Point Cloud Library (PCL) 1.12.1
ndt.hpp
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40
41#ifndef PCL_REGISTRATION_NDT_IMPL_H_
42#define PCL_REGISTRATION_NDT_IMPL_H_
43
44namespace pcl {
45
46template <typename PointSource, typename PointTarget>
48: target_cells_()
49, resolution_(1.0f)
50, step_size_(0.1)
51, outlier_ratio_(0.55)
52, gauss_d1_()
53, gauss_d2_()
54, trans_probability_()
55{
56 reg_name_ = "NormalDistributionsTransform";
57
58 // Initializes the gaussian fitting parameters (eq. 6.8) [Magnusson 2009]
59 const double gauss_c1 = 10.0 * (1 - outlier_ratio_);
60 const double gauss_c2 = outlier_ratio_ / pow(resolution_, 3);
61 const double gauss_d3 = -std::log(gauss_c2);
62 gauss_d1_ = -std::log(gauss_c1 + gauss_c2) - gauss_d3;
63 gauss_d2_ =
64 -2 * std::log((-std::log(gauss_c1 * std::exp(-0.5) + gauss_c2) - gauss_d3) /
65 gauss_d1_);
66
68 max_iterations_ = 35;
69}
70
71template <typename PointSource, typename PointTarget>
72void
74 PointCloudSource& output, const Eigen::Matrix4f& guess)
75{
76 nr_iterations_ = 0;
77 converged_ = false;
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 != Eigen::Matrix4f::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<float, 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 Eigen::Vector3f init_translation = eig_transformation.translation();
106 Eigen::Vector3f init_rotation = eig_transformation.rotation().eulerAngles(0, 1, 2);
107 transform << init_translation.cast<double>(), init_rotation.cast<double>();
108
109 Eigen::Matrix<double, 6, 6> hessian;
110
111 // Calculate derivates of initial transform vector, subsequent derivative calculations
112 // are done in the step length determination.
113 double score = computeDerivatives(score_gradient, hessian, output, transform);
114
115 while (!converged_) {
116 // Store previous transformation
117 previous_transformation_ = transformation_;
118
119 // Solve for decent direction using newton method, line 23 in Algorithm 2 [Magnusson
120 // 2009]
121 Eigen::JacobiSVD<Eigen::Matrix<double, 6, 6>> sv(
122 hessian, Eigen::ComputeFullU | Eigen::ComputeFullV);
123 // Negative for maximization as opposed to minimization
124 Eigen::Matrix<double, 6, 1> delta = sv.solve(-score_gradient);
125
126 // Calculate step length with guarnteed sufficient decrease [More, Thuente 1994]
127 double delta_norm = delta.norm();
128
129 if (delta_norm == 0 || std::isnan(delta_norm)) {
130 trans_probability_ = score / static_cast<double>(input_->size());
131 converged_ = delta_norm == 0;
132 return;
133 }
134
135 delta /= delta_norm;
136 delta_norm = computeStepLengthMT(transform,
137 delta,
138 delta_norm,
139 step_size_,
140 transformation_epsilon_ / 2,
141 score,
142 score_gradient,
143 hessian,
144 output);
145 delta *= delta_norm;
146
147 // Convert delta into matrix form
148 convertTransform(delta, transformation_);
149
150 transform += delta;
151
152 // Update Visualizer (untested)
153 if (update_visualizer_)
154 update_visualizer_(output, pcl::Indices(), *target_, pcl::Indices());
155
156 const double cos_angle =
157 0.5 * (transformation_.template block<3, 3>(0, 0).trace() - 1);
158 const double translation_sqr =
159 transformation_.template block<3, 1>(0, 3).squaredNorm();
160
161 nr_iterations_++;
162
163 if (nr_iterations_ >= max_iterations_ ||
164 ((transformation_epsilon_ > 0 && translation_sqr <= transformation_epsilon_) &&
165 (transformation_rotation_epsilon_ > 0 &&
166 cos_angle >= transformation_rotation_epsilon_)) ||
167 ((transformation_epsilon_ <= 0) &&
168 (transformation_rotation_epsilon_ > 0 &&
169 cos_angle >= transformation_rotation_epsilon_)) ||
170 ((transformation_epsilon_ > 0 && translation_sqr <= transformation_epsilon_) &&
171 (transformation_rotation_epsilon_ <= 0))) {
172 converged_ = true;
173 }
174 }
175
176 // Store transformation probability. The realtive differences within each scan
177 // registration are accurate but the normalization constants need to be modified for
178 // it to be globally accurate
179 trans_probability_ = score / static_cast<double>(input_->size());
180}
181
182template <typename PointSource, typename PointTarget>
183double
185 Eigen::Matrix<double, 6, 1>& score_gradient,
186 Eigen::Matrix<double, 6, 6>& hessian,
187 const PointCloudSource& trans_cloud,
188 const Eigen::Matrix<double, 6, 1>& transform,
189 bool compute_hessian)
190{
191 score_gradient.setZero();
192 hessian.setZero();
193 double score = 0;
194
195 // Precompute Angular Derivatives (eq. 6.19 and 6.21)[Magnusson 2009]
196 computeAngleDerivatives(transform);
197
198 // Update gradient and hessian for each point, line 17 in Algorithm 2 [Magnusson 2009]
199 for (std::size_t idx = 0; idx < input_->size(); idx++) {
200 // Transformed Point
201 const auto& x_trans_pt = trans_cloud[idx];
202
203 // Find neighbors (Radius search has been experimentally faster than direct neighbor
204 // checking.
