_inset
+
+, 
+\begin_inset Formula $c$
+\end_inset
+
+, and 
+\begin_inset Formula $g$
+\end_inset
+
+ are Lie group elements (Pose2, Pose3, Rot2, Rot3, Point2, Point3, 
+\emph on
+etc
+\emph default
+).
+ 
+\begin_inset Formula $\delta$
+\end_inset
+
+ is a set of Lie algebra coordinates (i.e.
+ linear update, linear delta, tangent space coordinates), and 
+\begin_inset Formula $\mathrm{X}$
+\end_inset
+
+ is a Lie group type (e.g.
+ Pose2).
+ 
+\begin_inset Formula $p$
+\end_inset
+
+ and 
+\begin_inset Formula $q$
+\end_inset
+
+ are the objects of Lie group actions (Point2, Point3, 
+\emph on
+etc
+\emph default
+).
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+At the core of most coordinate frame usage in GTSAM are geometry and Lie
+ group operations.
+ We explain the geometry and Lie group operations in GTSAM in terms of the
+ coordinate frame transformations they perform, detailed in Table
+\begin_inset space ~
+\end_inset
+
+
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "tab:Coordinate-frame-transformations"
+
+\end_inset
+
+.
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+\begin_inset Text
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+\begin_inset Formula $\delta=a.\mathbf{localCoordinates}\left(b\right)$
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+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="left" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\size footnotesize
+\begin_inset Formula $a.\mathbf{retract}\left(\delta\right)==b$
+\end_inset
+
+
+\begin_inset Newline newline
+\end_inset
+
+
+\begin_inset Formula $\mathbf{I}.\mathbf{localCoordinates}($
+\end_inset
+
+
+\begin_inset Newline newline
+\end_inset
+
+
+\begin_inset Formula $\quad a.\mathbf{between}\left(b\right))==\delta$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="left" valignment="top" bottomline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\size footnotesize
+\begin_inset Formula $b=a.\mathbf{retract}\left(\delta\right)$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="left" valignment="top" bottomline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\size footnotesize
+\begin_inset Formula $a$
+\end_inset
+
+ in 
+\begin_inset Formula $g$
+\end_inset
+
+
+\begin_inset Newline newline
+\end_inset
+
+
+\begin_inset Formula $\delta$
+\end_inset
+
+ in 
+\begin_inset Formula $a$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="left" valignment="top" bottomline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\size footnotesize
+\begin_inset Formula $b$
+\end_inset
+
+ in 
+\begin_inset Formula $g$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="left" valignment="top" bottomline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\size footnotesize
+\begin_inset Formula $a.\mathbf{compose}($
+\end_inset
+
+
+\begin_inset Newline newline
+\end_inset
+
+
+\begin_inset Formula $\quad\mathbf{I}.\mathbf{retract}\left(\delta\right))==b$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+</lyxtabular>
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Plain Layout
+\begin_inset Caption
+
+\begin_layout Plain Layout
+\begin_inset CommandInset label
+LatexCommand label
+name "tab:Coordinate-frames-manifold"
+
+\end_inset
+
+Coordinate frames for manifold tangent space operations.
+ Here, 
+\begin_inset Formula $a$
+\end_inset
+
+, 
+\begin_inset Formula $b$
+\end_inset
+
+, and 
+\begin_inset Formula $g$
+\end_inset
+
+ are manifold elements, 
+\begin_inset Formula $\delta$
+\end_inset
+
+ is a tangent space element, and 
+\begin_inset Formula $\mathrm{X}$
+\end_inset
+
+ is a Lie group type (e.g.
+ Pose2).
+ For the identities column, we assume the elements are also Lie group elements
+ with identity 
+\begin_inset Formula $\mathbf{I}$
+\end_inset
+
+.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+The manifold tangent space operations 
+\begin_inset Quotes eld
+\end_inset
+
+retract
+\begin_inset Quotes erd
+\end_inset
+
+ and 
+\begin_inset Quotes eld
+\end_inset
+
+local coordinates
+\begin_inset Quotes erd
+\end_inset
+
+ also work in the body frame for Lie group elements.
