Merge pull request #288 from borglab/feature/lines

Representation for lines in 3D - Line3

doc/math.lyx

@@ -1,7 +1,9 @@
-#LyX 2.1 created this file. For more info see http://www.lyx.org/
+#LyX 2.3 created this file. For more info see http://www.lyx.org/
-\lyxformat 474
+\lyxformat 544
 \begin_document
 \begin_header
+\save_transient_properties true
+\origin unavailable
 \textclass article
 \use_default_options false
 \begin_modules
@@ -14,16 +16,18 @@ theorems-ams-bytype
 \language_package default
 \inputencoding auto
 \fontencoding global
-\font_roman times
+\font_roman "times" "default"
-\font_sans default
+\font_sans "default" "default"
-\font_typewriter default
+\font_typewriter "default" "default"
-\font_math auto
+\font_math "auto" "auto"
 \font_default_family rmdefault
 \use_non_tex_fonts false
 \font_sc false
 \font_osf false
-\font_sf_scale 100
+\font_sf_scale 100 100
-\font_tt_scale 100
+\font_tt_scale 100 100
+\use_microtype false
+\use_dash_ligatures true
 \graphics default
 \default_output_format default
 \output_sync 0
@@ -53,6 +57,7 @@ theorems-ams-bytype
 \suppress_date false
 \justification true
 \use_refstyle 0
+\use_minted 0
 \index Index
 \shortcut idx
 \color #008000
@@ -65,7 +70,10 @@ theorems-ams-bytype
 \tocdepth 3
 \paragraph_separation indent
 \paragraph_indentation default
-\quotes_language english
+\is_math_indent 0
+\math_numbering_side default
+\quotes_style english
+\dynamic_quotes 0
 \papercolumns 1
 \papersides 1
 \paperpagestyle default
@@ -98,6 +106,11 @@ width "100col%"
 special "none"
 height "1in"
 height_special "totalheight"
+thickness "0.4pt"
+separation "3pt"
+shadowsize "4pt"
+framecolor "black"
+backgroundcolor "none"
 status collapsed
 
 \begin_layout Plain Layout
@@ -654,6 +667,7 @@ reference "eq:LocalBehavior"
 \begin_inset CommandInset citation
 LatexCommand cite
 key "Spivak65book"
+literal "true"
 
 \end_inset
 
@@ -934,6 +948,7 @@ See
 \begin_inset CommandInset citation
 LatexCommand cite
 key "Spivak65book"
+literal "true"
 
 \end_inset
 
@@ -1025,6 +1040,7 @@ See
 \begin_inset CommandInset citation
 LatexCommand cite
 key "Spivak65book"
+literal "true"
 
 \end_inset
 
@@ -2209,6 +2225,7 @@ instantaneous velocity
 LatexCommand cite
 after "page 51 for rotations, page 419 for SE(3)"
 key "Murray94book"
+literal "true"
 
 \end_inset
 
@@ -2965,7 +2982,7 @@ B^{T} & I_{3}\end{array}\right]
 \begin_layout Subsection
 \begin_inset CommandInset label
 LatexCommand label
-name "sub:Pushforward-of-Between"
+name "subsec:Pushforward-of-Between"
 
 \end_inset
 
@@ -3419,6 +3436,7 @@ A retraction
 \begin_inset CommandInset citation
 LatexCommand cite
 key "Absil07book"
+literal "true"
 
 \end_inset
 
@@ -3873,7 +3891,7 @@ BetweenFactor
 , derived in Section 
 \begin_inset CommandInset ref
 LatexCommand ref
-reference "sub:Pushforward-of-Between"
+reference "subsec:Pushforward-of-Between"
 
 \end_inset
 
@@ -4430,7 +4448,7 @@ In the case of
 \begin_inset Formula $\SOthree$
 \end_inset
 
- the vector space is  
+ the vector space is 
 \begin_inset Formula $\Rthree$
 \end_inset
 
@@ -4502,7 +4520,7 @@ reference "Th:InverseAction"
 \begin_layout Subsection
 \begin_inset CommandInset label
 LatexCommand label
-name "sub:3DAngularVelocities"
+name "subsec:3DAngularVelocities"
 
 \end_inset
 
@@ -4695,6 +4713,7 @@ Absil
 LatexCommand cite
 after "page 58"
 key "Absil07book"
+literal "true"
 