205 std::vector<TargetGridLeafConstPtr> neighborhood;
206 std::vector<float> distances;
207 target_cells_.radiusSearch(x_trans_pt, resolution_, neighborhood, distances);
208
209 for (const auto& cell : neighborhood) {
210 // Original Point
211 const auto& x_pt = (*input_)[idx];
212 const Eigen::Vector3d x = x_pt.getVector3fMap().template cast<double>();
213
214 // Denorm point, x_k' in Equations 6.12 and 6.13 [Magnusson 2009]
215 const Eigen::Vector3d x_trans =
216 x_trans_pt.getVector3fMap().template cast<double>() - cell->getMean();
217 // Inverse Covariance of Occupied Voxel
218 // Uses precomputed covariance for speed.
219 const Eigen::Matrix3d c_inv = cell->getInverseCov();
220
221 // Compute derivative of transform function w.r.t. transform vector, J_E and H_E
222 // 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
225 // Equations 6.10, 6.12 and 6.13, respectively [Magnusson 2009]
226 score +=
227 updateDerivatives(score_gradient, hessian, x_trans, c_inv, compute_hessian);
228 }
229 }
230 return score;
231}
232
233template <typename PointSource, typename PointTarget>
234void
236 const Eigen::Matrix<double, 6, 1>& transform, bool compute_hessian)
237{
238 // Simplified math for near 0 angles
239 const auto calculate_cos_sin = [](double angle, double& c, double& s) {
240 if (std::abs(angle) < 10e-5) {
241 c = 1.0;
242 s = 0.0;
243 }
244 else {
245 c = std::cos(angle);
246 s = std::sin(angle);
247 }
248 };
249
250 double cx, cy, cz, sx, sy, sz;
251 calculate_cos_sin(transform(3), cx, sx);
252 calculate_cos_sin(transform(4), cy, sy);
253 calculate_cos_sin(transform(5), cz, sz);
254
255 // Precomputed angular gradient components. Letters correspond to Equation 6.19
256 // [Magnusson 2009]
257 angular_jacobian_.setZero();
258 angular_jacobian_.row(0).noalias() = Eigen::Vector4d(
259 (-sx * sz + cx * sy * cz), (-sx * cz - cx * sy * sz), (-cx * cy), 1.0); // a
260 angular_jacobian_.row(1).noalias() = Eigen::Vector4d(
261 (cx * sz + sx * sy * cz), (cx * cz - sx * sy * sz), (-sx * cy), 1.0); // b
262 angular_jacobian_.row(2).noalias() =
263 Eigen::Vector4d((-sy * cz), sy * sz, cy, 1.0); // c
264 angular_jacobian_.row(3).noalias() =
265 Eigen::Vector4d(sx * cy * cz, (-sx * cy * sz), sx * sy, 1.0); // d
266 angular_jacobian_.row(4).noalias() =
267 Eigen::Vector4d((-cx * cy * cz), cx * cy * sz, (-cx * sy), 1.0); // e
268 angular_jacobian_.row(5).noalias() =
269 Eigen::Vector4d((-cy * sz), (-cy * cz), 0, 1.0); // f
270 angular_jacobian_.row(6).noalias() =
271 Eigen::Vector4d((cx * cz - sx * sy * sz), (-cx * sz - sx * sy * cz), 0, 1.0); // g
272 angular_jacobian_.row(7).noalias() =
273 Eigen::Vector4d((sx * cz + cx * sy * sz), (cx * sy * cz - sx * sz), 0, 1.0); // h
274
275 if (compute_hessian) {
276 // Precomputed angular hessian components. Letters correspond to Equation 6.21 and
277 // numbers correspond to row index [Magnusson 2009]
278 angular_hessian_.setZero();
279 angular_hessian_.row(0).noalias() = Eigen::Vector4d(
280 (-cx * sz - sx * sy * cz), (-cx * cz + sx * sy * sz), sx * cy, 0.0f); // a2
281 angular_hessian_.row(1).noalias() = Eigen::Vector4d(
282 (-sx * sz + cx * sy * cz), (-cx * sy * sz - sx * cz), (-cx * cy), 0.0f); // a3
283
284 angular_hessian_.row(2).noalias() =
285 Eigen::Vector4d((cx * cy * cz), (-cx * cy * sz), (cx * sy), 0.0f); // b2
286 angular_hessian_.row(3).noalias() =
287 Eigen::Vector4d((sx * cy * cz), (-sx * cy * sz), (sx * sy), 0.0f); // b3
288
289 // The sign of 'sx * sz' in c2 is incorrect in the thesis, and is fixed here.
