Merged in feature/sphere2_expmap (pull request #3)

Sphere2 Expmap

gtsam/geometry/Rot3.cpp

@@ -81,6 +81,17 @@ Sphere2 Rot3::rotate(const Sphere2& p,
   return q;
 }
 
+/* ************************************************************************* */
+Sphere2 Rot3::unrotate(const Sphere2& p,
+    boost::optional<Matrix&> HR, boost::optional<Matrix&> Hp) const {
+  Sphere2 q = unrotate(p.point3(Hp));
+  if (Hp)
+    (*Hp) = q.basis().transpose() * matrix().transpose () * (*Hp);
+  if (HR)
+    (*HR) = q.basis().transpose() * q.skew();
+  return q;
+}
+
 /* ************************************************************************* */
 Sphere2 Rot3::operator*(const Sphere2& p) const {
   return rotate(p);

gtsam/geometry/Rot3.h

@@ -331,6 +331,10 @@ namespace gtsam {
     Sphere2 rotate(const Sphere2& p, boost::optional<Matrix&> HR = boost::none,
         boost::optional<Matrix&> Hp = boost::none) const;
 
+    /// unrotate 3D direction from world frame to rotated coordinate frame
+    Sphere2 unrotate(const Sphere2& p, boost::optional<Matrix&> HR = boost::none,
+        boost::optional<Matrix&> Hp = boost::none) const;
+
     /// rotate 3D direction from rotated coordinate frame to world frame
     Sphere2 operator*(const Sphere2& p) const;
 

gtsam/geometry/Sphere2.cpp

@@ -14,6 +14,7 @@
  * @date Feb 02, 2011
  * @author Can Erdogan
  * @author Frank Dellaert
+ * @author Alex Trevor
  * @brief The Sphere2 class - basically a point on a unit sphere
  */
 
@@ -98,7 +99,7 @@ double Sphere2::distance(const Sphere2& q, boost::optional<Matrix&> H) const {
 }
 
 /* ************************************************************************* */
-Sphere2 Sphere2::retract(const Vector& v) const {
+Sphere2 Sphere2::retract(const Vector& v, Sphere2::CoordinatesMode mode) const {
 
   // Get the vector form of the point and the basis matrix
   Vector p = Point3::Logmap(p_);
@@ -106,35 +107,75 @@ Sphere2 Sphere2::retract(const Vector& v) const {
 
   // Compute the 3D xi_hat vector
   Vector xi_hat = v(0) * B.col(0) + v(1) * B.col(1);
-  Vector newPoint = p + xi_hat;
   
-  // Project onto the manifold, i.e. the closest point on the circle to the new location;
-  // same as putting it onto the unit circle
-  Vector projected = newPoint / newPoint.norm();
+  if (mode == Sphere2::EXPMAP) {
+    double xi_hat_norm = xi_hat.norm();
 
-  return Sphere2(Point3::Expmap(projected));
+    // Avoid nan
+    if ((xi_hat_norm == 0.0)) {
+      if (v.norm () == 0.0)
+        return Sphere2 (point3 ());
+      else
+        return Sphere2 (-point3 ());
+    }
+    
+    Vector exp_p_xi_hat = cos (xi_hat_norm) * p + sin(xi_hat_norm) * (xi_hat / xi_hat_norm);
+    return Sphere2(exp_p_xi_hat);
+  } else if (mode == Sphere2::RENORM) {
+    // Project onto the manifold, i.e. the closest point on the circle to the new location;
+    // same as putting it onto the unit circle
+    Vector newPoint = p + xi_hat;
+    Vector projected = newPoint / newPoint.norm();
+
+    return Sphere2(Point3::Expmap(projected));
+  } else {
+    assert (false);
+    exit (1);
+  }  
 }
 
 /* ************************************************************************* */
-Vector Sphere2::localCoordinates(const Sphere2& y) const {
+  Vector Sphere2::localCoordinates(const Sphere2& y, Sphere2::CoordinatesMode mode) const {
 
-  // Make sure that the angle different between x and y is less than 90. Otherwise,
-  // we can project x + xi_hat from the tangent space at x to y.
-  assert(y.p_.dot(p_) > 0.0 && "Can not retract from x to y.");
+  if (mode == Sphere2::EXPMAP) {
+    Matrix B = basis();
     
-  // Get the basis matrix
-  Matrix B = basis();
+    Vector p = Point3::Logmap(p_);
+    Vector q = Point3::Logmap(y.p_);
+    double theta = acos(p.transpose() * q);
 
-  // Create the vector forms of p and q (the Point3 of y).
-  Vector p = Point3::Logmap(p_);
-  Vector q = Point3::Logmap(y.p_);
+    // the below will be nan if theta == 0.0
+    if (p == q)
+      return (Vector (2) << 0, 0);
+    else if (p == -q)
+      return (Vector (2) << M_PI, 0);
     
