Point Cloud Library (PCL)  1.10.0
ndt.hpp
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40 
41 #ifndef PCL_REGISTRATION_NDT_IMPL_H_
42 #define PCL_REGISTRATION_NDT_IMPL_H_
43 
44 //////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
45 template<typename PointSource, typename PointTarget>
47  : target_cells_ ()
48  , resolution_ (1.0f)
49  , step_size_ (0.1)
50  , outlier_ratio_ (0.55)
51  , gauss_d1_ ()
52  , gauss_d2_ ()
53  , trans_probability_ ()
54 {
55  reg_name_ = "NormalDistributionsTransform";
56 
57  double gauss_c1, gauss_c2, gauss_d3;
58 
59  // Initializes the gaussian fitting parameters (eq. 6.8) [Magnusson 2009]
60  gauss_c1 = 10.0 * (1 - outlier_ratio_);
61  gauss_c2 = outlier_ratio_ / pow (resolution_, 3);
62  gauss_d3 = -std::log (gauss_c2);
63  gauss_d1_ = -std::log ( gauss_c1 + gauss_c2 ) - gauss_d3;
64  gauss_d2_ = -2 * std::log ((-std::log ( gauss_c1 * std::exp ( -0.5 ) + gauss_c2 ) - gauss_d3) / gauss_d1_);
65 
67  max_iterations_ = 35;
68 }
69 
70 //////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
71 template<typename PointSource, typename PointTarget> void
73 {
74  nr_iterations_ = 0;
75  converged_ = false;
76 
77  double gauss_c1, gauss_c2, gauss_d3;
78 
79  // Initializes the gaussian fitting parameters (eq. 6.8) [Magnusson 2009]
80  gauss_c1 = 10 * (1 - outlier_ratio_);
81  gauss_c2 = outlier_ratio_ / pow (resolution_, 3);
82  gauss_d3 = -std::log (gauss_c2);
83  gauss_d1_ = -std::log ( gauss_c1 + gauss_c2 ) - gauss_d3;
84  gauss_d2_ = -2 * std::log ((-std::log ( gauss_c1 * std::exp ( -0.5 ) + gauss_c2 ) - gauss_d3) / gauss_d1_);
85 
86  if (guess != Eigen::Matrix4f::Identity ())
87  {
88  // Initialise final transformation to the guessed one
89  final_transformation_ = guess;
90  // Apply guessed transformation prior to search for neighbours
91  transformPointCloud (output, output, guess);
92  }
93 
94  // Initialize Point Gradient and Hessian
95  point_gradient_.setZero ();
96  point_gradient_.block<3, 3>(0, 0).setIdentity ();
97  point_hessian_.setZero ();
98 
99  Eigen::Transform<float, 3, Eigen::Affine, Eigen::ColMajor> eig_transformation;
100  eig_transformation.matrix () = final_transformation_;
101 
102  // Convert initial guess matrix to 6 element transformation vector
103  Eigen::Matrix<double, 6, 1> p, delta_p, score_gradient;
104  Eigen::Vector3f init_translation = eig_transformation.translation ();
105  Eigen::Vector3f init_rotation = eig_transformation.rotation ().eulerAngles (0, 1, 2);
106  p << init_translation (0), init_translation (1), init_translation (2),
107  init_rotation (0), init_rotation (1), init_rotation (2);
108 
109  Eigen::Matrix<double, 6, 6> hessian;
110 
111  double score = 0;
112  double delta_p_norm;
113 
114  // Calculate derivates of initial transform vector, subsequent derivative calculations are done in the step length determination.