290 angular_hessian_.row(4).noalias() = Eigen::Vector4d(
291 (-sx * cz - cx * sy * sz), (sx * sz - cx * sy * cz), 0, 0.0f); // c2
292 angular_hessian_.row(5).noalias() = Eigen::Vector4d(
293 (cx * cz - sx * sy * sz), (-sx * sy * cz - cx * sz), 0, 0.0f); // c3
294
295 angular_hessian_.row(6).noalias() =
296 Eigen::Vector4d((-cy * cz), (cy * sz), (-sy), 0.0f); // d1
297 angular_hessian_.row(7).noalias() =
298 Eigen::Vector4d((-sx * sy * cz), (sx * sy * sz), (sx * cy), 0.0f); // d2
299 angular_hessian_.row(8).noalias() =
300 Eigen::Vector4d((cx * sy * cz), (-cx * sy * sz), (-cx * cy), 0.0f); // d3
301
302 angular_hessian_.row(9).noalias() =
303 Eigen::Vector4d((sy * sz), (sy * cz), 0, 0.0f); // e1
304 angular_hessian_.row(10).noalias() =
305 Eigen::Vector4d((-sx * cy * sz), (-sx * cy * cz), 0, 0.0f); // e2
306 angular_hessian_.row(11).noalias() =
307 Eigen::Vector4d((cx * cy * sz), (cx * cy * cz), 0, 0.0f); // e3
308
309 angular_hessian_.row(12).noalias() =
310 Eigen::Vector4d((-cy * cz), (cy * sz), 0, 0.0f); // f1
311 angular_hessian_.row(13).noalias() = Eigen::Vector4d(
312 (-cx * sz - sx * sy * cz), (-cx * cz + sx * sy * sz), 0, 0.0f); // f2
313 angular_hessian_.row(14).noalias() = Eigen::Vector4d(
314 (-sx * sz + cx * sy * cz), (-cx * sy * sz - sx * cz), 0, 0.0f); // f3
315 }
316}
317
318template <typename PointSource, typename PointTarget>
319void
321 const Eigen::Vector3d& x, bool compute_hessian)
322{
323 // Calculate first derivative of Transformation Equation 6.17 w.r.t. transform vector.
