Point Cloud Library (PCL)  1.14.0-dev
ndt.hpp
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40 
41 #ifndef PCL_REGISTRATION_NDT_IMPL_H_
42 #define PCL_REGISTRATION_NDT_IMPL_H_
43 
44 namespace pcl {
45 
46 template <typename PointSource, typename PointTarget, typename Scalar>
49 : target_cells_()
50 {
51  reg_name_ = "NormalDistributionsTransform";
52 
53  // Initializes the gaussian fitting parameters (eq. 6.8) [Magnusson 2009]
54  const double gauss_c1 = 10.0 * (1 - outlier_ratio_);
55  const double gauss_c2 = outlier_ratio_ / pow(resolution_, 3);
56  const double gauss_d3 = -std::log(gauss_c2);
57  gauss_d1_ = -std::log(gauss_c1 + gauss_c2) - gauss_d3;
58  gauss_d2_ =
59  -2 * std::log((-std::log(gauss_c1 * std::exp(-0.5) + gauss_c2) - gauss_d3) /
60  gauss_d1_);
61 
63  max_iterations_ = 35;
64 }
65 
66 template <typename PointSource, typename PointTarget, typename Scalar>
67 void
69  PointCloudSource& output, const Matrix4& guess)
70 {
71  nr_iterations_ = 0;
72  converged_ = false;
73  if (target_cells_.getCentroids()->empty()) {
74  PCL_ERROR("[%s::computeTransformation] Voxel grid is not searchable!\n",
75  getClassName().c_str());
76  return;
77  }
78 
79  // Initializes the gaussian fitting parameters (eq. 6.8) [Magnusson 2009]
80  const double gauss_c1 = 10 * (1 - outlier_ratio_);
81  const double gauss_c2 = outlier_ratio_ / pow(resolution_, 3);
82  const double gauss_d3 = -std::log(gauss_c2);
83  gauss_d1_ = -std::log(gauss_c1 + gauss_c2) - gauss_d3;
84  gauss_d2_ =
85  -2 * std::log((-std::log(gauss_c1 * std::exp(-0.5) + gauss_c2) - gauss_d3) /
86  gauss_d1_);
87 
88  if (guess != Matrix4::Identity()) {
89  // Initialise final transformation to the guessed one
90  final_transformation_ = guess;
91  // Apply guessed transformation prior to search for neighbours
92  transformPointCloud(output, output, guess);
93  }
94 
95  // Initialize Point Gradient and Hessian
96  point_jacobian_.setZero();
97  point_jacobian_.block<3, 3>(0, 0).setIdentity();
98  point_hessian_.setZero();
99 
100  Eigen::Transform<Scalar, 3, Eigen::Affine, Eigen::ColMajor> eig_transformation;
101  eig_transformation.matrix() = final_transformation_;
102 
103  // Convert initial guess matrix to 6 element transformation vector
104  Eigen::Matrix<double, 6, 1> transform, score_gradient;
105  Vector3 init_translation = eig_transformation.translation();
106  Vector3 init_rotation = eig_transformation.rotation().eulerAngles(0, 1, 2);
107  transform << init_translation.template cast<double>(),
108  init_rotation.template cast<double>();
109 
110  Eigen::Matrix<double, 6, 6> hessian;
111 
112  // Calculate derivates of initial transform vector, subsequent derivative calculations
113  // are done in the step length determination.
114  double score = computeDerivatives(score_gradient, hessian, output, transform);
115 
116  while (!converged_) {
117  // Store previous transformation
118  previous_transformation_ = transformation_;
119 
120  // Solve for decent direction using newton method, line 23 in Algorithm 2 [Magnusson
121  // 2009]
122  Eigen::JacobiSVD<Eigen::Matrix<double, 6, 6>> sv(
123  hessian, Eigen::ComputeFullU | Eigen::ComputeFullV);
124  // Negative for maximization as opposed to minimization
125  Eigen::Matrix<double, 6, 1> delta = sv.solve(-score_gradient);
126 
127  // Calculate step length with guaranteed sufficient decrease [More, Thuente 1994]
128  double delta_norm = delta.norm();
129 
130  if (delta_norm == 0 || std::isnan(delta_norm)) {
131  trans_likelihood_ = score / static_cast<double>(input_->size());
132  converged_ = delta_norm == 0;
133  return;
134  }
135 
136  delta /= delta_norm;
137  delta_norm = computeStepLengthMT(transform,
138  delta,
139  delta_norm,
140  step_size_,
141  transformation_epsilon_ / 2,