-  // Compute the basis coefficients [v0,v1] = (B'q)/(p'q).
-  double alpha = p.transpose() * q;
-  assert(alpha != 0.0);
-  Matrix coeffs = (B.transpose() * q) / alpha;
-  Vector result = Vector_(2, coeffs(0, 0), coeffs(1, 0));
-  return result;
+    Vector result_hat = (theta / sin(theta)) * (q - p * cos(theta));
+    Vector result = B.transpose() * result_hat;
+    
+    return result;
+  } else if (mode == Sphere2::RENORM) {
+    // Make sure that the angle different between x and y is less than 90. Otherwise,
+    // we can project x + xi_hat from the tangent space at x to y.
+    assert(y.p_.dot(p_) > 0.0 && "Can not retract from x to y.");
+    
+    // Get the basis matrix
+    Matrix B = basis();
+    
+    // Create the vector forms of p and q (the Point3 of y).
+    Vector p = Point3::Logmap(p_);
+    Vector q = Point3::Logmap(y.p_);
+    
+    // Compute the basis coefficients [v0,v1] = (B'q)/(p'q).
+    double alpha = p.transpose() * q;
+    assert(alpha != 0.0);
+    Matrix coeffs = (B.transpose() * q) / alpha;
+    Vector result = Vector_(2, coeffs(0, 0), coeffs(1, 0));
+    return result;
+  } else {
+    assert (false);
+    exit (1);
+  }
 }
 /* ************************************************************************* */
 

gtsam/geometry/Sphere2.h

@@ -14,6 +14,7 @@
  * @date Feb 02, 2011
  * @author Can Erdogan
  * @author Frank Dellaert
+ * @author Alex Trevor
  * @brief Develop a Sphere2 class - basically a point on a unit sphere
  */
 
@@ -22,6 +23,10 @@
 #include <gtsam/geometry/Point3.h>
 #include <gtsam/base/DerivedValue.h>
 
+#ifndef SPHERE2_DEFAULT_COORDINATES_MODE
+  #define SPHERE2_DEFAULT_COORDINATES_MODE Sphere2::RENORM
+#endif
+
 // (Cumbersome) forward declaration for random generator
 namespace boost {
 namespace random {
@@ -98,6 +103,13 @@ public:
     return p_;
   }
 
+  /// Return unit-norm Vector
+  Vector unitVector(boost::optional<Matrix&> H = boost::none) const {
+    if (H)
+      *H = basis();
+    return (p_.vector ());
+  }
+
   /// Signed, vector-valued error between two directions
   Vector error(const Sphere2& q,
       boost::optional<Matrix&> H = boost::none) const;
@@ -121,11 +133,16 @@ public:
     return 2;
   }
 
+  enum CoordinatesMode {
+    EXPMAP, ///< Use the exponential map to retract
+    RENORM ///< Retract with vector addtion and renormalize.
+  };
+
   /// The retract function
-  Sphere2 retract(const Vector& v) const;
+  Sphere2 retract(const Vector& v, Sphere2::CoordinatesMode mode = SPHERE2_DEFAULT_COORDINATES_MODE) const;
 
   /// The local coordinates function
-  Vector localCoordinates(const Sphere2& s) const;
+  Vector localCoordinates(const Sphere2& s, Sphere2::CoordinatesMode mode = SPHERE2_DEFAULT_COORDINATES_MODE) const;
 
   /// @}
 };

gtsam/geometry/tests/testSphere2.cpp

@@ -13,6 +13,8 @@
  * @file testSphere2.cpp
  * @date Feb 03, 2012
  * @author Can Erdogan
+ * @author Frank Dellaert
+ * @author Alex Trevor
  * @brief Tests the Sphere2 class
  */
 