 \end_inset
 
@@ -5395,6 +5414,7 @@ While not a Lie group, we can define an exponential map, which is given
 \begin_inset CommandInset citation
 LatexCommand cite
 key "Ma01ijcv"
+literal "true"
 
 \end_inset
 
@@ -5402,6 +5422,7 @@ key "Ma01ijcv"
 \begin_inset CommandInset href
 LatexCommand href
 name "http://stat.fsu.edu/~anuj/CVPR_Tutorial/Part2.pdf"
+literal "false"
 
 \end_inset
 
@@ -5605,6 +5626,7 @@ The exponential map uses
 \begin_inset CommandInset citation
 LatexCommand cite
 key "Absil07book"
+literal "true"
 
 \end_inset
 
@@ -6293,7 +6315,7 @@ d^{c}=R_{w}^{c}\left(d^{w}+(t^{w}v^{w})v^{w}-t^{w}\right)
 \end_layout
 
 \begin_layout Section
-Line3 (Ocaml)
+Line3
 \end_layout
 
 \begin_layout Standard
@@ -6345,6 +6367,14 @@ R'=R(I+\Omega)
 
 \end_inset
 
+
+\end_layout
+
+\begin_layout Subsection
+Projecting Line3
+\end_layout
+
+\begin_layout Standard
 Projecting a line to 2D can be done easily, as both 
 \begin_inset Formula $v$
 \end_inset
@@ -6430,13 +6460,21 @@ or the
 
 \end_layout
 
+\begin_layout Subsection
+Action of 
+\begin_inset Formula $\SEthree$
+\end_inset
+
+ on the line
+\end_layout
+
 \begin_layout Standard
 Transforming a 3D line 
 \begin_inset Formula $(R,(a,b))$
 \end_inset
 
  from a world coordinate frame to a camera frame 
-\begin_inset Formula $(R_{w}^{c},t^{w})$
+\begin_inset Formula $T_{c}^{w}=(R_{c}^{w},t^{w})$
 \end_inset
 
  is done by
@@ -6466,17 +6504,115 @@ b'=b-R_{2}^{T}t^{w}
 
 \end_inset
 
-Again, we need to redo the derivatives, as R is incremented from the right.
+where 
- The first argument is incremented from the left, but the result is incremented
+\begin_inset Formula $R_{1}$
- on the right:
+\end_inset
+
+ and 
+\begin_inset Formula $R_{2}$
+\end_inset
+
+ are the columns of 
+\begin_inset Formula $R$
+\end_inset
+
+ , as before.
+ 
+\end_layout
+
+\begin_layout Standard
+To find the derivatives, the transformation of a line 
+\begin_inset Formula $l^{w}=(R,a,b)$
+\end_inset
+
+ from world coordinates to a camera coordinate frame 
+\begin_inset Formula $T_{c}^{w}$
+\end_inset
+
+, specified in world coordinates, can be written as a function 
+\begin_inset Formula $f:\SEthree\times L\rightarrow L$
+\end_inset
+
+, as given above, i.e., 
 \begin_inset Formula 
-\begin{eqnarray*}
+\[
-R'(I+\Omega')=(AB)(I+\Omega') & = & (I+\Skew{S\omega})AB\\
+f(T_{c}^{w},l^{w})=\left(\left(R_{c}^{w}\right)^{T}R,a-R_{1}^{T}t^{w},b-R_{2}^{T}t^{w}\right).
-I+\Omega' & = & (AB)^{T}(I+\Skew{S\omega})(AB)\\
+\]
-\Omega' & = & R'^{T}\Skew{S\omega}R'\\
+
-\Omega' & = & \Skew{R'^{T}S\omega}\\
+\end_inset
-\omega' & = & R'^{T}S\omega
+
-\end{eqnarray*}
+Let us find the Jacobian 
+\begin_inset Formula $J_{1}$
+\end_inset
+
+ of 
+\begin_inset Formula $f$
+\end_inset
+
+ with respect to the first argument 
+\begin_inset Formula $T_{c}^{w}$
+\end_inset
+
+, which should obey
+\begin_inset Formula 
+\begin{align*}
+f(T_{c}^{w}e^{\xihat},l^{w}) & \approx f(T_{c}^{w},l^{w})+J_{1}\xi
+\end{align*}
+
+\end_inset
+
+Note that
+\begin_inset Formula 
+\[
+T_{c}^{w}e^{\xihat}\approx\left[\begin{array}{cc}
+R_{c}^{w}\left(I_{3}+\Skew{\omega}\right) & t^{w}+R_{c}^{w}v\\
+0 & 1
+\end{array}\right]
+\]
+
+\end_inset
+
+Let's write this out separately for each of 
+\begin_inset Formula $R,a,b$
+\end_inset
+
+:
+\begin_inset Formula 
+\begin{align*}
+\left(R_{c}^{w}\left(I_{3}+\Skew{\omega}\right)\right)^{T}R & \approx\left(R_{c}^{w}\right)^{T}R(I+\left[J_{R\omega}\omega\right]_{\times})\\
+a-R_{1}^{T}\left(t^{w}+R_{c}^{w}v\right) & \approx a-R_{1}^{T}t^{w}+J_{av}v\\
+b-R_{2}^{T}\left(t^{w}+R_{c}^{w}v\right) & \approx b-R_{2}^{T}t^{w}+J_{bv}v
+\end{align*}
+
+\end_inset
+
+Simplifying, we get:
+\begin_inset Formula 
+\begin{align*}
+-\Skew{\omega}R' & \approx R'\left[J_{R\omega}\omega\right]_{\times}\\
+-R_{1}^{T}R_{c}^{w} & \approx J_{av}\\
+-R_{2}^{T}R_{c}^{w} & \approx J_{bv}
+\end{align*}
+
+\end_inset
+
+which gives the expressions for 
+\begin_inset Formula $J_{av}$
+\end_inset
+
+ and 
+\begin_inset Formula $J_{bv}$
+\end_inset
+
+.
+ The top line can be further simplified:
+\begin_inset Formula 
+\begin{align*}
+-\Skew{\omega}R' & \approx R'\left[J_{R\omega}\omega\right]_{\times}\\
+-R'^{T}\Skew{\omega}R' & \approx\left[J_{R\omega}\omega\right]_{\times}\\
+-\Skew{R'^{T}\omega} & \approx\left[J_{R\omega}\omega\right]_{\times}\\
+-R'^{T} & \approx J_{R\omega}
+\end{align*}
 