142  score,
143  score_gradient,
144  hessian,
145  output);
146  delta *= delta_norm;
147 
148  // Convert delta into matrix form
149  convertTransform(delta, transformation_);
150 
151  transform += delta;
152 
153  // Update Visualizer (untested)
154  if (update_visualizer_)
155  update_visualizer_(output, pcl::Indices(), *target_, pcl::Indices());
156 
157  const double cos_angle =
158  0.5 * (transformation_.template block<3, 3>(0, 0).trace() - 1);
159  const double translation_sqr =
160  transformation_.template block<3, 1>(0, 3).squaredNorm();
161 
162  nr_iterations_++;
163 
164  if (nr_iterations_ >= max_iterations_ ||
165  ((transformation_epsilon_ > 0 && translation_sqr <= transformation_epsilon_) &&
166  (transformation_rotation_epsilon_ > 0 &&
167  cos_angle >= transformation_rotation_epsilon_)) ||
168  ((transformation_epsilon_ <= 0) &&
169  (transformation_rotation_epsilon_ > 0 &&
170  cos_angle >= transformation_rotation_epsilon_)) ||
171  ((transformation_epsilon_ > 0 && translation_sqr <= transformation_epsilon_) &&
172  (transformation_rotation_epsilon_ <= 0))) {
173  converged_ = true;
174  }
175  }
176 
177  // Store transformation likelihood. The relative differences within each scan
178  // registration are accurate but the normalization constants need to be modified for
179  // it to be globally accurate
180  trans_likelihood_ = score / static_cast<double>(input_->size());
181 }
182 
183 template <typename PointSource, typename PointTarget, typename Scalar>
184 double
186  Eigen::Matrix<double, 6, 1>& score_gradient,
187  Eigen::Matrix<double, 6, 6>& hessian,
188  const PointCloudSource& trans_cloud,
189  const Eigen::Matrix<double, 6, 1>& transform,
190  bool compute_hessian)
191 {
192  score_gradient.setZero();
193  hessian.setZero();
194  double score = 0;
195 
196  // Precompute Angular Derivatives (eq. 6.19 and 6.21)[Magnusson 2009]
197  computeAngleDerivatives(transform);
198 
199  // Update gradient and hessian for each point, line 17 in Algorithm 2 [Magnusson 2009]
200  for (std::size_t idx = 0; idx < input_->size(); idx++) {
201  // Transformed Point
202  const auto& x_trans_pt = trans_cloud[idx];
203 
204  // Find neighbors (Radius search has been experimentally faster than direct neighbor
205  // checking.
206  std::vector<TargetGridLeafConstPtr> neighborhood;
207  std::vector<float> distances;
208  target_cells_.radiusSearch(x_trans_pt, resolution_, neighborhood, distances);
209 
210  for (const auto& cell : neighborhood) {
211  // Original Point
212  const auto& x_pt = (*input_)[idx];
213  const Eigen::Vector3d x = x_pt.getVector3fMap().template cast<double>();
214 
215  // Denorm point, x_k' in Equations 6.12 and 6.13 [Magnusson 2009]
216  const Eigen::Vector3d x_trans =
217  x_trans_pt.getVector3fMap().template cast<double>() - cell->getMean();
218  // Inverse Covariance of Occupied Voxel
219  // Uses precomputed covariance for speed.
220  const Eigen::Matrix3d c_inv = cell->getInverseCov();
221 
222  // Compute derivative of transform function w.r.t. transform vector, J_E and H_E
223  // in Equations 6.18 and 6.20 [Magnusson 2009]
224  computePointDerivatives(x);
225  // Update score, gradient and hessian, lines 19-21 in Algorithm 2, according to
226  // Equations 6.10, 6.12 and 6.13, respectively [Magnusson 2009]
227  score +=
228  updateDerivatives(score_gradient, hessian, x_trans, c_inv, compute_hessian);
229  }
230  }
231  return score;
232 }
233 
234 template <typename PointSource, typename PointTarget, typename Scalar>
235 void
237  const Eigen::Matrix<double, 6, 1>& transform, bool compute_hessian)
238 {
239  // Simplified math for near 0 angles
240  const auto calculate_cos_sin = [](double angle, double& c, double& s) {
241  if (std::abs(angle) < 10e-5) {
242  c = 1.0;