115  score = computeDerivatives (score_gradient, hessian, output, p);
116 
117  while (!converged_)
118  {
119  // Store previous transformation
120  previous_transformation_ = transformation_;
121 
122  // Solve for decent direction using newton method, line 23 in Algorithm 2 [Magnusson 2009]
123  Eigen::JacobiSVD<Eigen::Matrix<double, 6, 6> > sv (hessian, Eigen::ComputeFullU | Eigen::ComputeFullV);
124  // Negative for maximization as opposed to minimization
125  delta_p = sv.solve (-score_gradient);
126 
127  //Calculate step length with guarnteed sufficient decrease [More, Thuente 1994]
128  delta_p_norm = delta_p.norm ();
129 
130  if (delta_p_norm == 0 || std::isnan(delta_p_norm))
131  {
132  trans_probability_ = score / static_cast<double> (input_->points.size ());
133  converged_ = delta_p_norm == delta_p_norm;
134  return;
135  }
136 
137  delta_p.normalize ();
138  delta_p_norm = computeStepLengthMT (p, delta_p, delta_p_norm, step_size_, transformation_epsilon_ / 2, score, score_gradient, hessian, output);
139  delta_p *= delta_p_norm;
140 
141 
142  transformation_ = (Eigen::Translation<float, 3> (static_cast<float> (delta_p (0)), static_cast<float> (delta_p (1)), static_cast<float> (delta_p (2))) *
143  Eigen::AngleAxis<float> (static_cast<float> (delta_p (3)), Eigen::Vector3f::UnitX ()) *
144  Eigen::AngleAxis<float> (static_cast<float> (delta_p (4)), Eigen::Vector3f::UnitY ()) *
145  Eigen::AngleAxis<float> (static_cast<float> (delta_p (5)), Eigen::Vector3f::UnitZ ())).matrix ();
146 
147 
148  p += delta_p;
149 
150  // Update Visualizer (untested)
151  if (update_visualizer_)
152  update_visualizer_ (output, std::vector<int>(), *target_, std::vector<int>() );
153 
154  double cos_angle = 0.5 * (transformation_.coeff (0, 0) + transformation_.coeff (1, 1) + transformation_.coeff (2, 2) - 1);
155  double translation_sqr = transformation_.coeff (0, 3) * transformation_.coeff (0, 3) +
156  transformation_.coeff (1, 3) * transformation_.coeff (1, 3) +
157  transformation_.coeff (2, 3) * transformation_.coeff (2, 3);
158 
159  nr_iterations_++;
160 
161  if (nr_iterations_ >= max_iterations_ ||
162  ((transformation_epsilon_ > 0 && translation_sqr <= transformation_epsilon_) && (transformation_rotation_epsilon_ > 0 && cos_angle >= transformation_rotation_epsilon_)) ||
163  ((transformation_epsilon_ <= 0) && (transformation_rotation_epsilon_ > 0 && cos_angle >= transformation_rotation_epsilon_)) ||
164  ((transformation_epsilon_ > 0 && translation_sqr <= transformation_epsilon_) && (transformation_rotation_epsilon_ <= 0)))
165  {
166  converged_ = true;
167  }
168  }
169 
170  // Store transformation probability. The realtive differences within each scan registration are accurate
171  // but the normalization constants need to be modified for it to be globally accurate
172  trans_probability_ = score / static_cast<double> (input_->points.size ());
173 }
174 
175 //////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
176 template<typename PointSource, typename PointTarget> double
178  Eigen::Matrix<double, 6, 6> &hessian,
179  PointCloudSource &trans_cloud,
180  Eigen::Matrix<double, 6, 1> &p,
181  bool compute_hessian)
182 {
183  // Original Point and Transformed Point
184  PointSource x_pt, x_trans_pt;
185  // Original Point and Transformed Point (for math)
186  Eigen::Vector3d x, x_trans;
187  // Occupied Voxel
189  // Inverse Covariance of Occupied Voxel
190  Eigen::Matrix3d c_inv;
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 (p);
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_->points.size (); idx++)
201  {
202  x_trans_pt = trans_cloud.points[idx];
203 
204  // Find nieghbors (Radius search has been experimentally faster than direct neighbor checking.
205  std::vector<TargetGridLeafConstPtr> neighborhood;
206  std::vector<float> distances;
207  target_cells_.radiusSearch (x_trans_pt, resolution_, neighborhood, distances);
208 
209  for (typename std::vector<TargetGridLeafConstPtr>::iterator neighborhood_it = neighborhood.begin (); neighborhood_it != neighborhood.end (); neighborhood_it++)
210  {
211  cell = *neighborhood_it;
212  x_pt = input_->points[idx];
213  x = Eigen::Vector3d (x_pt.x, x_pt.y, x_pt.z);
214 
215  x_trans = Eigen::Vector3d (x_trans_pt.x, x_trans_pt.y, x_trans_pt.z);
216 
217  // Denorm point, x_k' in Equations 6.12 and 6.13 [Magnusson 2009]
218  x_trans -= cell->getMean ();
219  // Uses precomputed covariance for speed.