 \end_inset
 
@@ -6687,6 +6823,7 @@ Spivak
 \begin_inset CommandInset citation
 LatexCommand cite
 key "Spivak65book"
+literal "true"
 
 \end_inset
 
@@ -6795,6 +6932,7 @@ The following is adapted from Appendix A in
 \begin_inset CommandInset citation
 LatexCommand cite
 key "Murray94book"
+literal "true"
 
 \end_inset
 
@@ -6924,6 +7062,7 @@ might be
 LatexCommand cite
 after "page 45"
 key "Hall00book"
+literal "true"
 
 \end_inset
 

gtsam/geometry/Line3.cpp

@@ -0,0 +1,120 @@
+#include <gtsam/geometry/Line3.h>
+
+namespace gtsam {
+
+Line3 Line3::retract(const Vector4 &v, OptionalJacobian<4, 4> Dp, OptionalJacobian<4, 4> Dv) const {
+  Vector3 w;
+  w << v[0], v[1], 0;
+  Rot3 incR;
+
+  if (Dp) {
+    Dp->setIdentity();
+    incR = Rot3::Expmap(w);
+    Dp->block<2, 2>(0, 0) = ((incR.matrix()).transpose()).block<2, 2>(0, 0);
+  }
+  if (Dv) {
+    Matrix3 Dw;
+    incR = Rot3::Expmap(w, Dw);
+    Dv->setIdentity();
+    Dv->block<2, 2>(0, 0) = Dw.block<2, 2>(0, 0);
+  } else {
+    incR = Rot3::Expmap(w);
+  }
+  Rot3 Rt = R_ * incR;
+  return Line3(Rt, a_ + v[2], b_ + v[3]);
+}
+
+Vector4 Line3::localCoordinates(const Line3 &q, OptionalJacobian<4, 4> Dp,
+                                OptionalJacobian<4, 4> Dq) const {
+  Vector3 omega;
+  Matrix3 D_log;
+  omega = Rot3::Logmap(R_.inverse() * q.R_, D_log);
+  if (Dp) {
+    Matrix3 D_log_wp = -((q.R_).matrix()).transpose() * R_.matrix();
+    Matrix3 Dwp = D_log * D_log_wp;
+    Dp->setIdentity();
+    Dp->block<2, 2>(0, 0) = Dwp.block<2, 2>(0, 0);
+    (*Dp)(2, 2) = -1;
+    (*Dp)(3, 3) = -1;
+  }
+  if (Dq) {
+    Dq->setIdentity();
+    Dq->block<2, 2>(0, 0) = D_log.block<2, 2>(0, 0);
+  }
+  Vector4 local;
+  local << omega[0], omega[1], q.a_ - a_, q.b_ - b_;
+  return local;
+}
+
+void Line3::print(const std::string &s) const {
+  std::cout << s << std::endl;
+  R_.print("R:\n");
+  std::cout << "a: " << a_ << ", b: " << b_ << std::endl;
+}
+
+bool Line3::equals(const Line3 &l2, double tol) const {
+  Vector4 diff = localCoordinates(l2);
+  return fabs(diff[0]) < tol && fabs(diff[1]) < tol
+      && fabs(diff[2]) < tol && fabs(diff[3]) < tol;
+}
+
+Unit3 Line3::project(OptionalJacobian<2, 4> Dline) const {
+  Vector3 V_0;
+  V_0 << -b_, a_, 0.0;
+
+  Unit3 l;
+  if (Dline) {
+    // Jacobian of the normalized Unit3 projected line with respect to
+    // un-normalized Vector3 projected line in homogeneous coordinates.
+    Matrix23 D_unit_line;
+    l = Unit3::FromPoint3(Point3(R_ * V_0), D_unit_line);
+    // Jacobian of the un-normalized Vector3 line with respect to
+    // input 3D line
+    Matrix34 D_vec_line = Matrix34::Zero();
+    D_vec_line.col(0) = a_ * R_.r3();
+    D_vec_line.col(1) = b_ * R_.r3();
+    D_vec_line.col(2) = R_.r2();
+    D_vec_line.col(3) = -R_.r1();
+    // Jacobian of output wrt input is the product of the two.
+    *Dline = D_unit_line * D_vec_line;
+  } else {
+    l = Unit3::FromPoint3(Point3(R_ * V_0));
+  }
+  return l;
+}
+
+Point3 Line3::point(double distance) const {
+  // defining "center" of the line to be the point where it
+  // intersects rotated XY axis
+  Point3 center(a_, b_, 0);
+  Point3 rotated_center = R_ * center;
+  return rotated_center + distance * R_.r3();
+}
+
+Line3 transformTo(const Pose3 &wTc, const Line3 &wL,
+                  OptionalJacobian<4, 6> Dpose, OptionalJacobian<4, 4> Dline) {
+  Rot3 wRc = wTc.rotation();
+  Rot3 cRw = wRc.inverse();
+  Rot3 cRl = cRw * wL.R_;
+
+  Vector2 w_ab;
+  Vector3 t = ((wL.R_).transpose() * wTc.translation());
+  Vector2 c_ab(wL.a_ - t[0], wL.b_ - t[1]);
+
+  if (Dpose) {
+    Matrix3 lRc = (cRl.matrix()).transpose();
+    Dpose->setZero();
+    // rotation
+    Dpose->block<2, 3>(0, 0) = -lRc.block<2, 3>(0, 0);
+    // translation
+    Dpose->block<2, 3>(2, 3) = -lRc.block<2, 3>(0, 0);
+  }
+  if (Dline) {
+    Dline->setIdentity();
+    (*Dline)(0, 3) = -t[2];
+    (*Dline)(1, 2) = t[2];
+  }
+  return Line3(cRl, c_ab[0], c_ab[1]);
+}
+
+}