243  s = 0.0;
244  }
245  else {
246  c = std::cos(angle);
247  s = std::sin(angle);
248  }
249  };
250 
251  double cx, cy, cz, sx, sy, sz;
252  calculate_cos_sin(transform(3), cx, sx);
253  calculate_cos_sin(transform(4), cy, sy);
254  calculate_cos_sin(transform(5), cz, sz);
255 
256  // Precomputed angular gradient components. Letters correspond to Equation 6.19
257  // [Magnusson 2009]
258  angular_jacobian_.setZero();
259  angular_jacobian_.row(0).noalias() = Eigen::Vector4d(
260  (-sx * sz + cx * sy * cz), (-sx * cz - cx * sy * sz), (-cx * cy), 1.0); // a
261  angular_jacobian_.row(1).noalias() = Eigen::Vector4d(
262  (cx * sz + sx * sy * cz), (cx * cz - sx * sy * sz), (-sx * cy), 1.0); // b
263  angular_jacobian_.row(2).noalias() =
264  Eigen::Vector4d((-sy * cz), sy * sz, cy, 1.0); // c
265  angular_jacobian_.row(3).noalias() =
266  Eigen::Vector4d(sx * cy * cz, (-sx * cy * sz), sx * sy, 1.0); // d
267  angular_jacobian_.row(4).noalias() =
268  Eigen::Vector4d((-cx * cy * cz), cx * cy * sz, (-cx * sy), 1.0); // e
269  angular_jacobian_.row(5).noalias() =
270  Eigen::Vector4d((-cy * sz), (-cy * cz), 0, 1.0); // f
271  angular_jacobian_.row(6).noalias() =
272  Eigen::Vector4d((cx * cz - sx * sy * sz), (-cx * sz - sx * sy * cz), 0, 1.0); // g
273  angular_jacobian_.row(7).noalias() =
274  Eigen::Vector4d((sx * cz + cx * sy * sz), (cx * sy * cz - sx * sz), 0, 1.0); // h
275 
276  if (compute_hessian) {
277  // Precomputed angular hessian components. Letters correspond to Equation 6.21 and
278  // numbers correspond to row index [Magnusson 2009]
279  angular_hessian_.setZero();
280  angular_hessian_.row(0).noalias() = Eigen::Vector4d(
281  (-cx * sz - sx * sy * cz), (-cx * cz + sx * sy * sz), sx * cy, 0.0f); // a2
282  angular_hessian_.row(1).noalias() = Eigen::Vector4d(
283  (-sx * sz + cx * sy * cz), (-cx * sy * sz - sx * cz), (-cx * cy), 0.0f); // a3
284 
285  angular_hessian_.row(2).noalias() =
286  Eigen::Vector4d((cx * cy * cz), (-cx * cy * sz), (cx * sy), 0.0f); // b2
287  angular_hessian_.row(3).noalias() =
288  Eigen::Vector4d((sx * cy * cz), (-sx * cy * sz), (sx * sy), 0.0f); // b3
289 
290  // The sign of 'sx * sz' in c2 is incorrect in the thesis, and is fixed here.
291  angular_hessian_.row(4).noalias() = Eigen::Vector4d(
292  (-sx * cz - cx * sy * sz), (sx * sz - cx * sy * cz), 0, 0.0f); // c2
293  angular_hessian_.row(5).noalias() = Eigen::Vector4d(
294  (cx * cz - sx * sy * sz), (-sx * sy * cz - cx * sz), 0, 0.0f); // c3
295 
296  angular_hessian_.row(6).noalias() =
297  Eigen::Vector4d((-cy * cz), (cy * sz), (-sy), 0.0f); // d1
298  angular_hessian_.row(7).noalias() =
299  Eigen::Vector4d((-sx * sy * cz), (sx * sy * sz), (sx * cy), 0.0f); // d2
300  angular_hessian_.row(8).noalias() =
301  Eigen::Vector4d((cx * sy * cz), (-cx * sy * sz), (-cx * cy), 0.0f); // d3
302 
303  angular_hessian_.row(9).noalias() =
304  Eigen::Vector4d((sy * sz), (sy * cz), 0, 0.0f); // e1
305  angular_hessian_.row(10).noalias() =
306  Eigen::Vector4d((-sx * cy * sz), (-sx * cy * cz), 0, 0.0f); // e2
307  angular_hessian_.row(11).noalias() =
308  Eigen::Vector4d((cx * cy * sz), (cx * cy * cz), 0, 0.0f); // e3
309 
310  angular_hessian_.row(12).noalias() =
311  Eigen::Vector4d((-cy * cz), (cy * sz), 0, 0.0f); // f1
312  angular_hessian_.row(13).noalias() = Eigen::Vector4d(
313  (-cx * sz - sx * sy * cz), (-cx * cz + sx * sy * sz), 0, 0.0f); // f2
314  angular_hessian_.row(14).noalias() = Eigen::Vector4d(
315  (-sx * sz + cx * sy * cz), (-cx * sy * sz - sx * cz), 0, 0.0f); // f3
316  }
317 }
318 
319 template <typename PointSource, typename PointTarget, typename Scalar>
320 void
322  const Eigen::Vector3d& x, bool compute_hessian)
323 {
324  // Calculate first derivative of Transformation Equation 6.17 w.r.t. transform vector.
