Unit3.cpp
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1 /* ----------------------------------------------------------------------------
2 
3  * Atlanta, Georgia 30332-0415
4  * All Rights Reserved
5  * GTSAM Copyright 2010, Georgia Tech Research Corporation,
6  * Authors: Frank Dellaert, et al. (see THANKS for the full author list)
7 
8  * See LICENSE for the license information
9 
10  * -------------------------------------------------------------------------- */
11 
12 /*
13  * @file Unit3.h
14  * @date Feb 02, 2011
15  * @author Can Erdogan
16  * @author Frank Dellaert
17  * @author Alex Trevor
18  * @author Zhaoyang Lv
19  * @brief The Unit3 class - basically a point on a unit sphere
20  */
21 
22 #include <gtsam/geometry/Unit3.h>
23 #include <gtsam/geometry/Point2.h>
24 #include <gtsam/config.h> // for GTSAM_USE_TBB
25 
26 #include <iostream>
27 #include <limits>
28 #include <cmath>
29 #include <vector>
30 
31 using namespace std;
32 
33 namespace gtsam {
34 
35 /* ************************************************************************* */
36 Unit3::Unit3(const Vector3& p) : p_(p.normalized()) {}
37 
38 Unit3::Unit3(double x, double y, double z) : p_(x, y, z) { p_.normalize(); }
39 
40 Unit3::Unit3(const Point2& p, double f) : p_(p.x(), p.y(), f) {
41  p_.normalize();
42 }
43 /* ************************************************************************* */
45  // 3*3 Derivative of representation with respect to point is 3*3:
46  Matrix3 D_p_point;
47  Unit3 direction;
48  direction.p_ = normalize(point, H ? &D_p_point : 0);
49  if (H)
50  *H << direction.basis().transpose() * D_p_point;
51  return direction;
52 }
53 
54 /* ************************************************************************* */
55 Unit3 Unit3::Random(std::mt19937& rng) {
56  // http://mathworld.wolfram.com/SpherePointPicking.html
57  // Adapted from implementation in boost, but using std <random>
58  std::uniform_real_distribution<double> uniform(-1.0, 1.0);
59  double sqsum;
60  double x, y;
61  do {
62  x = uniform(rng);
63  y = uniform(rng);
64  sqsum = x * x + y * y;
65  } while (sqsum > 1);
66  const double mult = 2 * sqrt(1 - sqsum);
67  return Unit3(x * mult, y * mult, 2 * sqsum - 1);
68 }
69 
70 /* ************************************************************************* */
71 // Get the axis of rotation with the minimum projected length of the point
72 static Point3 CalculateBestAxis(const Point3& n) {
73  double mx = std::abs(n.x()), my = std::abs(n.y()), mz = std::abs(n.z());
74  if ((mx <= my) && (mx <= mz)) {
75  return Point3(1.0, 0.0, 0.0);
76  } else if ((my <= mx) && (my <= mz)) {
77  return Point3(0.0, 1.0, 0.0);
78  } else {
79  return Point3(0, 0, 1);
80  }
81 }
82 
83 /* ************************************************************************* */
84 const Matrix32& Unit3::basis(OptionalJacobian<6, 2> H) const {
85 #ifdef GTSAM_USE_TBB
86  // NOTE(hayk): At some point it seemed like this reproducably resulted in
87  // deadlock. However, I don't know why and I can no longer reproduce it.
88  // It either was a red herring or there is still a latent bug left to debug.
89  std::unique_lock<std::mutex> lock(B_mutex_);
90 #endif
91 
92  const bool cachedBasis = static_cast<bool>(B_);
93  const bool cachedJacobian = static_cast<bool>(H_B_);
94 
95  if (H) {
96  if (!cachedJacobian) {
97  // Compute Jacobian. Recomputes B_
98  Matrix32 B;
99  Matrix62 jacobian;
100  Matrix33 H_B1_n, H_b1_B1, H_b2_n, H_b2_b1;
101 
102  // Choose the direction of the first basis vector b1 in the tangent plane
103  // by crossing n with the chosen axis.