gtsam/geometry/Line3.h

@@ -0,0 +1,135 @@
+/* ----------------------------------------------------------------------------
+ * GTSAM Copyright 2010, Georgia Tech Research Corporation,
+ * Atlanta, Georgia 30332-0415
+ * All Rights Reserved
+ * Authors: Frank Dellaert, et al. (see THANKS for the full author list)
+ * See LICENSE for the license information
+ * -------------------------------------------------------------------------- */
+
+/**
+ * @file    Line3.h
+ * @brief   4 dimensional manifold of 3D lines
+ * @author  Akshay Krishnan
+ * @author  Frank Dellaert
+ */
+// \callgraph
+
+#pragma once
+
+#include <gtsam/geometry/Rot3.h>
+#include <gtsam/geometry/Pose3.h>
+
+namespace gtsam {
+
+/**
+ * A 3D line (R,a,b) : (Rot3,Scalar,Scalar)
+ * @addtogroup geometry
+ * \nosubgrouping
+ */
+class Line3 {
+ private:
+  Rot3 R_;    // Rotation of line about x and y in world frame
+  double a_, b_;  // Intersection of line with the world x-y plane rotated by R_
+  // Also the closest point on line to origin
+ public:
+  enum { dimension = 4 };
+
+  /** Default constructor is the Z axis **/
+  Line3() :
+      a_(0), b_(0) {}
+
+  /** Constructor for general line from (R, a, b) **/
+  Line3(const Rot3 &R, const double a, const double b) :
+      R_(R), a_(a), b_(b) {}
+
+  /**
+   * The retract method maps from the tangent space back to the manifold.
+   * The method q = p + v, where p is this line, v is an increment along
+   * the tangent space and q is the resulting line.
+   * The tangent space for the rotation of a line is only two dimensional -
+   * rotation about x and y
+   * @param v: increment in tangent space
+   * @param Dp: increment of retraction with respect to this line
+   * @param Dv: Jacobian of retraction with respect to the increment
+   * @return q: resulting line after adding the increment and mapping to the manifold
+   */
+  Line3 retract(const Vector4 &v,
+                OptionalJacobian<4, 4> Dp = boost::none,
+                OptionalJacobian<4, 4> Dv = boost::none) const;
+
+  /**
+   * The localCoordinates method is the inverse of retract and finds the difference
+   * between two lines in the tangent space. It computes v = q - p where q is an input
+   * line, p is this line and v is their difference in the tangent space.
+   * @param q: Line3 on manifold
+   * @param Dp: OptionalJacobian of localCoordinates with respect to this line
+   * @param Dq: OptionalJacobian of localCoordinates with respect to this line
+   * @return v: difference in the tangent space
+   */
+  Vector4 localCoordinates(const Line3 &q,
+                           OptionalJacobian<4, 4> Dp = boost::none,
+                           OptionalJacobian<4, 4> Dq = boost::none) const;
+
+  /**
+   * Print R, a, b
+   * @param s: optional starting string
+   */
+  void print(const std::string &s = "") const;
+
+  /**
+   * Check if two lines are equal
+   * @param l2 - line to be compared
+   * @param tol : optional tolerance
+   * @return boolean - true if lines are equal
+   */
+  bool equals(const Line3 &l2, double tol = 10e-9) const;
+
+  /**
+   * Projecting a line to the image plane. Assumes this line is in camera frame.
+   * @param Dline: OptionalJacobian of projected line with respect to this line
+   * @return Unit3 - projected line in image plane, in homogenous coordinates.
+   * We use Unit3 since it is a manifold with the right dimension.
+   */
+  Unit3 project(OptionalJacobian<2, 4> Dline = boost::none) const;
+
+  /**
+   * Returns point on the line that is at a certain distance starting from the
+   * point where the rotated XY axis intersects the line.
+   * @param distance (can be positive or negative) - positive is the positive
+   * direction of the rotated z axis that forms the line. The default value of zero
+   * returns the point where the rotated XY axis intersects the line.
+   * @return Point on the line
+   */
+  Point3 point(double distance = 0) const;
+
+  /**
+   * Transform a line from world to camera frame
+   * @param wTc - Pose3 of camera in world frame
+   * @param wL - Line3 in world frame
+   * @param Dpose - OptionalJacobian of transformed line with respect to p
+   * @param Dline -  OptionalJacobian of transformed line with respect to l
+   * @return Transformed line in camera frame
+   */
+  friend Line3 transformTo(const Pose3 &wTc, const Line3 &wL,
+                           OptionalJacobian<4, 6> Dpose,
+                           OptionalJacobian<4, 4> Dline);
+};
+
+/**
+ * Transform a line from world to camera frame
+ * @param wTc - Pose3 of camera in world frame
+ * @param wL - Line3 in world frame
+ * @param Dpose - OptionalJacobian of transformed line with respect to p
+ * @param Dline -  OptionalJacobian of transformed line with respect to l
+ * @return Transformed line in camera frame
+ */
+Line3 transformTo(const Pose3 &wTc, const Line3 &wL,
+                  OptionalJacobian<4, 6> Dpose = boost::none,
+                  OptionalJacobian<4, 4> Dline = boost::none);
+
+template<>
+struct traits<Line3> : public internal::Manifold<Line3> {};
+
+template<>
+struct traits<const Line3> : public internal::Manifold<Line3> {};
+}