325  // Derivative w.r.t. ith element of transform vector corresponds to column i,
326  // Equation 6.18 and 6.19 [Magnusson 2009]
327  Eigen::Matrix<double, 8, 1> point_angular_jacobian =
328  angular_jacobian_ * Eigen::Vector4d(x[0], x[1], x[2], 0.0);
329  point_jacobian_(1, 3) = point_angular_jacobian[0];
330  point_jacobian_(2, 3) = point_angular_jacobian[1];
331  point_jacobian_(0, 4) = point_angular_jacobian[2];
332  point_jacobian_(1, 4) = point_angular_jacobian[3];
333  point_jacobian_(2, 4) = point_angular_jacobian[4];
334  point_jacobian_(0, 5) = point_angular_jacobian[5];
335  point_jacobian_(1, 5) = point_angular_jacobian[6];
336  point_jacobian_(2, 5) = point_angular_jacobian[7];
337 
338  if (compute_hessian) {
339  Eigen::Matrix<double, 15, 1> point_angular_hessian =
340  angular_hessian_ * Eigen::Vector4d(x[0], x[1], x[2], 0.0);
341 
342  // Vectors from Equation 6.21 [Magnusson 2009]
343  const Eigen::Vector3d a(0, point_angular_hessian[0], point_angular_hessian[1]);
344  const Eigen::Vector3d b(0, point_angular_hessian[2], point_angular_hessian[3]);
345  const Eigen::Vector3d c(0, point_angular_hessian[4], point_angular_hessian[5]);
346  const Eigen::Vector3d d = point_angular_hessian.block<3, 1>(6, 0);
347  const Eigen::Vector3d e = point_angular_hessian.block<3, 1>(9, 0);
348  const Eigen::Vector3d f = point_angular_hessian.block<3, 1>(12, 0);
349 
350  // Calculate second derivative of Transformation Equation 6.17 w.r.t. transform
351  // vector. Derivative w.r.t. ith and jth elements of transform vector corresponds to
352  // the 3x1 block matrix starting at (3i,j), Equation 6.20 and 6.21 [Magnusson 2009]
353  point_hessian_.block<3, 1>(9, 3) = a;
354  point_hessian_.block<3, 1>(12, 3) = b;
355  point_hessian_.block<3, 1>(15, 3) = c;
356  point_hessian_.block<3, 1>(9, 4) = b;
357  point_hessian_.block<3, 1>(12, 4) = d;
358  point_hessian_.block<3, 1>(15, 4) = e;
359  point_hessian_.block<3, 1>(9, 5) = c;
360  point_hessian_.block<3, 1>(12, 5) = e;
361  point_hessian_.block<3, 1>(15, 5) = f;
362  }
363 }
364 
365 template <typename PointSource, typename PointTarget, typename Scalar>
366 double
368  Eigen::Matrix<double, 6, 1>& score_gradient,
369  Eigen::Matrix<double, 6, 6>& hessian,
370  const Eigen::Vector3d& x_trans,
371  const Eigen::Matrix3d& c_inv,
372  bool compute_hessian) const
373 {
374  // e^(-d_2/2 * (x_k - mu_k)^T Sigma_k^-1 (x_k - mu_k)) Equation 6.9 [Magnusson 2009]
375  double e_x_cov_x = std::exp(-gauss_d2_ * x_trans.dot(c_inv * x_trans) / 2);
376  // Calculate likelihood of transformed points existence, Equation 6.9 [Magnusson
377  // 2009]
378  const double score_inc = -gauss_d1_ * e_x_cov_x;
379 
380  e_x_cov_x = gauss_d2_ * e_x_cov_x;
381 
382  // Error checking for invalid values.
383  if (e_x_cov_x > 1 || e_x_cov_x < 0 || std::isnan(e_x_cov_x)) {
384  return 0;
385  }
386 
387  // Reusable portion of Equation 6.12 and 6.13 [Magnusson 2009]
388  e_x_cov_x *= gauss_d1_;
389 
390  for (int i = 0; i < 6; i++) {
391  // Sigma_k^-1 d(T(x,p))/dpi, Reusable portion of Equation 6.12 and 6.13 [Magnusson
392  // 2009]
393  const Eigen::Vector3d cov_dxd_pi = c_inv * point_jacobian_.col(i);
394 
395  // Update gradient, Equation 6.12 [Magnusson 2009]
396  score_gradient(i) += x_trans.dot(cov_dxd_pi) * e_x_cov_x;
397 
398  if (compute_hessian) {
399  for (Eigen::Index j = 0; j < hessian.cols(); j++) {
400  // Update hessian, Equation 6.13 [Magnusson 2009]
401  hessian(i, j) +=
402  e_x_cov_x * (-gauss_d2_ * x_trans.dot(cov_dxd_pi) *
403  x_trans.dot(c_inv * point_jacobian_.col(j)) +
404  x_trans.dot(c_inv * point_hessian_.block<3, 1>(3 * i, j)) +
405  point_jacobian_.col(j).dot(cov_dxd_pi));
406  }
407  }
408  }
409 
410  return score_inc;
411 }
412 
413 template <typename PointSource, typename PointTarget, typename Scalar>
414 void
416  Eigen::Matrix<double, 6, 6>& hessian, const PointCloudSource& trans_cloud)
417 {
418  hessian.setZero();
419 
420  // Precompute Angular Derivatives unnecessary because only used after regular
421  // derivative calculation Update hessian for each point, line 17 in Algorithm 2
422  // [Magnusson 2009]
423  for (std::size_t idx = 0; idx < input_->size(); idx++) {
424  // Transformed Point
425  const auto& x_trans_pt = trans_cloud[idx];
426 
427  // Find neighbors (Radius search has been experimentally faster than direct neighbor
428  // checking.