324 // Derivative w.r.t. ith element of transform vector corresponds to column i,
325 // Equation 6.18 and 6.19 [Magnusson 2009]
326 Eigen::Matrix<double, 8, 1> point_angular_jacobian =
327 angular_jacobian_ * Eigen::Vector4d(x[0], x[1], x[2], 0.0);
328 point_jacobian_(1, 3) = point_angular_jacobian[0];
329 point_jacobian_(2, 3) = point_angular_jacobian[1];
330 point_jacobian_(0, 4) = point_angular_jacobian[2];
331 point_jacobian_(1, 4) = point_angular_jacobian[3];
332 point_jacobian_(2, 4) = point_angular_jacobian[4];
333 point_jacobian_(0, 5) = point_angular_jacobian[5];
334 point_jacobian_(1, 5) = point_angular_jacobian[6];
335 point_jacobian_(2, 5) = point_angular_jacobian[7];
336
337 if (compute_hessian) {
338 Eigen::Matrix<double, 15, 1> point_angular_hessian =
339 angular_hessian_ * Eigen::Vector4d(x[0], x[1], x[2], 0.0);
340
341 // Vectors from Equation 6.21 [Magnusson 2009]
342 const Eigen::Vector3d a(0, point_angular_hessian[0], point_angular_hessian[1]);
343 const Eigen::Vector3d b(0, point_angular_hessian[2], point_angular_hessian[3]);
344 const Eigen::Vector3d c(0, point_angular_hessian[4], point_angular_hessian[5]);
345 const Eigen::Vector3d d = point_angular_hessian.block<3, 1>(6, 0);
346 const Eigen::Vector3d e = point_angular_hessian.block<3, 1>(9, 0);
347 const Eigen::Vector3d f = point_angular_hessian.block<3, 1>(12, 0);
348
349 // Calculate second derivative of Transformation Equation 6.17 w.r.t. transform
350 // vector. Derivative w.r.t. ith and jth elements of transform vector corresponds to
351 // the 3x1 block matrix starting at (3i,j), Equation 6.20 and 6.21 [Magnusson 2009]
352 point_hessian_.block<3, 1>(9, 3) = a;
353 point_hessian_.block<3, 1>(12, 3) = b;
354 point_hessian_.block<3, 1>(15, 3) = c;
355 point_hessian_.block<3, 1>(9, 4) = b;
356 point_hessian_.block<3, 1>(12, 4) = d;
357 point_hessian_.block<3, 1>(15, 4) = e;
358 point_hessian_.block<3, 1>(9, 5) = c;
359 point_hessian_.block<3, 1>(12, 5) = e;
360 point_hessian_.block<3, 1>(15, 5) = f;
361 }
362}
363
364template <typename PointSource, typename PointTarget>
365double
367 Eigen::Matrix<double, 6, 1>& score_gradient,
368 Eigen::Matrix<double, 6, 6>& hessian,
369 const Eigen::Vector3d& x_trans,
370 const Eigen::Matrix3d& c_inv,
371 bool compute_hessian) const
372{
373 // e^(-d_2/2 * (x_k - mu_k)^T Sigma_k^-1 (x_k - mu_k)) Equation 6.9 [Magnusson 2009]
374 double e_x_cov_x = std::exp(-gauss_d2_ * x_trans.dot(c_inv * x_trans) / 2);
375 // Calculate probability of transformed points existence, Equation 6.9 [Magnusson
376 // 2009]
377 const double score_inc = -gauss_d1_ * e_x_cov_x;
378
379 e_x_cov_x = gauss_d2_ * e_x_cov_x;
380
381 // Error checking for invalid values.
382 if (e_x_cov_x > 1 || e_x_cov_x < 0 || std::isnan(e_x_cov_x)) {
383 return 0;
384 }
385
386 // Reusable portion of Equation 6.12 and 6.13 [Magnusson 2009]
387 e_x_cov_x *= gauss_d1_;
388
389 for (int i = 0; i < 6; i++) {
390 // Sigma_k^-1 d(T(x,p))/dpi, Reusable portion of Equation 6.12 and 6.13 [Magnusson
391 // 2009]
392 const Eigen::Vector3d cov_dxd_pi = c_inv * point_jacobian_.col(i);
393
394 // Update gradient, Equation 6.12 [Magnusson 2009]
395 score_gradient(i) += x_trans.dot(cov_dxd_pi) * e_x_cov_x;
396
397 if (compute_hessian) {
398 for (Eigen::Index j = 0; j < hessian.cols(); j++) {
399 // Update hessian, Equation 6.13 [Magnusson 2009]
400 hessian(i, j) +=
401 e_x_cov_x * (-gauss_d2_ * x_trans.dot(cov_dxd_pi) *
402 x_trans.dot(c_inv * point_jacobian_.col(j)) +
403 x_trans.dot(c_inv * point_hessian_.block<3, 1>(3 * i, j)) +
404 point_jacobian_.col(j).dot(cov_dxd_pi));
405 }
406 }
407 }
408
409 return score_inc;
410}
411
412template <typename PointSource, typename PointTarget>
413void
415 Eigen::Matrix<double, 6, 6>& hessian, const PointCloudSource& trans_cloud)
416{
417 hessian.setZero();
418
419 // Precompute Angular Derivatives unessisary because only used after regular
420 // derivative calculation Update hessian for each point, line 17 in Algorithm 2
421 // [Magnusson 2009]
422 for (std::size_t idx = 0; idx < input_->size(); idx++) {
423 // Transformed Point
424 const auto& x_trans_pt = trans_cloud[idx];
425
426 // Find nieghbors (Radius search has been experimentally faster than direct neighbor
427 // checking.