+ The tangent space coordinates given to 
+\begin_inset Quotes eld
+\end_inset
+
+retract
+\begin_inset Quotes erd
+\end_inset
+
+ should be in the body frame, not the global frame.
+ Similarly, the tangent space coordinates returned by 
+\begin_inset Quotes eld
+\end_inset
+
+local coordinates
+\begin_inset Quotes erd
+\end_inset
+
+ will be in the same body frame.
+ This is detailed in Table
+\begin_inset space ~
+\end_inset
+
+
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "tab:Coordinate-frames-manifold"
+
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Section
+Frame and Uncertainty Conventions For Built-in Factors
+\end_layout
+
+\begin_layout Standard
+\begin_inset Float table
+wide false
+sideways false
+status open
+
+\begin_layout Plain Layout
+\align center
+
+\size footnotesize
+\begin_inset Tabular
+<lyxtabular version="3" rows="7" columns="5">
+<features tabularvalignment="middle">
+<column alignment="left" valignment="top" width="2.5cm">
+<column alignment="left" valignment="top" width="2.75cm">
+<column alignment="left" valignment="top" width="1.25cm">
+<column alignment="left" valignment="top" width="1.75cm">
+<column alignment="left" valignment="top" width="1.75cm">
+<row>
+<cell alignment="left" valignment="top" topline="true" bottomline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\size footnotesize
+Name
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="left" valignment="top" topline="true" bottomline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\size footnotesize
+Residual
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="left" valignment="top" topline="true" bottomline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\size footnotesize
+Variables
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="left" valignment="top" topline="true" bottomline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\size footnotesize
+Measurement
+\end_layout
+
+\begin_layout Plain Layout
+
+\size footnotesize
+(
+\begin_inset Formula $z$
+\end_inset
+
+)
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="left" valignment="top" topline="true" bottomline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\size footnotesize
+Measurement Uncertainty
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="left" valignment="top" topline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\series bold
+\size footnotesize
+PriorFactor
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="left" valignment="top" topline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\size footnotesize
+\begin_inset Formula $z.\mathrm{localCoordinates}\left(x\right)$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="left" valignment="top" topline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\size footnotesize
+\begin_inset Formula $x$
+\end_inset
+
+ in 
+\begin_inset Formula $g$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="left" valignment="top" topline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\size footnotesize
+Ideal 
+\begin_inset Formula $x$
+\end_inset
+
+ in 
+\begin_inset Formula $g$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="left" valignment="top" topline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\size footnotesize
+In 
+\begin_inset Formula $z$
+\end_inset
+
+ / In 
+\begin_inset Formula $x$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="left" valignment="top" topline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\series bold
+\size footnotesize
+BetweenFactor
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="left" valignment="top" topline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\size footnotesize
+\begin_inset Formula $z.\mathrm{localCoordinates}($
+\end_inset
+
+
+\begin_inset Newline newline
+\end_inset
+
+
+\begin_inset Formula $\; x.\mathrm{between}\left(y\right))$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="left" valignment="top" topline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\size footnotesize
+\begin_inset Formula $x$
+\end_inset
+
+ in 
+\begin_inset Formula $g$
+\end_inset
+
+
+\begin_inset Newline newline
+\end_inset
+
+
+\begin_inset Formula $y$
+\end_inset
+
+ in 
+\begin_inset Formula $g$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="left" valignment="top" topline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\size footnotesize
+Ideal 
+\begin_inset Formula $y$
+\end_inset
+
+ in 
+\begin_inset Formula $x$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="left" valignment="top" topline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\size footnotesize
+In 
+\begin_inset Formula $z$
+\end_inset
+
+ / In 
+\begin_inset Formula $y$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="left" valignment="top" topline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\series bold
+\size footnotesize
+RangeFactor
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="left" valignment="top" topline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\size footnotesize