429  std::vector<TargetGridLeafConstPtr> neighborhood;
430  std::vector<float> distances;
431  target_cells_.radiusSearch(x_trans_pt, resolution_, neighborhood, distances);
432 
433  for (const auto& cell : neighborhood) {
434  // Original Point
435  const auto& x_pt = (*input_)[idx];
436  const Eigen::Vector3d x = x_pt.getVector3fMap().template cast<double>();
437 
438  // Denorm point, x_k' in Equations 6.12 and 6.13 [Magnusson 2009]
439  const Eigen::Vector3d x_trans =
440  x_trans_pt.getVector3fMap().template cast<double>() - cell->getMean();
441  // Inverse Covariance of Occupied Voxel
442  // Uses precomputed covariance for speed.
443  const Eigen::Matrix3d c_inv = cell->getInverseCov();
444 
445  // Compute derivative of transform function w.r.t. transform vector, J_E and H_E
446  // in Equations 6.18 and 6.20 [Magnusson 2009]
447  computePointDerivatives(x);
448  // Update hessian, lines 21 in Algorithm 2, according to Equations 6.10, 6.12
449  // and 6.13, respectively [Magnusson 2009]
450  updateHessian(hessian, x_trans, c_inv);
451  }
452  }
453 }
454 
455 template <typename PointSource, typename PointTarget, typename Scalar>
456 void
458  Eigen::Matrix<double, 6, 6>& hessian,
459  const Eigen::Vector3d& x_trans,
460  const Eigen::Matrix3d& c_inv) const
461 {
462  // e^(-d_2/2 * (x_k - mu_k)^T Sigma_k^-1 (x_k - mu_k)) Equation 6.9 [Magnusson 2009]
463  double e_x_cov_x =
464  gauss_d2_ * std::exp(-gauss_d2_ * x_trans.dot(c_inv * x_trans) / 2);
465 
466  // Error checking for invalid values.
467  if (e_x_cov_x > 1 || e_x_cov_x < 0 || std::isnan(e_x_cov_x)) {
468  return;
469  }
470 
471  // Reusable portion of Equation 6.12 and 6.13 [Magnusson 2009]
472  e_x_cov_x *= gauss_d1_;
473 
474  for (int i = 0; i < 6; i++) {
475  // Sigma_k^-1 d(T(x,p))/dpi, Reusable portion of Equation 6.12 and 6.13 [Magnusson
476  // 2009]
477  const Eigen::Vector3d cov_dxd_pi = c_inv * point_jacobian_.col(i);
478 
479  for (Eigen::Index j = 0; j < hessian.cols(); j++) {
480  // Update hessian, Equation 6.13 [Magnusson 2009]
481  hessian(i, j) +=
482  e_x_cov_x * (-gauss_d2_ * x_trans.dot(cov_dxd_pi) *
483  x_trans.dot(c_inv * point_jacobian_.col(j)) +
484  x_trans.dot(c_inv * point_hessian_.block<3, 1>(3 * i, j)) +
485  point_jacobian_.col(j).dot(cov_dxd_pi));
486  }
487  }
488 }
489 
490 template <typename PointSource, typename PointTarget, typename Scalar>
491 bool
493  double& a_l,
494  double& f_l,
495  double& g_l,
496  double& a_u,
497  double& f_u,
498  double& g_u,
499  double a_t,
500  double f_t,
501  double g_t) const
502 {
503  // Case U1 in Update Algorithm and Case a in Modified Update Algorithm [More, Thuente
504  // 1994]
505  if (f_t > f_l) {
506  a_u = a_t;
507  f_u = f_t;
508  g_u = g_t;
509  return false;
510  }
511  // Case U2 in Update Algorithm and Case b in Modified Update Algorithm [More, Thuente
512  // 1994]
513  if (g_t * (a_l - a_t) > 0) {
514  a_l = a_t;
515  f_l = f_t;
516  g_l = g_t;
517  return false;
518  }
519  // Case U3 in Update Algorithm and Case c in Modified Update Algorithm [More, Thuente
520  // 1994]
521  if (g_t * (a_l - a_t) < 0) {
522  a_u = a_l;
523  f_u = f_l;
524  g_u = g_l;
525 
526  a_l = a_t;
527  f_l = f_t;
528  g_l = g_t;
529  return false;
530  }
531  // Interval Converged
532  return true;
533 }
534 
535 template <typename PointSource, typename PointTarget, typename Scalar>
536 double
538  double a_l,
539  double f_l,
540  double g_l,
541  double a_u,
542  double f_u,
543  double g_u,
544  double a_t,
545  double f_t,
546  double g_t) const
547 {
548  if (a_t == a_l && a_t == a_u) {
549  return a_t;
550  }
551 
552  // Endpoints condition check [More, Thuente 1994], p.299 - 300
553  enum class EndpointsCondition { Case1, Case2, Case3, Case4 };
554  EndpointsCondition condition;
555 
556  if (a_t == a_l) {
557  condition = EndpointsCondition::Case4;
558  }
559  else if (f_t > f_l) {