gtsam/geometry/tests/testLine3.cpp

@@ -0,0 +1,155 @@
+#include <CppUnitLite/TestHarness.h>
+#include <gtsam/base/Testable.h>
+#include <gtsam/base/numericalDerivative.h>
+#include <gtsam/geometry/Line3.h>
+#include <gtsam/slam/expressions.h>
+#include <gtsam/slam/PriorFactor.h>
+#include <gtsam/nonlinear/ExpressionFactor.h>
+#include <gtsam/nonlinear/expressionTesting.h>
+
+using namespace gtsam;
+
+GTSAM_CONCEPT_TESTABLE_INST(Line3)
+GTSAM_CONCEPT_MANIFOLD_INST(Line3)
+
+static const Line3 l(Rot3(), 1, 1);
+
+// Testing equals function of Line3
+TEST(Line3, equals) {
+  Line3 l_same = l;
+  EXPECT(l.equals(l_same));
+  Line3 l2(Rot3(), 1, 2);
+  EXPECT(!l.equals(l2));
+}
+
+// testing localCoordinates along 4 dimensions
+TEST(Line3, localCoordinates) {
+  // l1 and l differ only in a_
+  Line3 l1(Rot3(), 2, 1);
+  Vector4 v1(0, 0, -1, 0);
+  CHECK(assert_equal(l1.localCoordinates(l), v1));
+
+  // l2 and l differ only in b_
+  Line3 l2(Rot3(), 1, 2);
+  Vector4 v2(0, 0, 0, -1);
+  CHECK(assert_equal(l2.localCoordinates(l), v2));
+
+  // l3 and l differ in R_x
+  Rot3 r3 = Rot3::Expmap(Vector3(M_PI / 4, 0, 0));
+  Line3 l3(r3, 1, 1);
+  Vector4 v3(-M_PI / 4, 0, 0, 0);
+  CHECK(assert_equal(l3.localCoordinates(l), v3));
+
+  // l4 and l differ in R_y
+  Rot3 r4 = Rot3::Expmap(Vector3(0, M_PI / 4, 0));
+  Line3 l4(r4, 1, 1);
+  Vector4 v4(0, -M_PI / 4, 0, 0);
+  CHECK(assert_equal(l4.localCoordinates(l), v4));
+
+  // Jacobians
+  Line3 l5(Rot3::Expmap(Vector3(M_PI / 3, -M_PI / 4, 0)), -0.8, 2);
+  Values val;
+  val.insert(1, l);
+  val.insert(2, l5);
+  Line3_ l_(1);
+  Line3_ l5_(2);
+  SharedNoiseModel model = noiseModel::Isotropic::Sigma(4, 0.1);
+  Vector4_ local_(l5_, &Line3::localCoordinates, l_);
+  ExpressionFactor<Vector4> f(model, l5.localCoordinates(l), local_);
+  EXPECT_CORRECT_FACTOR_JACOBIANS(f, val, 1e-5, 1e-7);
+}
+
+// testing retract along 4 dimensions
+TEST(Line3, retract) {
+  // l1 and l differ only in a_
+  Vector4 v1(0, 0, 0, 1);
+  Line3 l1(Rot3(), 1, 2);
+  EXPECT(l1.equals(l.retract(v1)));
+
+  // l2 and l differ only in b_
+  Vector4 v2(0, 0, 1, 0);
+  Line3 l2(Rot3(), 2, 1);
+  EXPECT(l2.equals(l.retract(v2)));
+
+  // l3 and l differ in R_x
+  Vector4 v3(M_PI / 4, 0, 0, 0);
+  Rot3 r3;
+  r3 = r3.Expmap(Vector3(M_PI / 4, 0, 0));