104  const Point3 n(p_), axis = CalculateBestAxis(n);
105  const Point3 B1 = gtsam::cross(n, axis, &H_B1_n);
106 
107  // Normalize result to get a unit vector: b1 = B1 / |B1|.
108  B.col(0) = normalize(B1, &H_b1_B1);
109 
110  // Get the second basis vector b2, which is orthogonal to n and b1.
111  B.col(1) = gtsam::cross(n, B.col(0), &H_b2_n, &H_b2_b1);
112 
113  // Chain rule tomfoolery to compute the jacobian.
114  const Matrix32& H_n_p = B;
115  jacobian.block<3, 2>(0, 0) = H_b1_B1 * H_B1_n * H_n_p;
116  auto H_b1_p = jacobian.block<3, 2>(0, 0);
117  jacobian.block<3, 2>(3, 0) = H_b2_n * H_n_p + H_b2_b1 * H_b1_p;
118 
119  // Cache the result and jacobian
120  H_B_ = (jacobian);
121  B_ = (B);
122  }
123 
124  // Return cached jacobian, possibly computed just above
125  *H = *H_B_;
126  }
127 
128  if (!cachedBasis) {
129  // Same calculation as above, without derivatives.
130  // Done after H block, as that possibly computes B_ for the first time
131  Matrix32 B;
132 
133  const Point3 n(p_), axis = CalculateBestAxis(n);
134  const Point3 B1 = gtsam::cross(n, axis);
135  B.col(0) = normalize(B1);
136  B.col(1) = gtsam::cross(n, B.col(0));
137  B_ = (B);
138  }
139 
140  return *B_;
141 }
142 
143 /* ************************************************************************* */
145  if (H)
146  *H = basis();
147  return Point3(p_);
148 }
149 
150 /* ************************************************************************* */
152  if (H)
153  *H = basis();
154  return p_;
155 }
156 
157 /* ************************************************************************* */
158 std::ostream& operator<<(std::ostream& os, const Unit3& pair) {
159  os << pair.p_ << endl;
160  return os;
161 }
162 
163 /* ************************************************************************* */
164 void Unit3::print(const std::string& s) const {
165  cout << s << ":" << p_ << endl;
166 }
167 
168 /* ************************************************************************* */
169 Matrix3 Unit3::skew() const {
170  return skewSymmetric(p_.x(), p_.y(), p_.z());
171 }
172 
173 /* ************************************************************************* */
175  OptionalJacobian<1, 2> H_q) const {
176  // Get the unit vectors of each, and the derivative.
177  Matrix32 H_pn_p;
178  Point3 pn = point3(H_p ? &H_pn_p : nullptr);
179 
180  Matrix32 H_qn_q;
181  const Point3 qn = q.point3(H_q ? &H_qn_q : nullptr);
182 
183  // Compute the dot product of the Point3s.
184  Matrix13 H_dot_pn, H_dot_qn;
185  double d = gtsam::dot(pn, qn, H_p ? &H_dot_pn : nullptr, H_q ? &H_dot_qn : nullptr);
186 
187  if (H_p) {
188  (*H_p) << H_dot_pn * H_pn_p;
189  }
190 
191  if (H_q) {
192  (*H_q) = H_dot_qn * H_qn_q;
193  }
194 
195  return d;
196 }
197 
198 /* ************************************************************************* */
200  // 2D error is equal to B'*q, as B is 3x2 matrix and q is 3x1
201  const Vector2 xi = basis().transpose() * q.p_;
202  if (H_q) {
203  *H_q = basis().transpose() * q.basis();
204  }
205  return xi;
206 }
207 
208 /* ************************************************************************* */
210  OptionalJacobian<2, 2> H_q) const {
211  // Get the point3 of this, and the derivative.