560  condition = EndpointsCondition::Case1;
561  }
562  else if (g_t * g_l < 0) {
563  condition = EndpointsCondition::Case2;
564  }
565  else if (std::fabs(g_t) <= std::fabs(g_l)) {
566  condition = EndpointsCondition::Case3;
567  }
568  else {
569  condition = EndpointsCondition::Case4;
570  }
571 
572  switch (condition) {
573  case EndpointsCondition::Case1: {
574  // Calculate the minimizer of the cubic that interpolates f_l, f_t, g_l and g_t
575  // Equation 2.4.52 [Sun, Yuan 2006]
576  const double z = 3 * (f_t - f_l) / (a_t - a_l) - g_t - g_l;
577  const double w = std::sqrt(z * z - g_t * g_l);
578  // Equation 2.4.56 [Sun, Yuan 2006]
579  const double a_c = a_l + (a_t - a_l) * (w - g_l - z) / (g_t - g_l + 2 * w);
580 
581  // Calculate the minimizer of the quadratic that interpolates f_l, f_t and g_l
582  // Equation 2.4.2 [Sun, Yuan 2006]
583  const double a_q =
584  a_l - 0.5 * (a_l - a_t) * g_l / (g_l - (f_l - f_t) / (a_l - a_t));
585 
586  if (std::fabs(a_c - a_l) < std::fabs(a_q - a_l)) {
587  return a_c;
588  }
589  return 0.5 * (a_q + a_c);
590  }
591 
592  case EndpointsCondition::Case2: {
593  // Calculate the minimizer of the cubic that interpolates f_l, f_t, g_l and g_t
594  // Equation 2.4.52 [Sun, Yuan 2006]
595  const double z = 3 * (f_t - f_l) / (a_t - a_l) - g_t - g_l;
596  const double w = std::sqrt(z * z - g_t * g_l);
597  // Equation 2.4.56 [Sun, Yuan 2006]
598  const double a_c = a_l + (a_t - a_l) * (w - g_l - z) / (g_t - g_l + 2 * w);
599 
600  // Calculate the minimizer of the quadratic that interpolates f_l, g_l and g_t
601  // Equation 2.4.5 [Sun, Yuan 2006]
602  const double a_s = a_l - (a_l - a_t) / (g_l - g_t) * g_l;
603 
604  if (std::fabs(a_c - a_t) >= std::fabs(a_s - a_t)) {
605  return a_c;
606  }
607  return a_s;
608  }
609 
610  case EndpointsCondition::Case3: {
611  // Calculate the minimizer of the cubic that interpolates f_l, f_t, g_l and g_t
612  // Equation 2.4.52 [Sun, Yuan 2006]
613  const double z = 3 * (f_t - f_l) / (a_t - a_l) - g_t - g_l;
614  const double w = std::sqrt(z * z - g_t * g_l);
615  const double a_c = a_l + (a_t - a_l) * (w - g_l - z) / (g_t - g_l + 2 * w);
616 
617  // Calculate the minimizer of the quadratic that interpolates g_l and g_t
618  // Equation 2.4.5 [Sun, Yuan 2006]
619  const double a_s = a_l - (a_l - a_t) / (g_l - g_t) * g_l;
620 
621  double a_t_next;
622 
623  if (std::fabs(a_c - a_t) < std::fabs(a_s - a_t)) {
624  a_t_next = a_c;
625  }
626  else {
627  a_t_next = a_s;
628  }
629 
630  if (a_t > a_l) {
631  return std::min(a_t + 0.66 * (a_u - a_t), a_t_next);
632  }
633  return std::max(a_t + 0.66 * (a_u - a_t), a_t_next);
634  }
635 
636  default:
637  case EndpointsCondition::Case4: {
638  // Calculate the minimizer of the cubic that interpolates f_u, f_t, g_u and g_t
639  // Equation 2.4.52 [Sun, Yuan 2006]
640  const double z = 3 * (f_t - f_u) / (a_t - a_u) - g_t - g_u;
641  const double w = std::sqrt(z * z - g_t * g_u);
642  // Equation 2.4.56 [Sun, Yuan 2006]
643  return a_u + (a_t - a_u) * (w - g_u - z) / (g_t - g_u + 2 * w);
644  }
645  }
646 }
647 
648 template <typename PointSource, typename PointTarget, typename Scalar>
649 double
651  const Eigen::Matrix<double, 6, 1>& x,
652  Eigen::Matrix<double, 6, 1>& step_dir,
653  double step_init,
654  double step_max,
655  double step_min,
656  double& score,
657  Eigen::Matrix<double, 6, 1>& score_gradient,
658  Eigen::Matrix<double, 6, 6>& hessian,
659  PointCloudSource& trans_cloud)
660 {
661  // Set the value of phi(0), Equation 1.3 [More, Thuente 1994]
662  const double phi_0 = -score;
663  // Set the value of phi'(0), Equation 1.3 [More, Thuente 1994]
664  double d_phi_0 = -(score_gradient.dot(step_dir));
665 
666  if (d_phi_0 >= 0) {
667  // Not a decent direction
668  if (d_phi_0 == 0) {
669  return 0;
670  }
671  // Reverse step direction and calculate optimal step.