428 std::vector<TargetGridLeafConstPtr> neighborhood;
429 std::vector<float> distances;
430 target_cells_.radiusSearch(x_trans_pt, resolution_, neighborhood, distances);
431
432 for (const auto& cell : neighborhood) {
433 // Original Point
434 const auto& x_pt = (*input_)[idx];
435 const Eigen::Vector3d x = x_pt.getVector3fMap().template cast<double>();
436
437 // Denorm point, x_k' in Equations 6.12 and 6.13 [Magnusson 2009]
438 const Eigen::Vector3d x_trans =
439 x_trans_pt.getVector3fMap().template cast<double>() - cell->getMean();
440 // Inverse Covariance of Occupied Voxel
441 // Uses precomputed covariance for speed.
442 const Eigen::Matrix3d c_inv = cell->getInverseCov();
443
444 // Compute derivative of transform function w.r.t. transform vector, J_E and H_E
445 // in Equations 6.18 and 6.20 [Magnusson 2009]
446 computePointDerivatives(x);
447 // Update hessian, lines 21 in Algorithm 2, according to Equations 6.10, 6.12
448 // and 6.13, respectively [Magnusson 2009]
449 updateHessian(hessian, x_trans, c_inv);
450 }
451 }
452}
453
454template <typename PointSource, typename PointTarget>
455void
457 Eigen::Matrix<double, 6, 6>& hessian,
458 const Eigen::Vector3d& x_trans,
459 const Eigen::Matrix3d& c_inv) const
460{
461 // e^(-d_2/2 * (x_k - mu_k)^T Sigma_k^-1 (x_k - mu_k)) Equation 6.9 [Magnusson 2009]
462 double e_x_cov_x =
463 gauss_d2_ * std::exp(-gauss_d2_ * x_trans.dot(c_inv * x_trans) / 2);
464
465 // Error checking for invalid values.
466 if (e_x_cov_x > 1 || e_x_cov_x < 0 || std::isnan(e_x_cov_x)) {
467 return;
468 }
469
470 // Reusable portion of Equation 6.12 and 6.13 [Magnusson 2009]
471 e_x_cov_x *= gauss_d1_;
472
473 for (int i = 0; i < 6; i++) {
474 // Sigma_k^-1 d(T(x,p))/dpi, Reusable portion of Equation 6.12 and 6.13 [Magnusson
475 // 2009]
476 const Eigen::Vector3d cov_dxd_pi = c_inv * point_jacobian_.col(i);
477
478 for (Eigen::Index j = 0; j < hessian.cols(); j++) {
479 // Update hessian, Equation 6.13 [Magnusson 2009]
480 hessian(i, j) +=
481 e_x_cov_x * (-gauss_d2_ * x_trans.dot(cov_dxd_pi) *
482 x_trans.dot(c_inv * point_jacobian_.col(j)) +
483 x_trans.dot(c_inv * point_hessian_.block<3, 1>(3 * i, j)) +
484 point_jacobian_.col(j).dot(cov_dxd_pi));
485 }
486 }
487}
488
489template <typename PointSource, typename PointTarget>
490bool
492 double& a_l,
493 double& f_l,
494 double& g_l,
495 double& a_u,
496 double& f_u,
497 double& g_u,
498 double a_t,
499 double f_t,
500 double g_t) const
501{
502 // Case U1 in Update Algorithm and Case a in Modified Update Algorithm [More, Thuente
503 // 1994]
504 if (f_t > f_l) {
505 a_u = a_t;
506 f_u = f_t;
507 g_u = g_t;
508 return false;
509 }
510 // Case U2 in Update Algorithm and Case b in Modified Update Algorithm [More, Thuente
511 // 1994]
512 if (g_t * (a_l - a_t) > 0) {
513 a_l = a_t;
514 f_l = f_t;
515 g_l = g_t;
516 return false;
517 }
518 // Case U3 in Update Algorithm and Case c in Modified Update Algorithm [More, Thuente
519 // 1994]
520 if (g_t * (a_l - a_t) < 0) {
521 a_u = a_l;
522 f_u = f_l;
523 g_u = g_l;
524
525 a_l = a_t;
526 f_l = f_t;
527 g_l = g_t;
528 return false;
529 }
530 // Interval Converged
531 return true;
532}
533
534template <typename PointSource, typename PointTarget>
535double
537 double a_l,
538 double f_l,
539 double g_l,
540 double a_u,
541 double f_u,
542 double g_u,
543 double a_t,
544 double f_t,
545 double g_t) const
546{
547 if (a_t == a_l && a_t == a_u) {
548 return a_t;
549 }
550
551 // Endpoints condition check [More, Thuente 1994], p.299 - 300
552 enum class EndpointsCondition { Case1, Case2, Case3, Case4 };
553 EndpointsCondition condition;
554
555 if (a_t == a_l) {
556 condition = EndpointsCondition::Case4;
557 }
558 else if (f_t > f_l) {