+\begin_inset Formula $x.\mathrm{range}\left(y\right)-z$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\size footnotesize
+\begin_inset Formula $x$
+\end_inset
+
+ in 
+\begin_inset Formula $g$
+\end_inset
+
+
+\begin_inset Newline newline
+\end_inset
+
+
+\begin_inset Formula $y$
+\end_inset
+
+ in 
+\begin_inset Formula $g$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="left" valignment="top" topline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\size footnotesize
+Euclidean distance
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="left" valignment="top" topline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\size footnotesize
+In 
+\begin_inset Formula $z$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="left" valignment="top" topline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\series bold
+\size footnotesize
+BearingFactor
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="left" valignment="top" topline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\size footnotesize
+\begin_inset Formula $z.\mathrm{localCoordinates}($
+\end_inset
+
+
+\begin_inset Newline newline
+\end_inset
+
+
+\begin_inset Formula $\; x.\mathrm{bearing}\left(y\right))$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\size footnotesize
+\begin_inset Formula $x$
+\end_inset
+
+ in 
+\begin_inset Formula $g$
+\end_inset
+
+
+\begin_inset Newline newline
+\end_inset
+
+
+\begin_inset Formula $y$
+\end_inset
+
+ in 
+\begin_inset Formula $g$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="left" valignment="top" topline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\size footnotesize
+Bearing of 
+\begin_inset Formula $y$
+\end_inset
+
+ position in frame 
+\begin_inset Formula $x$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="left" valignment="top" topline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\size footnotesize
+In 
+\begin_inset Formula $z$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="left" valignment="top" topline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\series bold
+\size footnotesize
+GenericProjection
+\begin_inset Newline newline
+\end_inset
+
+
+\begin_inset space ~
+\end_inset
+
+
+\begin_inset space ~
+\end_inset
+
+Factor
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="left" valignment="top" topline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\size footnotesize
+\begin_inset Formula $K^{-1}\left(P\left(x^{-1}p\right)\right)-z$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\size footnotesize
+\begin_inset Formula $x$
+\end_inset
+
+ in 
+\begin_inset Formula $g$
+\end_inset
+
+
+\begin_inset Newline newline
+\end_inset
+
+
+\begin_inset Formula $p$
+\end_inset
+
+ in 
+\begin_inset Formula $g$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="left" valignment="top" topline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\size footnotesize
+Perspective projection of 
+\begin_inset Formula $p$
+\end_inset
+
+ in 
+\begin_inset Formula $x$
+\end_inset
+
+.
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="left" valignment="top" topline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\size footnotesize
+In 
+\begin_inset Formula $z$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="left" valignment="top" topline="true" bottomline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\series bold
+\size footnotesize
+GeneralSFM
+\begin_inset Newline newline
+\end_inset
+
+
+\begin_inset space ~
+\end_inset
+
+
+\begin_inset space ~
+\end_inset
+
+Factor
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="left" valignment="top" topline="true" bottomline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\size footnotesize
+\begin_inset Formula $K^{-1}\left(P\left(x^{-1}p\right)\right)-z$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\size footnotesize
+\begin_inset Formula $x$
+\end_inset
+
+ in 
+\begin_inset Formula $g$
+\end_inset
+
+
+\begin_inset Newline newline
+\end_inset
+
+
+\begin_inset Formula $p$
+\end_inset
+
+ in 
+\begin_inset Formula $g$
+\end_inset
+
+
+\begin_inset Newline newline
+\end_inset
+
+Parameters
+\begin_inset Newline newline
+\end_inset
+
+
+\begin_inset space ~
+\end_inset
+
+
+\begin_inset space ~
+\end_inset
+
+of 
+\begin_inset Formula $K$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="left" valignment="top" topline="true" bottomline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\size footnotesize
+Perspective projection of 
+\begin_inset Formula $p$
+\end_inset
+
+ in 
+\begin_inset Formula $x$
+\end_inset
+
+.
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="left" valignment="top" topline="true" bottomline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\size footnotesize
+In 
+\begin_inset Formula $z$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+</lyxtabular>
+
+\end_inset
+
+
+\size default
+
+\begin_inset Caption
+
+\begin_layout Plain Layout
+Measurement functions and coordinate frames of factors provided with GTSAM.