672  d_phi_0 *= -1;
673  step_dir *= -1;
674  }
675 
676  // The Search Algorithm for T(mu) [More, Thuente 1994]
677 
678  constexpr int max_step_iterations = 10;
679  int step_iterations = 0;
680 
681  // Sufficient decrease constant, Equation 1.1 [More, Thuete 1994]
682  constexpr double mu = 1.e-4;
683  // Curvature condition constant, Equation 1.2 [More, Thuete 1994]
684  constexpr double nu = 0.9;
685 
686  // Initial endpoints of Interval I,
687  double a_l = 0, a_u = 0;
688 
689  // Auxiliary function psi is used until I is determined ot be a closed interval,
690  // Equation 2.1 [More, Thuente 1994]
691  double f_l = auxilaryFunction_PsiMT(a_l, phi_0, phi_0, d_phi_0, mu);
692  double g_l = auxilaryFunction_dPsiMT(d_phi_0, d_phi_0, mu);
693 
694  double f_u = auxilaryFunction_PsiMT(a_u, phi_0, phi_0, d_phi_0, mu);
695  double g_u = auxilaryFunction_dPsiMT(d_phi_0, d_phi_0, mu);
696 
697  // Check used to allow More-Thuente step length calculation to be skipped by making
698  // step_min == step_max
699  bool interval_converged = (step_max - step_min) < 0, open_interval = true;
700 
701  double a_t = step_init;
702  a_t = std::min(a_t, step_max);
703  a_t = std::max(a_t, step_min);
704 
705  Eigen::Matrix<double, 6, 1> x_t = x + step_dir * a_t;
706 
707  // Convert x_t into matrix form
708  convertTransform(x_t, final_transformation_);
709 
710  // New transformed point cloud
711  transformPointCloud(*input_, trans_cloud, final_transformation_);
712 
713  // Updates score, gradient and hessian. Hessian calculation is unnecessary but
714  // testing showed that most step calculations use the initial step suggestion and
715  // recalculation the reusable portions of the hessian would entail more computation
716  // time.
717  score = computeDerivatives(score_gradient, hessian, trans_cloud, x_t, true);
718 
719  // Calculate phi(alpha_t)
720  double phi_t = -score;
721  // Calculate phi'(alpha_t)
722  double d_phi_t = -(score_gradient.dot(step_dir));
723 
724  // Calculate psi(alpha_t)
725  double psi_t = auxilaryFunction_PsiMT(a_t, phi_t, phi_0, d_phi_0, mu);
726  // Calculate psi'(alpha_t)
727  double d_psi_t = auxilaryFunction_dPsiMT(d_phi_t, d_phi_0, mu);
728 
729  // Iterate until max number of iterations, interval convergence or a value satisfies
730  // the sufficient decrease, Equation 1.1, and curvature condition, Equation 1.2 [More,
731  // Thuente 1994]
732  while (!interval_converged && step_iterations < max_step_iterations &&
733  !(psi_t <= 0 /*Sufficient Decrease*/ &&
734  d_phi_t <= -nu * d_phi_0 /*Curvature Condition*/)) {
735  // Use auxiliary function if interval I is not closed
736  if (open_interval) {
737  a_t = trialValueSelectionMT(a_l, f_l, g_l, a_u, f_u, g_u, a_t, psi_t, d_psi_t);
738  }
739  else {
740  a_t = trialValueSelectionMT(a_l, f_l, g_l, a_u, f_u, g_u, a_t, phi_t, d_phi_t);
741  }
742 
743  a_t = std::min(a_t, step_max);
744  a_t = std::max(a_t, step_min);
745 
746  x_t = x + step_dir * a_t;
747 
748  // Convert x_t into matrix form
749  convertTransform(x_t, final_transformation_);
750 
751  // New transformed point cloud
752  // Done on final cloud to prevent wasted computation
753  transformPointCloud(*input_, trans_cloud, final_transformation_);
754 
755  // Updates score, gradient. Values stored to prevent wasted computation.
756  score = computeDerivatives(score_gradient, hessian, trans_cloud, x_t, false);
757 
758  // Calculate phi(alpha_t+)
759  phi_t = -score;
760  // Calculate phi'(alpha_t+)
761  d_phi_t = -(score_gradient.dot(step_dir));
762 
763  // Calculate psi(alpha_t+)
764  psi_t = auxilaryFunction_PsiMT(a_t, phi_t, phi_0, d_phi_0, mu);
765  // Calculate psi'(alpha_t+)
766  d_psi_t = auxilaryFunction_dPsiMT(d_phi_t, d_phi_0, mu);
767 
768  // Check if I is now a closed interval
769  if (open_interval && (psi_t <= 0 && d_psi_t >= 0)) {
770  open_interval = false;
771 
772  // Converts f_l and g_l from psi to phi
773  f_l += phi_0 - mu * d_phi_0 * a_l;
774  g_l += mu * d_phi_0;
775 
776  // Converts f_u and g_u from psi to phi
777  f_u += phi_0 - mu * d_phi_0 * a_u;
778  g_u += mu * d_phi_0;
779  }
780 
781  if (open_interval) {
782  // Update interval end points using Updating Algorithm [More, Thuente 1994]
783  interval_converged =
784  updateIntervalMT(a_l, f_l, g_l, a_u, f_u, g_u, a_t, psi_t, d_psi_t);
785  }
786  else {
787  // Update interval end points using Modified Updating Algorithm [More, Thuente
788  // 1994]
789  interval_converged =
790  updateIntervalMT(a_l, f_l, g_l, a_u, f_u, g_u, a_t, phi_t, d_phi_t);
791  }
792 
793  step_iterations++;
794  }
795 
796  // If inner loop was run then hessian needs to be calculated.