+  Line3 l3(r3, 1, 1);
+  EXPECT(l3.equals(l.retract(v3)));
+
+  // l4 and l differ in R_y
+  Vector4 v4(0, M_PI / 4, 0, 0);
+  Rot3 r4 = Rot3::Expmap(Vector3(0, M_PI / 4, 0));
+  Line3 l4(r4, 1, 1);
+  EXPECT(l4.equals(l.retract(v4)));
+
+  // Jacobians
+  Vector4 v5(M_PI / 3, -M_PI / 4, -0.4, 1.2); // arbitrary vector
+  Values val;
+  val.insert(1, l);
+  val.insert(2, v5);
+  Line3_ l_(1);
+  Vector4_ v5_(2);
+  SharedNoiseModel model = noiseModel::Isotropic::Sigma(4, 0.1);
+  Line3_ retract_(l_, &Line3::retract, v5_);
+  ExpressionFactor<Line3> f(model, l.retract(v5), retract_);
+  EXPECT_CORRECT_FACTOR_JACOBIANS(f, val, 1e-5, 1e-7);
+}
+
+// testing manifold property - Retract(p, Local(p,q)) == q
+TEST(Line3, retractOfLocalCoordinates) {
+  Rot3 r2 = Rot3::Expmap(Vector3(M_PI / 4, M_PI / 3, 0));
+  Line3 l2(r2, 5, 9);
+  EXPECT(assert_equal(l.retract(l.localCoordinates(l2)), l2));
+}
+
+// testing manifold property - Local(p, Retract(p, v)) == v
+TEST(Line3, localCoordinatesOfRetract) {
+  Vector4 r2(2.3, 0.987, -3, 4);
+  EXPECT(assert_equal(l.localCoordinates(l.retract(r2)), r2));
+}
+
+// transform from world to camera test
+TEST(Line3, transformToExpressionJacobians) {
+  Rot3 r = Rot3::Expmap(Vector3(0, M_PI / 3, 0));
+  Vector3 t(0, 0, 0);
+  Pose3 p(r, t);
+
+  Line3 l_c(r.inverse(), 1, 1);
+  Line3 l_w(Rot3(), 1, 1);
+  EXPECT(l_c.equals(transformTo(p, l_w)));
+
+  Line3_ l_(1);
+  Pose3_ p_(2);
+  Values val;
+  val.insert(1, l_w);
+  val.insert(2, p);
+
+  SharedNoiseModel model = noiseModel::Isotropic::Sigma(4, 0.1);
+  ExpressionFactor<Line3> f(model, transformTo(p, l_w), transformTo(p_, l_));
+  EXPECT_CORRECT_FACTOR_JACOBIANS(f, val, 1e-5, 1e-7);
+}
+
+// projection in camera frame test
+TEST(Line3, projection) {
+  Rot3 r = Rot3::Expmap(Vector3(0, 0, 0));
+  Line3 wL(r, 1, 1);
+
+  Unit3 expected = Unit3::FromPoint3(Point3(-1, 1, 0));
+  EXPECT(expected.equals(wL.project()));
+
+  Values val;
+  val.insert(1, wL);
+  Line3_ wL_(1);
+
+  SharedNoiseModel model = noiseModel::Isotropic::Sigma(2, 0.1);
+  Unit3_ projected_(wL_, &Line3::project);
+  ExpressionFactor<Unit3> f(model, expected, projected_);
+  EXPECT_CORRECT_FACTOR_JACOBIANS(f, val, 1e-5, 1e-7);
+}
+
+int main() {
+  TestResult tr;
+  return TestRegistry::runAllTests(tr);
+}

