41 #ifndef PCL_REGISTRATION_NDT_IMPL_H_
42 #define PCL_REGISTRATION_NDT_IMPL_H_
46 template <
typename Po
intSource,
typename Po
intTarget>
51 , outlier_ratio_(0.55)
54 , trans_probability_()
56 reg_name_ =
"NormalDistributionsTransform";
61 const double gauss_d3 = -std::log(gauss_c2);
62 gauss_d1_ = -std::log(gauss_c1 + gauss_c2) - gauss_d3;
64 -2 * std::log((-std::log(gauss_c1 * std::exp(-0.5) + gauss_c2) - gauss_d3) /
71 template <
typename Po
intSource,
typename Po
intTarget>
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;
85 -2 * std::log((-std::log(gauss_c1 * std::exp(-0.5) + gauss_c2) - gauss_d3) /
88 if (guess != Eigen::Matrix4f::Identity()) {
90 final_transformation_ = guess;
96 point_jacobian_.setZero();
97 point_jacobian_.block<3, 3>(0, 0).setIdentity();
98 point_hessian_.setZero();
100 Eigen::Transform<float, 3, Eigen::Affine, Eigen::ColMajor> eig_transformation;
101 eig_transformation.matrix() = final_transformation_;
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>();
109 Eigen::Matrix<double, 6, 6> hessian;
113 double score = computeDerivatives(score_gradient, hessian, output, transform);
115 while (!converged_) {
117 previous_transformation_ = transformation_;
121 Eigen::JacobiSVD<Eigen::Matrix<double, 6, 6>> sv(
122 hessian, Eigen::ComputeFullU | Eigen::ComputeFullV);
124 Eigen::Matrix<double, 6, 1> delta = sv.solve(-score_gradient);
127 double delta_norm = delta.norm();
129 if (delta_norm == 0 || std::isnan(delta_norm)) {
130 trans_probability_ = score /
static_cast<double>(input_->size());
131 converged_ = delta_norm == 0;
136 delta_norm = computeStepLengthMT(transform,
140 transformation_epsilon_ / 2,
148 convertTransform(delta, transformation_);
153 if (update_visualizer_)
154 update_visualizer_(output, std::vector<int>(), *target_, std::vector<int>());
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();
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))) {
179 trans_probability_ = score /
static_cast<double>(input_->size());
182 template <
typename Po
intSource,
typename Po
intTarget>
185 Eigen::Matrix<double, 6, 1>& score_gradient,
186 Eigen::Matrix<double, 6, 6>& hessian,
188 const Eigen::Matrix<double, 6, 1>& transform,
189 bool compute_hessian)
191 score_gradient.setZero();
196 computeAngleDerivatives(transform);
199 for (std::size_t idx = 0; idx < input_->size(); idx++) {
201 const auto& x_trans_pt = trans_cloud[idx];
205 std::vector<TargetGridLeafConstPtr> neighborhood;
206 std::vector<float> distances;
207 target_cells_.radiusSearch(x_trans_pt, resolution_, neighborhood, distances);
209 for (
const auto& cell : neighborhood) {
211 const auto& x_pt = (*input_)[idx];
212 const Eigen::Vector3d x = x_pt.getVector3fMap().template cast<double>();
215 const Eigen::Vector3d x_trans =
216 x_trans_pt.getVector3fMap().template cast<double>() - cell->getMean();
219 const Eigen::Matrix3d c_inv = cell->getInverseCov();