559 condition = EndpointsCondition::Case1;
560 }
561 else if (g_t * g_l < 0) {
562 condition = EndpointsCondition::Case2;
563 }
564 else if (std::fabs(g_t) <= std::fabs(g_l)) {
565 condition = EndpointsCondition::Case3;
566 }
567 else {
568 condition = EndpointsCondition::Case4;
569 }
570
571 switch (condition) {
572 case EndpointsCondition::Case1: {
573 // Calculate the minimizer of the cubic that interpolates f_l, f_t, g_l and g_t
574 // Equation 2.4.52 [Sun, Yuan 2006]
575 const double z = 3 * (f_t - f_l) / (a_t - a_l) - g_t - g_l;
576 const double w = std::sqrt(z * z - g_t * g_l);
577 // Equation 2.4.56 [Sun, Yuan 2006]
578 const double a_c = a_l + (a_t - a_l) * (w - g_l - z) / (g_t - g_l + 2 * w);
579
580 // Calculate the minimizer of the quadratic that interpolates f_l, f_t and g_l
581 // Equation 2.4.2 [Sun, Yuan 2006]
582 const double a_q =
583 a_l - 0.5 * (a_l - a_t) * g_l / (g_l - (f_l - f_t) / (a_l - a_t));
584
585 if (std::fabs(a_c - a_l) < std::fabs(a_q - a_l)) {
586 return a_c;
587 }
588 return 0.5 * (a_q + a_c);
589 }
590
591 case EndpointsCondition::Case2: {
592 // Calculate the minimizer of the cubic that interpolates f_l, f_t, g_l and g_t
593 // Equation 2.4.52 [Sun, Yuan 2006]
594 const double z = 3 * (f_t - f_l) / (a_t - a_l) - g_t - g_l;
595 const double w = std::sqrt(z * z - g_t * g_l);
596 // Equation 2.4.56 [Sun, Yuan 2006]
597 const double a_c = a_l + (a_t - a_l) * (w - g_l - z) / (g_t - g_l + 2 * w);
598
599 // Calculate the minimizer of the quadratic that interpolates f_l, g_l and g_t
600 // Equation 2.4.5 [Sun, Yuan 2006]
601 const double a_s = a_l - (a_l - a_t) / (g_l - g_t) * g_l;
602
603 if (std::fabs(a_c - a_t) >= std::fabs(a_s - a_t)) {
604 return a_c;
605 }
606 return a_s;
607 }
608
609 case EndpointsCondition::Case3: {
610 // Calculate the minimizer of the cubic that interpolates f_l, f_t, g_l and g_t
611 // Equation 2.4.52 [Sun, Yuan 2006]
612 const double z = 3 * (f_t - f_l) / (a_t - a_l) - g_t - g_l;
613 const double w = std::sqrt(z * z - g_t * g_l);
614 const double a_c = a_l + (a_t - a_l) * (w - g_l - z) / (g_t - g_l + 2 * w);
615
616 // Calculate the minimizer of the quadratic that interpolates g_l and g_t
617 // Equation 2.4.5 [Sun, Yuan 2006]
618 const double a_s = a_l - (a_l - a_t) / (g_l - g_t) * g_l;
619
620 double a_t_next;
621
622 if (std::fabs(a_c - a_t) < std::fabs(a_s - a_t)) {
623 a_t_next = a_c;
624 }
625 else {
626 a_t_next = a_s;
627 }
628
629 if (a_t > a_l) {
630 return std::min(a_t + 0.66 * (a_u - a_t), a_t_next);
631 }
632 return std::max(a_t + 0.66 * (a_u - a_t), a_t_next);
633 }
634
635 default:
636 case EndpointsCondition::Case4: {
637 // Calculate the minimizer of the cubic that interpolates f_u, f_t, g_u and g_t
638 // Equation 2.4.52 [Sun, Yuan 2006]
639 const double z = 3 * (f_t - f_u) / (a_t - a_u) - g_t - g_u;
640 const double w = std::sqrt(z * z - g_t * g_u);
641 // Equation 2.4.56 [Sun, Yuan 2006]
642 return a_u + (a_t - a_u) * (w - g_u - z) / (g_t - g_u + 2 * w);
643 }
644 }
645}
646
647template <typename PointSource, typename PointTarget>
648double
650 const Eigen::Matrix<double, 6, 1>& x,
651 Eigen::Matrix<double, 6, 1>& step_dir,
652 double step_init,
653 double step_max,
654 double step_min,
655 double& score,
656 Eigen::Matrix<double, 6, 1>& score_gradient,
657 Eigen::Matrix<double, 6, 6>& hessian,
658 PointCloudSource& trans_cloud)
659{
660 // Set the value of phi(0), Equation 1.3 [More, Thuente 1994]
661 const double phi_0 = -score;
662 // Set the value of phi'(0), Equation 1.3 [More, Thuente 1994]
663 double d_phi_0 = -(score_gradient.dot(step_dir));
664
665 if (d_phi_0 >= 0) {
666 // Not a decent direction
667 if (d_phi_0 == 0) {
668 return 0;
669 }
670 // Reverse step direction and calculate optimal step.