220  c_inv = cell->getInverseCov ();
221 
222  // Compute derivative of transform function w.r.t. transform vector, J_E and H_E in Equations 6.18 and 6.20 [Magnusson 2009]
223  computePointDerivatives (x);
224  // Update score, gradient and hessian, lines 19-21 in Algorithm 2, according to Equations 6.10, 6.12 and 6.13, respectively [Magnusson 2009]
225  score += updateDerivatives (score_gradient, hessian, x_trans, c_inv, compute_hessian);
226 
227  }
228  }
229  return (score);
230 }
231 
232 //////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
233 template<typename PointSource, typename PointTarget> void
235 {
236  // Simplified math for near 0 angles
237  double cx, cy, cz, sx, sy, sz;
238  if (std::abs (p (3)) < 10e-5)
239  {
240  //p(3) = 0;
241  cx = 1.0;
242  sx = 0.0;
243  }
244  else
245  {
246  cx = std::cos (p (3));
247  sx = sin (p (3));
248  }
249  if (std::abs (p (4)) < 10e-5)
250  {
251  //p(4) = 0;
252  cy = 1.0;
253  sy = 0.0;
254  }
255  else
256  {
257  cy = std::cos (p (4));
258  sy = sin (p (4));
259  }
260 
261  if (std::abs (p (5)) < 10e-5)
262  {
263  //p(5) = 0;
264  cz = 1.0;
265  sz = 0.0;
266  }
267  else
268  {
269  cz = std::cos (p (5));
270  sz = sin (p (5));
271  }
272 
273  // Precomputed angular gradiant components. Letters correspond to Equation 6.19 [Magnusson 2009]
274  j_ang_a_ << (-sx * sz + cx * sy * cz), (-sx * cz - cx * sy * sz), (-cx * cy);
275  j_ang_b_ << (cx * sz + sx * sy * cz), (cx * cz - sx * sy * sz), (-sx * cy);
276  j_ang_c_ << (-sy * cz), sy * sz, cy;
277  j_ang_d_ << sx * cy * cz, (-sx * cy * sz), sx * sy;
278  j_ang_e_ << (-cx * cy * cz), cx * cy * sz, (-cx * sy);
279  j_ang_f_ << (-cy * sz), (-cy * cz), 0;
280  j_ang_g_ << (cx * cz - sx * sy * sz), (-cx * sz - sx * sy * cz), 0;
281  j_ang_h_ << (sx * cz + cx * sy * sz), (cx * sy * cz - sx * sz), 0;
282 
283  if (compute_hessian)
284  {
285  // Precomputed angular hessian components. Letters correspond to Equation 6.21 and numbers correspond to row index [Magnusson 2009]
286  h_ang_a2_ << (-cx * sz - sx * sy * cz), (-cx * cz + sx * sy * sz), sx * cy;
287  h_ang_a3_ << (-sx * sz + cx * sy * cz), (-cx * sy * sz - sx * cz), (-cx * cy);
288 
289  h_ang_b2_ << (cx * cy * cz), (-cx * cy * sz), (cx * sy);
290  h_ang_b3_ << (sx * cy * cz), (-sx * cy * sz), (sx * sy);
291 
292  h_ang_c2_ << (-sx * cz - cx * sy * sz), (sx * sz - cx * sy * cz), 0;
293  h_ang_c3_ << (cx * cz - sx * sy * sz), (-sx * sy * cz - cx * sz), 0;
294 
295  h_ang_d1_ << (-cy * cz), (cy * sz), (sy);
296  h_ang_d2_ << (-sx * sy * cz), (sx * sy * sz), (sx * cy);
297  h_ang_d3_ << (cx * sy * cz), (-cx * sy * sz), (-cx * cy);
298 
299  h_ang_e1_ << (sy * sz), (sy * cz), 0;
300  h_ang_e2_ << (-sx * cy * sz), (-sx * cy * cz), 0;
301  h_ang_e3_ << (cx * cy * sz), (cx * cy * cz), 0;
302 
303  h_ang_f1_ << (-cy * cz), (cy * sz), 0;
304  h_ang_f2_ << (-cx * sz - sx * sy * cz), (-cx * cz + sx * sy * sz), 0;
305  h_ang_f3_ << (-sx * sz + cx * sy * cz), (-cx * sy * sz - sx * cz), 0;
306  }
307 }
308 
309 //////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
310 template<typename PointSource, typename PointTarget> void
312 {
313  // Calculate first derivative of Transformation Equation 6.17 w.r.t. transform vector p.
314  // Derivative w.r.t. ith element of transform vector corresponds to column i, Equation 6.18 and 6.19 [Magnusson 2009]
315  point_gradient_ (1, 3) = x.dot (j_ang_a_);
316  point_gradient_ (2, 3) = x.dot (j_ang_b_);
317  point_gradient_ (0, 4) = x.dot (j_ang_c_);
318  point_gradient_ (1, 4) = x.dot (j_ang_d_);
319  point_gradient_ (2, 4) = x.dot (j_ang_e_);
320  point_gradient_ (0, 5) = x.dot (j_ang_f_);
321  point_gradient_ (1, 5) = x.dot (j_ang_g_);
322  point_gradient_ (2, 5) = x.dot (j_ang_h_);
323 
324  if (compute_hessian)
325  {
326  // Vectors from Equation 6.21 [Magnusson 2009]
327  Eigen::Vector3d a, b, c, d, e, f;
328 
329  a << 0, x.dot (h_ang_a2_), x.dot (h_ang_a3_);
330  b << 0, x.dot (h_ang_b2_), x.dot (h_ang_b3_);
331  c << 0, x.dot (h_ang_c2_), x.dot (h_ang_c3_);
332  d << x.dot (h_ang_d1_), x.dot (h_ang_d2_), x.dot (h_ang_d3_);
333  e << x.dot (h_ang_e1_), x.dot (h_ang_e2_), x.dot (h_ang_e3_);
334  f << x.dot (h_ang_f1_), x.dot (h_ang_f2_), x.dot (h_ang_f3_);
335 
336  // Calculate second derivative of Transformation Equation 6.17 w.r.t. transform vector p.