223 computePointDerivatives(x);
227 updateDerivatives(score_gradient, hessian, x_trans, c_inv, compute_hessian);
233 template <
typename Po
intSource,
typename Po
intTarget>
236 const Eigen::Matrix<double, 6, 1>& transform,
bool compute_hessian)
239 const auto calculate_cos_sin = [](
double angle,
double& c,
double& s) {
240 if (std::abs(angle) < 10e-5) {
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);
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);
260 angular_jacobian_.row(1).noalias() = Eigen::Vector4d(
261 (cx * sz + sx * sy * cz), (cx * cz - sx * sy * sz), (-sx * cy), 1.0);
262 angular_jacobian_.row(2).noalias() =
263 Eigen::Vector4d((-sy * cz), sy * sz, cy, 1.0);
264 angular_jacobian_.row(3).noalias() =
265 Eigen::Vector4d(sx * cy * cz, (-sx * cy * sz), sx * sy, 1.0);
266 angular_jacobian_.row(4).noalias() =
267 Eigen::Vector4d((-cx * cy * cz), cx * cy * sz, (-cx * sy), 1.0);
268 angular_jacobian_.row(5).noalias() =
269 Eigen::Vector4d((-cy * sz), (-cy * cz), 0, 1.0);
270 angular_jacobian_.row(6).noalias() =
271 Eigen::Vector4d((cx * cz - sx * sy * sz), (-cx * sz - sx * sy * cz), 0, 1.0);
272 angular_jacobian_.row(7).noalias() =
273 Eigen::Vector4d((sx * cz + cx * sy * sz), (cx * sy * cz - sx * sz), 0, 1.0);
275 if (compute_hessian) {
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);
281 angular_hessian_.row(1).noalias() = Eigen::Vector4d(
282 (-sx * sz + cx * sy * cz), (-cx * sy * sz - sx * cz), (-cx * cy), 0.0f);
284 angular_hessian_.row(2).noalias() =
285 Eigen::Vector4d((cx * cy * cz), (-cx * cy * sz), (cx * sy), 0.0f);
286 angular_hessian_.row(3).noalias() =
287 Eigen::Vector4d((sx * cy * cz), (-sx * cy * sz), (sx * sy), 0.0f);
289 angular_hessian_.row(4).noalias() = Eigen::Vector4d(
290 (-sx * cz - cx * sy * sz), (sx * sz - cx * sy * cz), 0, 0.0f);
291 angular_hessian_.row(5).noalias() = Eigen::Vector4d(
292 (cx * cz - sx * sy * sz), (-sx * sy * cz - cx * sz), 0, 0.0f);
294 angular_hessian_.row(6).noalias() =
295 Eigen::Vector4d((-cy * cz), (cy * sz), (sy), 0.0f);
296 angular_hessian_.row(7).noalias() =
297 Eigen::Vector4d((-sx * sy * cz), (sx * sy * sz), (sx * cy), 0.0f);
298 angular_hessian_.row(8).noalias() =
299 Eigen::Vector4d((cx * sy * cz), (-cx * sy * sz), (-cx * cy), 0.0f);
301 angular_hessian_.row(9).noalias() =
302 Eigen::Vector4d((sy * sz), (sy * cz), 0, 0.0f);
303 angular_hessian_.row(10).noalias() =
304 Eigen::Vector4d((-sx * cy * sz), (-sx * cy * cz), 0, 0.0f);
305 angular_hessian_.row(11).noalias() =
306 Eigen::Vector4d((cx * cy * sz), (cx * cy * cz), 0, 0.0f);
308 angular_hessian_.row(12).noalias() =
309 Eigen::Vector4d((-cy * cz), (cy * sz), 0, 0.0f);
310 angular_hessian_.row(13).noalias() = Eigen::Vector4d(
311 (-cx * sz - sx * sy * cz), (-cx * cz + sx * sy * sz), 0, 0.0f);
312 angular_hessian_.row(14).noalias() = Eigen::Vector4d(
313 (-sx * sz + cx * sy * cz), (-cx * sy * sz - sx * cz), 0, 0.0f);
317 template <