671 d_phi_0 *= -1;
672 step_dir *= -1;
673 }
674
675 // The Search Algorithm for T(mu) [More, Thuente 1994]
676
677 const int max_step_iterations = 10;
678 int step_iterations = 0;
679
680 // Sufficient decreace constant, Equation 1.1 [More, Thuete 1994]
681 const double mu = 1.e-4;
682 // Curvature condition constant, Equation 1.2 [More, Thuete 1994]
683 const double nu = 0.9;
684
685 // Initial endpoints of Interval I,
686 double a_l = 0, a_u = 0;
687
688 // Auxiliary function psi is used until I is determined ot be a closed interval,
689 // Equation 2.1 [More, Thuente 1994]
690 double f_l = auxilaryFunction_PsiMT(a_l, phi_0, phi_0, d_phi_0, mu);
691 double g_l = auxilaryFunction_dPsiMT(d_phi_0, d_phi_0, mu);
692
693 double f_u = auxilaryFunction_PsiMT(a_u, phi_0, phi_0, d_phi_0, mu);
694 double g_u = auxilaryFunction_dPsiMT(d_phi_0, d_phi_0, mu);
695
696 // Check used to allow More-Thuente step length calculation to be skipped by making
697 // step_min == step_max
698 bool interval_converged = (step_max - step_min) < 0, open_interval = true;
699
700 double a_t = step_init;
701 a_t = std::min(a_t, step_max);
702 a_t = std::max(a_t, step_min);
703
704 Eigen::Matrix<double, 6, 1> x_t = x + step_dir * a_t;
705
706 // Convert x_t into matrix form
707 convertTransform(x_t, final_transformation_);
708
709 // New transformed point cloud
710 transformPointCloud(*input_, trans_cloud, final_transformation_);
711
712 // Updates score, gradient and hessian. Hessian calculation is unessisary but testing
713 // showed that most step calculations use the initial step suggestion and
714 // recalculation the reusable portions of the hessian would intail more computation
715 // time.
716 score = computeDerivatives(score_gradient, hessian, trans_cloud, x_t, true);
717
718 // Calculate phi(alpha_t)
719 double phi_t = -score;
720 // Calculate phi'(alpha_t)
721 double d_phi_t = -(score_gradient.dot(step_dir));
722
723 // Calculate psi(alpha_t)
724 double psi_t = auxilaryFunction_PsiMT(a_t, phi_t, phi_0, d_phi_0, mu);
725 // Calculate psi'(alpha_t)
726 double d_psi_t = auxilaryFunction_dPsiMT(d_phi_t, d_phi_0, mu);
727
728 // Iterate until max number of iterations, interval convergance or a value satisfies
729 // the sufficient decrease, Equation 1.1, and curvature condition, Equation 1.2 [More,
730 // Thuente 1994]
731 while (!interval_converged && step_iterations < max_step_iterations &&
732 !(psi_t <= 0 /*Sufficient Decrease*/ &&
733 d_phi_t <= -nu * d_phi_0 /*Curvature Condition*/)) {
734 // Use auxiliary function if interval I is not closed
735 if (open_interval) {
736 a_t = trialValueSelectionMT(a_l, f_l, g_l, a_u, f_u, g_u, a_t, psi_t, d_psi_t);
737 }
738 else {
739 a_t = trialValueSelectionMT(a_l, f_l, g_l, a_u, f_u, g_u, a_t, phi_t, d_phi_t);
740 }
741
742 a_t = std::min(a_t, step_max);
743 a_t = std::max(a_t, step_min);
744
745 x_t = x + step_dir * a_t;
746
747 // Convert x_t into matrix form
748 convertTransform(x_t, final_transformation_);
749
750 // New transformed point cloud
751 // Done on final cloud to prevent wasted computation
752 transformPointCloud(*input_, trans_cloud, final_transformation_);
753
754 // Updates score, gradient. Values stored to prevent wasted computation.