337  // Derivative w.r.t. ith and jth elements of transform vector corresponds to the 3x1 block matrix starting at (3i,j), Equation 6.20 and 6.21 [Magnusson 2009]
338  point_hessian_.block<3, 1>(9, 3) = a;
339  point_hessian_.block<3, 1>(12, 3) = b;
340  point_hessian_.block<3, 1>(15, 3) = c;
341  point_hessian_.block<3, 1>(9, 4) = b;
342  point_hessian_.block<3, 1>(12, 4) = d;
343  point_hessian_.block<3, 1>(15, 4) = e;
344  point_hessian_.block<3, 1>(9, 5) = c;
345  point_hessian_.block<3, 1>(12, 5) = e;
346  point_hessian_.block<3, 1>(15, 5) = f;
347  }
348 }
349 
350 //////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
351 template<typename PointSource, typename PointTarget> double
353  Eigen::Matrix<double, 6, 6> &hessian,
354  Eigen::Vector3d &x_trans, Eigen::Matrix3d &c_inv,
355  bool compute_hessian)
356 {
357  Eigen::Vector3d cov_dxd_pi;
358  // e^(-d_2/2 * (x_k - mu_k)^T Sigma_k^-1 (x_k - mu_k)) Equation 6.9 [Magnusson 2009]
359  double e_x_cov_x = std::exp (-gauss_d2_ * x_trans.dot (c_inv * x_trans) / 2);
360  // Calculate probability of transformed points existence, Equation 6.9 [Magnusson 2009]
361  double score_inc = -gauss_d1_ * e_x_cov_x;
362 
363  e_x_cov_x = gauss_d2_ * e_x_cov_x;
364 
365  // Error checking for invalid values.
366  if (e_x_cov_x > 1 || e_x_cov_x < 0 || std::isnan(e_x_cov_x))
367  return (0);
368 
369  // Reusable portion of Equation 6.12 and 6.13 [Magnusson 2009]
370  e_x_cov_x *= gauss_d1_;
371 
372 
373  for (int i = 0; i < 6; i++)
374  {
375  // Sigma_k^-1 d(T(x,p))/dpi, Reusable portion of Equation 6.12 and 6.13 [Magnusson 2009]
376  cov_dxd_pi = c_inv * point_gradient_.col (i);
377 
378  // Update gradient, Equation 6.12 [Magnusson 2009]
379  score_gradient (i) += x_trans.dot (cov_dxd_pi) * e_x_cov_x;
380 
381  if (compute_hessian)
382  {
383  for (Eigen::Index j = 0; j < hessian.cols (); j++)
384  {
385  // Update hessian, Equation 6.13 [Magnusson 2009]
386  hessian (i, j) += e_x_cov_x * (-gauss_d2_ * x_trans.dot (cov_dxd_pi) * x_trans.dot (c_inv * point_gradient_.col (j)) +
387  x_trans.dot (c_inv * point_hessian_.block<3, 1>(3 * i, j)) +
388  point_gradient_.col (j).dot (cov_dxd_pi) );
389  }
390  }
391  }
392 
393  return (score_inc);
394 }
395 
396 //////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
397 template<typename PointSource, typename PointTarget> void
399  PointCloudSource &trans_cloud, Eigen::Matrix<double, 6, 1> &)
400 {
401  // Original Point and Transformed Point
402  PointSource x_pt, x_trans_pt;
403  // Original Point and Transformed Point (for math)
404  Eigen::Vector3d x, x_trans;
405  // Occupied Voxel
407  // Inverse Covariance of Occupied Voxel
408  Eigen::Matrix3d c_inv;
409 
410  hessian.setZero ();
411 
412  // Precompute Angular Derivatives unessisary because only used after regular derivative calculation
413 
414  // Update hessian for each point, line 17 in Algorithm 2 [Magnusson 2009]
415  for (std::size_t idx = 0; idx < input_->points.size (); idx++)
416  {
417  x_trans_pt = trans_cloud.points[idx];
418 
419  // Find nieghbors (Radius search has been experimentally faster than direct neighbor checking.
420  std::vector<TargetGridLeafConstPtr> neighborhood;
421  std::vector<float> distances;
422  target_cells_.radiusSearch (x_trans_pt, resolution_, neighborhood, distances);
423 
424  for (typename std::vector<TargetGridLeafConstPtr>::iterator neighborhood_it = neighborhood.begin (); neighborhood_it != neighborhood.end (); neighborhood_it++)
425  {
426  cell = *neighborhood_it;
427 
428  {
429  x_pt = input_->points[idx];
430  x = Eigen::Vector3d (x_pt.x, x_pt.y, x_pt.z);
431 
432  x_trans = Eigen::Vector3d (x_trans_pt.x, x_trans_pt.y, x_trans_pt.z);
433 
434  // Denorm point, x_k' in Equations 6.12 and 6.13 [Magnusson 2009]
435  x_trans -= cell->getMean ();
436  // Uses precomputed covariance for speed.