797  // Hessian is unnecessary for step length determination but gradients are required
798  // so derivative and transform data is stored for the next iteration.
799  if (step_iterations) {
800  computeHessian(hessian, trans_cloud);
801  }
802 
803  return a_t;
804 }
805 
806 } // namespace pcl
807 
808 #endif // PCL_REGISTRATION_NDT_IMPL_H_
void computePointDerivatives(const Eigen::Vector3d &x, bool compute_hessian=true)
Compute point derivatives.
Definition: ndt.hpp:321
virtual void computeTransformation(PointCloudSource &output)
Estimate the transformation and returns the transformed source (input) as output.
Definition: ndt.h:289
double updateDerivatives(Eigen::Matrix< double, 6, 1 > &score_gradient, Eigen::Matrix< double, 6, 6 > &hessian, const Eigen::Vector3d &x_trans, const Eigen::Matrix3d &c_inv, bool compute_hessian=true) const
Compute individual point contributions to derivatives of likelihood function w.r.t.
Definition: ndt.hpp:367
double computeDerivatives(Eigen::Matrix< double, 6, 1 > &score_gradient, Eigen::Matrix< double, 6, 6 > &hessian, const PointCloudSource &trans_cloud, const Eigen::Matrix< double, 6, 1 > &transform, bool compute_hessian=true)
Compute derivatives of likelihood function w.r.t.
Definition: ndt.hpp:185
void computeAngleDerivatives(const Eigen::Matrix< double, 6, 1 > &transform, bool compute_hessian=true)
Precompute angular components of derivatives.
Definition: ndt.hpp:236
typename Registration< PointSource, PointTarget, Scalar >::PointCloudSource PointCloudSource
Definition: ndt.h:70
NormalDistributionsTransform()
Constructor.
Definition: ndt.hpp:48
bool updateIntervalMT(double &a_l, double &f_l, double &g_l, double &a_u, double &f_u, double &g_u, double a_t, double f_t, double g_t) const
Update interval of possible step lengths for More-Thuente method, in More-Thuente (1994)
Definition: ndt.hpp:492
void computeHessian(Eigen::Matrix< double, 6, 6 > &hessian, const PointCloudSource &trans_cloud)
Compute hessian of likelihood function w.r.t.
Definition: ndt.hpp:415
typename Eigen::Matrix< Scalar, 3, 1 > Vector3
Definition: ndt.h:97
float resolution_
The side length of voxels.
Definition: ndt.h:578
typename Registration< PointSource, PointTarget, Scalar >::Matrix4 Matrix4
Definition: ndt.h:98
double outlier_ratio_
The ratio of outliers of points w.r.t.
Definition: ndt.h:585
void updateHessian(Eigen::Matrix< double, 6, 6 > &hessian, const Eigen::Vector3d &x_trans, const Eigen::Matrix3d &c_inv) const
Compute individual point contributions to hessian of likelihood function w.r.t.
Definition: ndt.hpp:457
double gauss_d1_
The normalization constants used fit the point distribution to a normal distribution,...
Definition: ndt.h:589
double trialValueSelectionMT(double a_l, double f_l, double g_l, double a_u, double f_u, double g_u, double a_t, double f_t, double g_t) const
Select new trial value for More-Thuente method.
Definition: ndt.hpp:537
double computeStepLengthMT(const Eigen::Matrix< double, 6, 1 > &transform, Eigen::Matrix< double, 6, 1 > &step_dir, double step_init, double step_max, double step_min, double &score, Eigen::Matrix< double, 6, 1 > &score_gradient, Eigen::Matrix< double, 6, 6 > &hessian, PointCloudSource &trans_cloud)
Compute line search step length and update transform and likelihood derivatives using More-Thuente me...
Definition: ndt.hpp:650
std::string reg_name_
The registration method name.
Definition: registration.h:548
int max_iterations_
The maximum number of iterations the internal optimization should run for.
Definition: registration.h:563
double transformation_epsilon_
The maximum difference between two consecutive transformations in order to consider convergence (user...
Definition: registration.h:585
void transformPointCloud(const pcl::PointCloud< PointT > &cloud_in, pcl::PointCloud< PointT > &cloud_out, const Eigen::Matrix< Scalar, 4, 4 > &transform, bool copy_all_fields)
Apply a rigid transform defined by a 4x4 matrix.
Definition: transforms.hpp:221
IndicesAllocator<> Indices
Type used for indices in PCL.
Definition: types.h:133