755 score = computeDerivatives(score_gradient, hessian, trans_cloud, x_t, false);
756
757 // Calculate phi(alpha_t+)
758 phi_t = -score;
759 // Calculate phi'(alpha_t+)
760 d_phi_t = -(score_gradient.dot(step_dir));
761
762 // Calculate psi(alpha_t+)
763 psi_t = auxilaryFunction_PsiMT(a_t, phi_t, phi_0, d_phi_0, mu);
764 // Calculate psi'(alpha_t+)
765 d_psi_t = auxilaryFunction_dPsiMT(d_phi_t, d_phi_0, mu);
766
767 // Check if I is now a closed interval
768 if (open_interval && (psi_t <= 0 && d_psi_t >= 0)) {
769 open_interval = false;
770
771 // Converts f_l and g_l from psi to phi
772 f_l += phi_0 - mu * d_phi_0 * a_l;
773 g_l += mu * d_phi_0;
774
775 // Converts f_u and g_u from psi to phi
776 f_u += phi_0 - mu * d_phi_0 * a_u;
777 g_u += mu * d_phi_0;
778 }
779
780 if (open_interval) {
781 // Update interval end points using Updating Algorithm [More, Thuente 1994]
782 interval_converged =
783 updateIntervalMT(a_l, f_l, g_l, a_u, f_u, g_u, a_t, psi_t, d_psi_t);
784 }
785 else {
786 // Update interval end points using Modified Updating Algorithm [More, Thuente
787 // 1994]
788 interval_converged =
789 updateIntervalMT(a_l, f_l, g_l, a_u, f_u, g_u, a_t, phi_t, d_phi_t);
790 }
791
792 step_iterations++;
793 }
794
795 // If inner loop was run then hessian needs to be calculated.
796 // Hessian is unnessisary for step length determination but gradients are required
797 // so derivative and transform data is stored for the next iteration.
798 if (step_iterations) {
799 computeHessian(hessian, trans_cloud);
800 }
801
802 return a_t;
803}
804
805} // namespace pcl
806
807#endif // PCL_REGISTRATION_NDT_IMPL_H_
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:491
void updateHessian(Eigen::Matrix< double, 6, 6 > &hessian, const Eigen::Vector3d &x_trans, const Eigen::Matrix3d &c_inv) const
Compute individual point contirbutions to hessian of probability function w.r.t.
Definition: ndt.hpp:456
void computePointDerivatives(const Eigen::Vector3d &x, bool compute_hessian=true)
Compute point derivatives.
Definition: ndt.hpp:320
virtual void computeTransformation(PointCloudSource &output)
Estimate the transformation and returns the transformed source (input) as output.
Definition: ndt.h:238
typename Registration< PointSource, PointTarget >::PointCloudSource PointCloudSource
Definition: ndt.h:66
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 probability function w.r.t.
Definition: ndt.hpp:184
double outlier_ratio_
The ratio of outliers of points w.r.t.
Definition: ndt.h:480
double gauss_d1_
The normalization constants used fit the point distribution to a normal distribution,...
Definition: ndt.h:484
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 contirbutions to derivatives of probability function w.r....
Definition: ndt.hpp:366
float resolution_
The side length of voxels.
Definition: ndt.h:473
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:536
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 probability derivatives using More-Thuente m...
Definition: ndt.hpp:649
void computeAngleDerivatives(const Eigen::Matrix< double, 6, 1 > &transform, bool compute_hessian=true)
Precompute anglular components of derivatives.
Definition: ndt.hpp:235
void computeHessian(Eigen::Matrix< double, 6, 6 > &hessian, const PointCloudSource &trans_cloud)
Compute hessian of probability function w.r.t.
Definition: ndt.hpp:414
NormalDistributionsTransform()
Constructor.
Definition: ndt.hpp:47
std::string reg_name_
The registration method name.
Definition: registration.h:558
int max_iterations_
The maximum number of iterations the internal optimization should run for.
Definition: registration.h:573
double transformation_epsilon_
The maximum difference between two consecutive transformations in order to consider convergence (user...
Definition: registration.h:595
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