437  c_inv = cell->getInverseCov ();
438 
439  // Compute derivative of transform function w.r.t. transform vector, J_E and H_E in Equations 6.18 and 6.20 [Magnusson 2009]
440  computePointDerivatives (x);
441  // Update hessian, lines 21 in Algorithm 2, according to Equations 6.10, 6.12 and 6.13, respectively [Magnusson 2009]
442  updateHessian (hessian, x_trans, c_inv);
443  }
444  }
445  }
446 }
447 
448 //////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
449 template<typename PointSource, typename PointTarget> void
450 pcl::NormalDistributionsTransform<PointSource, PointTarget>::updateHessian (Eigen::Matrix<double, 6, 6> &hessian, Eigen::Vector3d &x_trans, Eigen::Matrix3d &c_inv)
451 {
452  Eigen::Vector3d cov_dxd_pi;
453  // e^(-d_2/2 * (x_k - mu_k)^T Sigma_k^-1 (x_k - mu_k)) Equation 6.9 [Magnusson 2009]
454  double e_x_cov_x = gauss_d2_ * std::exp (-gauss_d2_ * x_trans.dot (c_inv * x_trans) / 2);
455 
456  // Error checking for invalid values.
457  if (e_x_cov_x > 1 || e_x_cov_x < 0 || std::isnan(e_x_cov_x))
458  return;
459 
460  // Reusable portion of Equation 6.12 and 6.13 [Magnusson 2009]
461  e_x_cov_x *= gauss_d1_;
462 
463  for (int i = 0; i < 6; i++)
464  {
465  // Sigma_k^-1 d(T(x,p))/dpi, Reusable portion of Equation 6.12 and 6.13 [Magnusson 2009]
466  cov_dxd_pi = c_inv * point_gradient_.col (i);
467 
468  for (Eigen::Index j = 0; j < hessian.cols (); j++)
469  {
470  // Update hessian, Equation 6.13 [Magnusson 2009]
471  hessian (i, j) += e_x_cov_x * (-gauss_d2_ * x_trans.dot (cov_dxd_pi) * x_trans.dot (c_inv * point_gradient_.col (j)) +
472  x_trans.dot (c_inv * point_hessian_.block<3, 1>(3 * i, j)) +
473  point_gradient_.col (j).dot (cov_dxd_pi) );
474  }
475  }
476 
477 }
478 
479 //////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
480 template<typename PointSource, typename PointTarget> bool
482  double &a_u, double &f_u, double &g_u,
483  double a_t, double f_t, double g_t)
484 {
485  // Case U1 in Update Algorithm and Case a in Modified Update Algorithm [More, Thuente 1994]
486  if (f_t > f_l)
487  {
488  a_u = a_t;
489  f_u = f_t;
490  g_u = g_t;
491  return (false);
492  }
493  // Case U2 in Update Algorithm and Case b in Modified Update Algorithm [More, Thuente 1994]
494  if (g_t * (a_l - a_t) > 0)
495  {
496  a_l = a_t;
497  f_l = f_t;
498  g_l = g_t;
499  return (false);
500  }
501  // Case U3 in Update Algorithm and Case c in Modified Update Algorithm [More, Thuente 1994]
502  if (g_t * (a_l - a_t) < 0)
503  {
504  a_u = a_l;
505  f_u = f_l;
506  g_u = g_l;
507 
508  a_l = a_t;
509  f_l = f_t;
510  g_l = g_t;
511  return (false);
512  }
513  // Interval Converged
514  return (true);
515 }
516 
517 //////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
518 template<typename PointSource, typename PointTarget> double
520  double a_u, double f_u, double g_u,
521  double a_t, double f_t, double g_t)
522 {
523  // Case 1 in Trial Value Selection [More, Thuente 1994]
524  if (f_t > f_l)
525  {
526  // Calculate the minimizer of the cubic that interpolates f_l, f_t, g_l and g_t
527  // Equation 2.4.52 [Sun, Yuan 2006]
528  double z = 3 * (f_t - f_l) / (a_t - a_l) - g_t - g_l;
529  double w = std::sqrt (z * z - g_t * g_l);
530  // Equation 2.4.56 [Sun, Yuan 2006]
531  double a_c = a_l + (a_t - a_l) * (w - g_l - z) / (g_t - g_l + 2 * w);
532 
533  // Calculate the minimizer of the quadratic that interpolates f_l, f_t and g_l
534  // Equation 2.4.2 [Sun, Yuan 2006]
535  double a_q = a_l - 0.5 * (a_l - a_t) * g_l / (g_l - (f_l - f_t) / (a_l - a_t));
536 
537  if (std::fabs (a_c - a_l) < std::fabs (a_q - a_l))
538  return (a_c);
539  return (0.5 * (a_q + a_c));
540  }
541  // Case 2 in Trial Value Selection [More, Thuente 1994]
542  if (g_t * g_l < 0)
543  {
544  // Calculate the minimizer of the cubic that interpolates f_l, f_t, g_l and g_t
545  // Equation 2.4.52 [Sun, Yuan 2006]
546  double z = 3 * (f_t - f_l) / (a_t - a_l) - g_t - g_l;
547  double w = std::sqrt (z * z - g_t * g_l);
548  // Equation 2.4.56 [Sun, Yuan 2006]
549  double a_c = a_l + (a_t - a_l) * (w - g_l - z) / (g_t - g_l + 2 * w);
550 
551  // Calculate the minimizer of the quadratic that interpolates f_l, g_l and g_t
552  // Equation 2.4.5 [Sun, Yuan 2006]
553  double a_s = a_l - (a_l - a_t) / (g_l - g_t) * g_l;
554 