212  Matrix32 H_qn_q;
213  const Point3 qn = q.point3(H_q ? &H_qn_q : nullptr);
214 
215  // 2D error here is projecting q into the tangent plane of this (p).
216  Matrix62 H_B_p;
217  Matrix23 Bt = basis(H_p ? &H_B_p : nullptr).transpose();
218  Vector2 xi = Bt * qn;
219 
220  if (H_p) {
221  // Derivatives of each basis vector.
222  const Matrix32& H_b1_p = H_B_p.block<3, 2>(0, 0);
223  const Matrix32& H_b2_p = H_B_p.block<3, 2>(3, 0);
224 
225  // Derivatives of the two entries of xi wrt the basis vectors.
226  const Matrix13 H_xi1_b1 = qn.transpose();
227  const Matrix13 H_xi2_b2 = qn.transpose();
228 
229  // Assemble dxi/dp = dxi/dB * dB/dp.
230  const Matrix12 H_xi1_p = H_xi1_b1 * H_b1_p;
231  const Matrix12 H_xi2_p = H_xi2_b2 * H_b2_p;
232  *H_p << H_xi1_p, H_xi2_p;
233  }
234 
235  if (H_q) {
236  // dxi/dq is given by dxi/dqu * dqu/dq, where qu is the unit vector of q.
237  const Matrix23 H_xi_qu = Bt;
238  *H_q = H_xi_qu * H_qn_q;
239  }
240 
241  return xi;
242 }
243 
244 /* ************************************************************************* */
246  Matrix2 H_xi_q;
247  const Vector2 xi = error(q, H ? &H_xi_q : nullptr);
248  const double theta = xi.norm();
249  if (H)
250  *H = (xi.transpose() / theta) * H_xi_q;
251  return theta;
252 }
253 
254 /* ************************************************************************* */
256  // Compute the 3D xi_hat vector
257  const Vector3 xi_hat = basis() * v;
258  const double theta = xi_hat.norm();
259  const double c = std::cos(theta);
260 
261  // Treat case of very small v differently.
262  Matrix23 H_from_point;
263  if (theta < std::numeric_limits<double>::epsilon()) {
264  const Unit3 exp_p_xi_hat = Unit3::FromPoint3(c * p_ + xi_hat,
265  H? &H_from_point : nullptr);
266  if (H) { // Jacobian
267  *H = H_from_point *
268  (-p_ * xi_hat.transpose() + Matrix33::Identity()) * basis();
269  }
270  return exp_p_xi_hat;
271  }
272 
273  const double st = std::sin(theta) / theta;
274  const Unit3 exp_p_xi_hat = Unit3::FromPoint3(c * p_ + xi_hat * st,
275  H? &H_from_point : nullptr);
276  if (H) { // Jacobian
277  *H = H_from_point *
278  (p_ * -st * xi_hat.transpose() + st * Matrix33::Identity() +
279  xi_hat * ((c - st) / std::pow(theta, 2)) * xi_hat.transpose()) * basis();
280  }
281  return exp_p_xi_hat;
282 }
283 
284 /* ************************************************************************* */
286  const double x = p_.dot(other.p_);
287  // Crucial quantity here is y = theta/sin(theta) with theta=acos(x)
288  // Now, y = acos(x) / sin(acos(x)) = acos(x)/sqrt(1-x^2)
289  // We treat the special case 1 and -1 below
290  const double x2 = x * x;
291  const double z = 1 - x2;
292  double y;
294  if (x > 0) // first order expansion at x=1
295  y = 1.0 - (x - 1.0) / 3.0;
296  else // cop out
297  return Vector2(M_PI, 0.0);
298  } else {
299  // no special case
300  y = acos(x) / sqrt(z);
301  }
302  return basis().transpose() * y * (other.p_ - x * p_);
303 }
304 /* ************************************************************************* */
305 
306 } // namespace gtsam
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