@@ -75,10 +77,35 @@ TEST(Sphere2, rotate) {
   }
 }
 
+//*******************************************************************************
+static Sphere2 unrotate_(const Rot3& R, const Sphere2& p) {
+  return R.unrotate (p);
+}
+
+TEST(Sphere2, unrotate) {
+  Rot3 R = Rot3::yaw(-M_PI/4.0);
+  Sphere2 p(1, 0, 0);
+  Sphere2 expected = Sphere2(1, 1, 0);
+  Sphere2 actual = R.unrotate (p);
+  EXPECT(assert_equal(expected, actual, 1e-8));
+  Matrix actualH, expectedH;
+  // Use numerical derivatives to calculate the expected Jacobian
+  {
+    expectedH = numericalDerivative21(unrotate_, R, p);
+    R.unrotate(p, actualH, boost::none);
+    EXPECT(assert_equal(expectedH, actualH, 1e-9));
+  }
+  {
+    expectedH = numericalDerivative22(unrotate_, R, p);
+    R.unrotate(p, boost::none, actualH);
+    EXPECT(assert_equal(expectedH, actualH, 1e-9));
+  }
+}
+
 //*******************************************************************************
 TEST(Sphere2, error) {
-  Sphere2 p(1, 0, 0), q = p.retract((Vector(2) << 0.5, 0)), //
-  r = p.retract((Vector(2) << 0.8, 0));
+  Sphere2 p(1, 0, 0), q = p.retract((Vector(2) << 0.5, 0), Sphere2::RENORM), //
+  r = p.retract((Vector(2) << 0.8, 0), Sphere2::RENORM);
   EXPECT(assert_equal((Vector(2) << 0, 0), p.error(p), 1e-8));
   EXPECT(assert_equal((Vector(2) << 0.447214, 0), p.error(q), 1e-5));
   EXPECT(assert_equal((Vector(2) << 0.624695, 0), p.error(r), 1e-5));
@@ -101,8 +128,8 @@ TEST(Sphere2, error) {
 
 //*******************************************************************************
 TEST(Sphere2, distance) {
-  Sphere2 p(1, 0, 0), q = p.retract((Vector(2) << 0.5, 0)), //
-  r = p.retract((Vector(2) << 0.8, 0));
+  Sphere2 p(1, 0, 0), q = p.retract((Vector(2) << 0.5, 0), Sphere2::RENORM), //
+  r = p.retract((Vector(2) << 0.8, 0), Sphere2::RENORM);
   EXPECT_DOUBLES_EQUAL(0, p.distance(p), 1e-8);
   EXPECT_DOUBLES_EQUAL(0.44721359549995798, p.distance(q), 1e-8);
   EXPECT_DOUBLES_EQUAL(0.6246950475544244, p.distance(r), 1e-8);
@@ -146,9 +173,20 @@ TEST(Sphere2, retract) {
   Vector v(2);
   v << 0.5, 0;
   Sphere2 expected(Point3(1, 0, 0.5));
-  Sphere2 actual = p.retract(v);
+  Sphere2 actual = p.retract(v, Sphere2::RENORM);
   EXPECT(assert_equal(expected, actual, 1e-8));
-  EXPECT(assert_equal(v, p.localCoordinates(actual), 1e-8));
+  EXPECT(assert_equal(v, p.localCoordinates(actual, Sphere2::RENORM), 1e-8));
+}
+
+//*******************************************************************************
+TEST(Sphere2, retract_expmap) {
+  Sphere2 p;
+  Vector v(2);
+  v << (M_PI/2.0), 0;
+  Sphere2 expected(Point3(0, 0, 1));
+  Sphere2 actual = p.retract(v, Sphere2::EXPMAP);
+  EXPECT(assert_equal(expected, actual, 1e-8));
+  EXPECT(assert_equal(v, p.localCoordinates(actual, Sphere2::EXPMAP), 1e-8));
 }
 
 //*******************************************************************************
@@ -198,6 +236,39 @@ TEST(Sphere2, localCoordinates_retract) {
   }
 }
 
+//*******************************************************************************
+// Let x and y be two Sphere2's.
+// The equality x.localCoordinates(x.retract(v)) == v should hold.
+TEST(Sphere2, localCoordinates_retract_expmap) {
+  
+  size_t numIterations = 10000;
+  Vector minSphereLimit = Vector_(3, -1.0, -1.0, -1.0), maxSphereLimit =
+      Vector_(3, 1.0, 1.0, 1.0);
+  Vector minXiLimit = Vector_(2, -M_PI, -M_PI), maxXiLimit = Vector_(2, M_PI, M_PI);
+  for (size_t i = 0; i < numIterations; i++) {
+
+    // Sleep for the random number generator (TODO?: Better create all of them first).
+    sleep(0);
+
+    // Create the two Sphere2s.
+    // Unlike the above case, we can use any two sphers.
+    Sphere2 s1(Point3(randomVector(minSphereLimit, maxSphereLimit)));
+//      Sphere2 s2 (Point3(randomVector(minSphereLimit, maxSphereLimit)));
+    Vector v12 = randomVector(minXiLimit, maxXiLimit);
+    
+    // Magnitude of the rotation can be at most pi
+    if (v12.norm () > M_PI)
+      v12 = v12 / M_PI;
+    Sphere2 s2 = s1.retract(v12);
+
+    // Check if the local coordinates and retract return the same results.
+    Vector actual_v12 = s1.localCoordinates(s2);
+    EXPECT(assert_equal(v12, actual_v12, 1e-3));
+    Sphere2 actual_s2 = s1.retract(actual_v12);
+    EXPECT(assert_equal(s2, actual_s2, 1e-3));
+  }
+}
+
 //*******************************************************************************
 //TEST( Pose2, between )
 //{