555  if (std::fabs (a_c - a_t) >= std::fabs (a_s - a_t))
556  return (a_c);
557  return (a_s);
558  }
559  // Case 3 in Trial Value Selection [More, Thuente 1994]
560  if (std::fabs (g_t) <= std::fabs (g_l))
561  {
562  // Calculate the minimizer of the cubic that interpolates f_l, f_t, g_l and g_t
563  // Equation 2.4.52 [Sun, Yuan 2006]
564  double z = 3 * (f_t - f_l) / (a_t - a_l) - g_t - g_l;
565  double w = std::sqrt (z * z - g_t * g_l);
566  double a_c = a_l + (a_t - a_l) * (w - g_l - z) / (g_t - g_l + 2 * w);
567 
568  // Calculate the minimizer of the quadratic that interpolates g_l and g_t
569  // Equation 2.4.5 [Sun, Yuan 2006]
570  double a_s = a_l - (a_l - a_t) / (g_l - g_t) * g_l;
571 
572  double a_t_next;
573 
574  if (std::fabs (a_c - a_t) < std::fabs (a_s - a_t))
575  a_t_next = a_c;
576  else
577  a_t_next = a_s;
578 
579  if (a_t > a_l)
580  return (std::min (a_t + 0.66 * (a_u - a_t), a_t_next));
581  return (std::max (a_t + 0.66 * (a_u - a_t), a_t_next));
582  }
583  // Case 4 in Trial Value Selection [More, Thuente 1994]
584  // Calculate the minimizer of the cubic that interpolates f_u, f_t, g_u and g_t
585  // Equation 2.4.52 [Sun, Yuan 2006]
586  double z = 3 * (f_t - f_u) / (a_t - a_u) - g_t - g_u;
587  double w = std::sqrt (z * z - g_t * g_u);
588  // Equation 2.4.56 [Sun, Yuan 2006]
589  return (a_u + (a_t - a_u) * (w - g_u - z) / (g_t - g_u + 2 * w));
590 }
591 
592 //////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
593 template<typename PointSource, typename PointTarget> double
594 pcl::NormalDistributionsTransform<PointSource, PointTarget>::computeStepLengthMT (const Eigen::Matrix<double, 6, 1> &x, Eigen::Matrix<double, 6, 1> &step_dir, double step_init, double step_max,
595  double step_min, double &score, Eigen::Matrix<double, 6, 1> &score_gradient, Eigen::Matrix<double, 6, 6> &hessian,
596  PointCloudSource &trans_cloud)
597 {
598  // Set the value of phi(0), Equation 1.3 [More, Thuente 1994]
599  double phi_0 = -score;
600  // Set the value of phi'(0), Equation 1.3 [More, Thuente 1994]
601  double d_phi_0 = -(score_gradient.dot (step_dir));
602 
603  Eigen::Matrix<double, 6, 1> x_t;
604 
605  if (d_phi_0 >= 0)
606  {
607  // Not a decent direction
608  if (d_phi_0 == 0)
609  return 0;
610  // Reverse step direction and calculate optimal step.
611  d_phi_0 *= -1;
612  step_dir *= -1;
613 
614  }
615 
616  // The Search Algorithm for T(mu) [More, Thuente 1994]
617 
618  int max_step_iterations = 10;
619  int step_iterations = 0;
620 
621  // Sufficient decreace constant, Equation 1.1 [More, Thuete 1994]
622  double mu = 1.e-4;
623  // Curvature condition constant, Equation 1.2 [More, Thuete 1994]
624  double nu = 0.9;
625 
626  // Initial endpoints of Interval I,
627  double a_l = 0, a_u = 0;
628 
629  // Auxiliary function psi is used until I is determined ot be a closed interval, Equation 2.1 [More, Thuente 1994]
630  double f_l = auxilaryFunction_PsiMT (a_l, phi_0, phi_0, d_phi_0, mu);
631  double g_l = auxilaryFunction_dPsiMT (d_phi_0, d_phi_0, mu);
632 
633  double f_u = auxilaryFunction_PsiMT (a_u, phi_0, phi_0, d_phi_0, mu);
634  double g_u = auxilaryFunction_dPsiMT (d_phi_0, d_phi_0, mu);
635 
636  // Check used to allow More-Thuente step length calculation to be skipped by making step_min == step_max
637  bool interval_converged = (step_max - step_min) > 0, open_interval = true;
638 
639  double a_t = step_init;
640  a_t = std::min (a_t, step_max);
641  a_t = std::max (a_t, step_min);
642 
643  x_t = x + step_dir * a_t;
644 
645  final_transformation_ = (Eigen::Translation<float, 3>(static_cast<float> (x_t (0)), static_cast<float> (x_t (1)), static_cast<float> (x_t (2))) *
646  Eigen::AngleAxis<float> (static_cast<float> (x_t (3)), Eigen::Vector3f::UnitX ()) *
647  Eigen::AngleAxis<float> (static_cast<float> (x_t (4)), Eigen::Vector3f::UnitY ()) *
648  Eigen::AngleAxis<float> (static_cast<float> (x_t (5)), Eigen::Vector3f::UnitZ ())).matrix ();
649 
650  // New transformed point cloud
651  transformPointCloud (*input_, trans_cloud, final_transformation_);
652 
653  // Updates score, gradient and hessian. Hessian calculation is unessisary but testing showed that most step calculations use the
654  // initial step suggestion and recalculation the reusable portions of the hessian would intail more computation time.