+ To simplify notation, 
+\begin_inset Formula $K$
+\end_inset
+
+ is a camera calibration function converting pixels to normalized image
+ coordinates, and 
+\begin_inset Formula $P$
+\end_inset
+
+ is the pinhole projection function.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+All built-in GTSAM factors follow a consistent coordinate frame convention
+ (though fundamentally how a measurement and its uncertainty are specified
+ depends on the measurement model described by a factor).
+ In all built-in GTSAM factors, the 
+\emph on
+noise model
+\emph default
+, i.e.
+ the measurement uncertainty, should be specified in the coordinate frame
+ of the measurement itself.
+ This is part of a convention in GTSAM that tangent-space quantities (like
+ Gaussian noise models and update delta vectors) are always in the coordinate
+ frame of the element owning the tangent space.
+\end_layout
+
+\begin_layout Subsection
+PriorFactor
+\end_layout
+
+\begin_layout Standard
+A PriorFactor is a simple unary prior.
+ It encodes a direct measurement of the value of a variable 
+\begin_inset Formula $x$
+\end_inset
+
+, with the specified mean 
+\begin_inset Formula $z$
+\end_inset
+
+ and uncertainty, such that 
+\begin_inset Formula $z.\mathrm{between}\left(x\right)$
+\end_inset
+
+ is distributed according to the specified noise model.
+ From this definition and the definition of 
+\series bold
+between
+\series default
+ in Table
+\begin_inset space ~
+\end_inset
+
+
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "tab:Coordinate-frame-transformations"
+
+\end_inset
+
+, the measurement itself should be specified in the frame with respect to
+ which 
+\begin_inset Formula $x$
+\end_inset
+
+ is specified, while the uncertainty is specified in the coordinate frame
+ of the measurement, or equivalently, in frame 
+\begin_inset Formula $x$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Subsection
+BetweenFactor
+\end_layout
+
+\begin_layout Standard
+A BetweenFactor is a measurement on the relative transformation between
+ two variables.
+ A BetweenFactor on variables 
+\begin_inset Formula $x$
+\end_inset
+
+ and 
+\begin_inset Formula $y$
+\end_inset
+
+ with measurement 
+\begin_inset Formula $z$
+\end_inset
+
+ implies that 
+\begin_inset Formula $z.\mathrm{between}\left(x.\mathrm{between}\left(y\right)\right)$
+\end_inset
+
+ is distributed according to the specified noise model.
+ This definition, along with that of 
+\series bold
+between
+\series default
+ in Table
+\begin_inset space ~
+\end_inset
+
+
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "tab:Coordinate-frame-transformations"
+
+\end_inset
+
+, implies that the measurement is in frame 
+\begin_inset Formula $x$
+\end_inset
+
+, i.e.
+ it measures 
+\begin_inset Formula $y$
+\end_inset
+
+ in 
+\begin_inset Formula $x$
+\end_inset
+
+, and that the uncertainty is in the frame of the measurement, or equivalently,
+ in frame 
+\begin_inset Formula $y$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Subsection
+RangeFactor
+\end_layout
+
+\begin_layout Standard
+A RangeFactor measures the Euclidean distance either between two poses,
+ a pose and a point, or two points.
+ The range is a scalar, specified to be distributed according to the specified
+ noise model.
+\end_layout
+
+\begin_layout Subsection
+BearingFactor
+\end_layout
+
+\begin_layout Standard
+A BearingFactor measures the bearing (angle) of the 
+\emph on
+position
+\emph default
+ of a pose or point 
+\begin_inset Formula $y$
+\end_inset
+
+ as observed from a pose 
+\begin_inset Formula $x$
+\end_inset
+
+.
+ The orientation of 
+\begin_inset Formula $x$
+\end_inset
+
+ affects the measurement prediction.
+ Though, if 
+\begin_inset Formula $y$
+\end_inset
+
+ is a pose, it's orientation does not matter.
+ The noise model specifies the distribution of the bearing, in radians.