gtsam/slam/expressions.h

@@ -11,6 +11,7 @@
 #include <gtsam/geometry/Pose2.h>
 #include <gtsam/geometry/Cal3_S2.h>
 #include <gtsam/geometry/Cal3Bundler.h>
+#include <gtsam/geometry/Line3.h>
 #include <gtsam/geometry/PinholeCamera.h>
 
 namespace gtsam {
@@ -31,6 +32,7 @@ typedef Expression<Point3> Point3_;
 typedef Expression<Unit3> Unit3_;
 typedef Expression<Rot3> Rot3_;
 typedef Expression<Pose3> Pose3_;
+typedef Expression<Line3> Line3_;
 
 inline Point3_ transformTo(const Pose3_& x, const Point3_& p) {
   return Point3_(x, &Pose3::transformTo, p);
@@ -40,6 +42,12 @@ inline Point3_ transformFrom(const Pose3_& x, const Point3_& p) {
   return Point3_(x, &Pose3::transformFrom, p);
 }
 
+inline Line3_ transformTo(const Pose3_ &wTc, const Line3_ &wL) {
+  Line3 (*f)(const Pose3 &, const Line3 &,
+             OptionalJacobian<4, 6>, OptionalJacobian<4, 4>) = &transformTo;
+  return Line3_(f, wTc, wL);
+}
+
 namespace internal {
 // define getter that returns value rather than reference
 inline Rot3 rotation(const Pose3& pose, OptionalJacobian<3, 6> H) {