655  score = computeDerivatives (score_gradient, hessian, trans_cloud, x_t, true);
656 
657  // Calculate phi(alpha_t)
658  double phi_t = -score;
659  // Calculate phi'(alpha_t)
660  double d_phi_t = -(score_gradient.dot (step_dir));
661 
662  // Calculate psi(alpha_t)
663  double psi_t = auxilaryFunction_PsiMT (a_t, phi_t, phi_0, d_phi_0, mu);
664  // Calculate psi'(alpha_t)
665  double d_psi_t = auxilaryFunction_dPsiMT (d_phi_t, d_phi_0, mu);
666 
667  // Iterate until max number of iterations, interval convergance or a value satisfies the sufficient decrease, Equation 1.1, and curvature condition, Equation 1.2 [More, Thuente 1994]
668  while (!interval_converged && step_iterations < max_step_iterations && !(psi_t <= 0 /*Sufficient Decrease*/ && d_phi_t <= -nu * d_phi_0 /*Curvature Condition*/))
669  {
670  // Use auxiliary function if interval I is not closed
671  if (open_interval)
672  {
673  a_t = trialValueSelectionMT (a_l, f_l, g_l,
674  a_u, f_u, g_u,
675  a_t, psi_t, d_psi_t);
676  }
677  else
678  {
679  a_t = trialValueSelectionMT (a_l, f_l, g_l,
680  a_u, f_u, g_u,
681  a_t, phi_t, d_phi_t);
682  }
683 
684  a_t = std::min (a_t, step_max);
685  a_t = std::max (a_t, step_min);
686 
687  x_t = x + step_dir * a_t;
688 
689  final_transformation_ = (Eigen::Translation<float, 3> (static_cast<float> (x_t (0)), static_cast<float> (x_t (1)), static_cast<float> (x_t (2))) *
690  Eigen::AngleAxis<float> (static_cast<float> (x_t (3)), Eigen::Vector3f::UnitX ()) *
691  Eigen::AngleAxis<float> (static_cast<float> (x_t (4)), Eigen::Vector3f::UnitY ()) *
692  Eigen::AngleAxis<float> (static_cast<float> (x_t (5)), Eigen::Vector3f::UnitZ ())).matrix ();
693 
694  // New transformed point cloud
695  // Done on final cloud to prevent wasted computation
696  transformPointCloud (*input_, trans_cloud, final_transformation_);
697 
698  // Updates score, gradient. Values stored to prevent wasted computation.
699  score = computeDerivatives (score_gradient, hessian, trans_cloud, x_t, false);
700 
701  // Calculate phi(alpha_t+)
702  phi_t = -score;
703  // Calculate phi'(alpha_t+)
704  d_phi_t = -(score_gradient.dot (step_dir));
705 
706  // Calculate psi(alpha_t+)
707  psi_t = auxilaryFunction_PsiMT (a_t, phi_t, phi_0, d_phi_0, mu);
708  // Calculate psi'(alpha_t+)
709  d_psi_t = auxilaryFunction_dPsiMT (d_phi_t, d_phi_0, mu);
710 
711  // Check if I is now a closed interval
712  if (open_interval && (psi_t <= 0 && d_psi_t >= 0))
713  {
714  open_interval = false;
715 
716  // Converts f_l and g_l from psi to phi
717  f_l += phi_0 - mu * d_phi_0 * a_l;
718  g_l += mu * d_phi_0;
719 
720  // Converts f_u and g_u from psi to phi
721  f_u += phi_0 - mu * d_phi_0 * a_u;
722  g_u += mu * d_phi_0;
723  }
724 
725  if (open_interval)
726  {
727  // Update interval end points using Updating Algorithm [More, Thuente 1994]
728  interval_converged = updateIntervalMT (a_l, f_l, g_l,
729  a_u, f_u, g_u,
730  a_t, psi_t, d_psi_t);
731  }
732  else
733  {
734  // Update interval end points using Modified Updating Algorithm [More, Thuente 1994]
735  interval_converged = updateIntervalMT (a_l, f_l, g_l,
736  a_u, f_u, g_u,
737  a_t, phi_t, d_phi_t);
738  }
739 
740  step_iterations++;
741  }
742 
743  // If inner loop was run then hessian needs to be calculated.