+\end_layout
+
+\begin_layout Subsection
+GenericProjectionFactor
+\end_layout
+
+\begin_layout Standard
+A GenericProjectionFactor measures the pixel coordinates of a landmark 
+\begin_inset Formula $p$
+\end_inset
+
+ projected into a camera 
+\begin_inset Formula $x$
+\end_inset
+
+ with the calibration function 
+\begin_inset Formula $K$
+\end_inset
+
+ that converts pixels to normalized image coordinates.
+ The measurement 
+\begin_inset Formula $z$
+\end_inset
+
+ is specified in real pixel coordinates (thanks to the 
+\begin_inset Quotes eld
+\end_inset
+
+uncalibration
+\begin_inset Quotes erd
+\end_inset
+
+ function 
+\begin_inset Formula $K^{-1}$
+\end_inset
+
+ used in the residual).
+ In a GenericProjectionFactor, the calibration is fixed.
+ On the other hand, GeneralSFMFactor allows the calibration parameters to
+ be optimized as variables.
+\end_layout
+
+\begin_layout Subsection
+GeneralSFMFactor
+\end_layout
+
+\begin_layout Standard
+A GeneralSFMFactor is the same as a GenericProjectionFactor except that
+ a GeneralSFMFactor also allows the parameters of the calibration function
+ 
+\begin_inset Formula $K$
+\end_inset
+
+ to be optimized as variables, instead of having them fixed.
+ A GeneralSFMFactor measures the pixel coordinates of a landmark 
+\begin_inset Formula $p$
+\end_inset
+
+ projected into a camera 
+\begin_inset Formula $x$
+\end_inset
+
+ with the calibration function 
+\begin_inset Formula $K$
+\end_inset
+
+ that converts pixels to normalized image coordinates.
+ The measurement 
+\begin_inset Formula $z$
+\end_inset
+
+ is specified in real pixel coordinates (thanks to the 
+\begin_inset Quotes eld
+\end_inset
+
+uncalibration
+\begin_inset Quotes erd
+\end_inset
+
+ function 
+\begin_inset Formula $K^{-1}$
+\end_inset
+
+ used in the residual).
+\end_layout
+
+\begin_layout Standard
+\begin_inset Note Note
+status collapsed
+
+\begin_layout Section
+Noise models of prior factors
+\end_layout
+
+\begin_layout Plain Layout
+The simplest way to describe noise models is by an example.
+ Let's take a prior factor on a 3D pose 
+\begin_inset Formula $x\in\SE 3$
+\end_inset
+
+, 
+\family typewriter
+Pose3
+\family default
+ in GTSAM.
+ Let 
+\begin_inset Formula $z\in\SE 3$
+\end_inset
+
+ be the expected pose, i.e.
+ the zero-error solution for the prior factor.
+ The 
+\emph on
+unwhitened error
+\emph default
+ (the error vector not accounting for the noise model) is
+\begin_inset Formula 
+\[
+h\left(x\right)=\log\left(z^{-1}x\right)\text{,}
+\]
+
+\end_inset
+
+where 
+\begin_inset Formula $\cdot^{-1}$
+\end_inset
+
+ is the Lie group inverse and 
+\begin_inset Formula $\log\cdot$
+\end_inset
+
+ is the logarithm map on 
+\begin_inset Formula $\SE 3$
+\end_inset
+
+.
+ The full factor error, including the noise model, is
+\begin_inset Formula 
+\[
+e\left(x\right)=\norm{h\left(x\right)}_{\Sigma}^{2}=h\left(x\right)^{\t}\Sigma^{-1}h\left(x\right)\text{.}
+\]
+
+\end_inset
+
+[ Skipping details of the derivation for now, for lack of time to get a
+ useful answer out quickly ]
+\end_layout
+
+\begin_layout Plain Layout
+The density induced by a noise model on the prior factor is Gaussian in
+ the tangent space about the linearization point.
+ Suppose that the pose is linearized at 
+\begin_inset Formula $\lin x\in\SE 3$
+\end_inset
+
+, which we assume is near to 
+\begin_inset Formula $z$
+\end_inset
+
+.
+ Let 
+\begin_inset Formula $\delta x\in\R 6$
+\end_inset
+
+ be an update vector in local coordinates (a twist).
+ Then, the factor error in terms of the update vector 
+\begin_inset Formula $\delta x$
+\end_inset
+
+ is
+\begin_inset Formula 
+\[
+e\left(\delta x\right)=\norm{h\left(\lin x\exp\delta x\right)}_{\Sigma}^{2}
+\]
+
+\end_inset
+
+We can see why the covariance 
+\begin_inset Formula $\Sigma$
+\end_inset
+
+ is in the body frame of 
+\begin_inset Formula $x$
+\end_inset
+
+ by looking at the linearized error function,
+\begin_inset Formula 
+\begin{align*}
+e\left(\delta x\right) & \approx\norm{\log\left(z^{-1}\lin x\exp\delta x\right)}_{\Sigma}^{2}\\
+ & \approx\norm{\log\left(z^{-1}\lin x\right)+\delta x}_{\Sigma}^{2}
+\end{align*}
+
+\end_inset
+
+Here we see that the update 
+\begin_inset Formula $\exp\delta x$
+\end_inset
+
+ from the linear step 
+\begin_inset Formula $\delta x$
+\end_inset
+
+ is applied in the body frame of 
+\begin_inset Formula $\lin x$
+\end_inset
+
+, because of the ordering 
+\begin_inset Formula $\lin x\exp\delta x$
+\end_inset
+
+.
+ Furthermore, 
+\begin_inset Formula $z^{-1}\lin x$
+\end_inset
+
+ is a constant term, so we can also see that the covariance 
+\begin_inset Formula $\Sigma$
+\end_inset
+
+ is actually applied to the linear update vector 
+\begin_inset Formula $\delta x$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Plain Layout
+This means that to draw random pose samples, we actually draw random samples
+ of 
+\begin_inset Formula $\delta x$
+\end_inset
+
+ with zero mean and covariance 
+\begin_inset Formula $\Sigma$
+\end_inset
+
+, i.e.
+\begin_inset Formula 
+\[
+\delta x\sim\mathcal{N}\left(0,\:\Sigma\right)\text{.}
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Noise models of between factors
+\end_layout
+
+\begin_layout Plain Layout
+The noise model of a BetweenFactor is a bit more complicated.
+ The unwhitened error is
+\begin_inset Formula 
+\[
+h\left(x_{1},x_{2}\right)=\log\left(z^{-1}x_{1}^{-1}x_{2}\right)\text{,}
+\]
+
+\end_inset
+
+where 
+\begin_inset Formula $z$
+\end_inset
+
+ is the expected relative pose between 
+\begin_inset Formula $x_{1}$
+\end_inset
+
+ and 
+\begin_inset Formula $x_{2}$
+\end_inset
+
+, i.e.
+ the factor has zero error when 
+\begin_inset Formula $x_{1}z=x_{2}$
+\end_inset
+
+.
+ If we consider the density on the second pose 
+\begin_inset Formula $x_{2}$
+\end_inset
+
+ induced by holding the first pose 
+\begin_inset Formula $x_{1}$
+\end_inset
+
+ fixed, we can see that the covariance is applied to the linear update in
+ the body frame of the second pose 
+\begin_inset Formula $x_{2}$
+\end_inset
+
+,
+\begin_inset Formula 
+\[
+e\left(\delta x_{2}\right)\approx\norm{\log\left(z^{-1}x_{1}^{-1}x_{2}\exp\delta x_{2}\right)}_{\Sigma}^{2}.
+\]
+
+\end_inset
+
+If we hold the second pose fixed, the covariance is applied as follows (actually
+, what frame is it in now??)
+\begin_inset Formula 
+\begin{align*}
+e\left(\delta x_{1}\right) & \approx\norm{\log\left(z^{-1}\left(x_{1}\exp\delta x_{1}\right)^{-1}x_{2}\right)}_{\Sigma}^{2}\\
+ & =\norm{\log\left(z^{-1}\exp-\delta x_{1}x_{1}^{-1}x_{2}\right)}_{\Sigma}^{2}
+\end{align*}
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\end_body
+\end_document
diff --git a/doc/gtsam-coordinate-frames.pdf b/doc/gtsam-coordinate-frames.pdf
new file mode 100644
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