744  // Hessian is unnessisary for step length determination but gradients are required
745  // so derivative and transform data is stored for the next iteration.
746  if (step_iterations)
747  computeHessian (hessian, trans_cloud, x_t);
748 
749  return (a_t);
750 }
751 
752 #endif // PCL_REGISTRATION_NDT_IMPL_H_
pcl::NormalDistributionsTransform::PointCloudSource
typename Registration< PointSource, PointTarget >::PointCloudSource PointCloudSource
Definition: ndt.h:67
pcl::NormalDistributionsTransform::computeAngleDerivatives
void computeAngleDerivatives(Eigen::Matrix< double, 6, 1 > &p, bool compute_hessian=true)
Precompute anglular components of derivatives.
Definition: ndt.hpp:234
pcl::NormalDistributionsTransform::gauss_d2_
double gauss_d2_
Definition: ndt.h:433
pcl::Registration< PointSource, PointTarget >::max_iterations_
int max_iterations_
The maximum number of iterations the internal optimization should run for.
Definition: registration.h:503
pcl::NormalDistributionsTransform::resolution_
float resolution_
The side length of voxels.
Definition: ndt.h:424
pcl::NormalDistributionsTransform::updateIntervalMT
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)
Update interval of possible step lengths for More-Thuente method, in More-Thuente (1994)
Definition: ndt.hpp:481
pcl::transformPointCloud
void transformPointCloud(const pcl::PointCloud< PointT > &cloud_in, pcl::PointCloud< PointT > &cloud_out, const Eigen::Transform< Scalar, 3, Eigen::Affine > &transform, bool copy_all_fields=true)
Apply an affine transform defined by an Eigen Transform.
Definition: transforms.hpp:215
pcl::Registration< PointSource, PointTarget >::transformation_epsilon_
double transformation_epsilon_
The maximum difference between two consecutive transformations in order to consider convergence (user...
Definition: registration.h:523
pcl::NormalDistributionsTransform::computeStepLengthMT
double computeStepLengthMT(const Eigen::Matrix< double, 6, 1 > &x, 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 probability derivatives using More-Thuente m...
Definition: ndt.hpp:594
pcl::NormalDistributionsTransform::computeTransformation
virtual void computeTransformation(PointCloudSource &output)
Estimate the transformation and returns the transformed source (input) as output.
Definition: ndt.h:239
pcl::NormalDistributionsTransform::gauss_d1_
double gauss_d1_
The normalization constants used fit the point distribution to a normal distribution,...
Definition: ndt.h:433
pcl::NormalDistributionsTransform::NormalDistributionsTransform
NormalDistributionsTransform()
Constructor.
Definition: ndt.hpp:46
pcl::NormalDistributionsTransform::updateDerivatives
double updateDerivatives(Eigen::Matrix< double, 6, 1 > &score_gradient, Eigen::Matrix< double, 6, 6 > &hessian, Eigen::Vector3d &x_trans, Eigen::Matrix3d &c_inv, bool compute_hessian=true)
Compute individual point contirbutions to derivatives of probability function w.r....
Definition: ndt.hpp:352
pcl::NormalDistributionsTransform::computePointDerivatives
void computePointDerivatives(Eigen::Vector3d &x, bool compute_hessian=true)
Compute point derivatives.
Definition: ndt.hpp:311
pcl::NormalDistributionsTransform::computeHessian
void computeHessian(Eigen::Matrix< double, 6, 6 > &hessian, PointCloudSource &trans_cloud, Eigen::Matrix< double, 6, 1 > &p)
Compute hessian of probability function w.r.t.
Definition: ndt.hpp:398
pcl::NormalDistributionsTransform::computeDerivatives
double computeDerivatives(Eigen::Matrix< double, 6, 1 > &score_gradient, Eigen::Matrix< double, 6, 6 > &hessian, PointCloudSource &trans_cloud, Eigen::Matrix< double, 6, 1 > &p, bool compute_hessian=true)
Compute derivatives of probability function w.r.t.
Definition: ndt.hpp:177
pcl::NormalDistributionsTransform::outlier_ratio_
double outlier_ratio_
The ratio of outliers of points w.r.t.
Definition: ndt.h:430
pcl::NormalDistributionsTransform::trialValueSelectionMT
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)
Select new trial value for More-Thuente method.
Definition: ndt.hpp:519
pcl::NormalDistributionsTransform::TargetGridLeafConstPtr
typename TargetGrid::LeafConstPtr TargetGridLeafConstPtr
Typename of const pointer to searchable voxel grid leaf.
Definition: ndt.h:85
pcl::NormalDistributionsTransform::updateHessian
void updateHessian(Eigen::Matrix< double, 6, 6 > &hessian, Eigen::Vector3d &x_trans, Eigen::Matrix3d &c_inv)
Compute individual point contirbutions to hessian of probability function w.r.t.
Definition: ndt.hpp:450
pcl::Registration< PointSource, PointTarget >::reg_name_
std::string reg_name_
The registration method name.
Definition: registration.h:489