typename Po
intSource,
typename Po
intTarget>
320 const Eigen::Vector3d& x,
bool compute_hessian)
325 Eigen::Matrix<double, 8, 1> point_angular_jacobian =
326 angular_jacobian_ * Eigen::Vector4d(x[0], x[1], x[2], 0.0);
327 point_jacobian_(1, 3) = point_angular_jacobian[0];
328 point_jacobian_(2, 3) = point_angular_jacobian[1];
329 point_jacobian_(0, 4) = point_angular_jacobian[2];
330 point_jacobian_(1, 4) = point_angular_jacobian[3];
331 point_jacobian_(2, 4) = point_angular_jacobian[4];
332 point_jacobian_(0, 5) = point_angular_jacobian[5];
333 point_jacobian_(1, 5) = point_angular_jacobian[6];
334 point_jacobian_(2, 5) = point_angular_jacobian[7];
336 if (compute_hessian) {
337 Eigen::Matrix<double, 15, 1> point_angular_hessian =
338 angular_hessian_ * Eigen::Vector4d(x[0], x[1], x[2], 0.0);
341 const Eigen::Vector3d a(0, point_angular_hessian[0], point_angular_hessian[1]);
342 const Eigen::Vector3d b(0, point_angular_hessian[2], point_angular_hessian[3]);
343 const Eigen::Vector3d c(0, point_angular_hessian[4], point_angular_hessian[5]);
344 const Eigen::Vector3d d = point_angular_hessian.block<3, 1>(6, 0);
345 const Eigen::Vector3d e = point_angular_hessian.block<3, 1>(9, 0);
346 const Eigen::Vector3d f = point_angular_hessian.block<3, 1>(12, 0);
351 point_hessian_.block<3, 1>(9, 3) = a;
352 point_hessian_.block<3, 1>(12, 3) = b;
353 point_hessian_.block<3, 1>(15, 3) = c;
354 point_hessian_.block<3, 1>(9, 4) = b;
355 point_hessian_.block<3, 1>(12, 4) = d;
356 point_hessian_.block<3, 1>(15, 4) = e;
357 point_hessian_.block<3, 1>(9, 5) = c;
358 point_hessian_.block<3, 1>(12, 5) = e;
359 point_hessian_.block<3, 1>(15, 5) = f;
363 template <
typename Po
intSource,
typename Po
intTarget>
366 Eigen::Matrix<double, 6, 1>& score_gradient,
367 Eigen::Matrix<double, 6, 6>& hessian,
368 const Eigen::Vector3d& x_trans,
369 const Eigen::Matrix3d& c_inv,
370 bool compute_hessian)
const
373 double e_x_cov_x = std::exp(-gauss_d2_ * x_trans.dot(c_inv * x_trans) / 2);
376 const double score_inc = -gauss_d1_ * e_x_cov_x;
378 e_x_cov_x = gauss_d2_ * e_x_cov_x;
381 if (e_x_cov_x > 1 || e_x_cov_x < 0 || std::isnan(e_x_cov_x)) {
386 e_x_cov_x *= gauss_d1_;
388 for (
int i = 0; i < 6; i++) {
391 const Eigen::Vector3d cov_dxd_pi = c_inv * point_jacobian_.col(i);
394 score_gradient(i) += x_trans.dot(cov_dxd_pi) * e_x_cov_x;
396 if (compute_hessian) {
397 for (Eigen::Index j = 0; j < hessian.cols(); j++) {
400 e_x_cov_x * (-gauss_d2_ * x_trans.dot(cov_dxd_pi) *
401 x_trans.dot(c_inv * point_jacobian_.col(j)) +
402 x_trans.dot(c_inv * point_hessian_.block<3, 1>(3 * i, j)) +
403 point_jacobian_.col(j).dot(cov_dxd_pi));
411 template <
typename Po
intSource,
typename Po
intTarget>
421 for (std::size_t idx = 0; idx < input_->size(); idx++) {
423 const auto& x_trans_pt = trans_cloud[idx];
427 std::vector<TargetGridLeafConstPtr> neighborhood;
428 std::vector<float> distances;
429 target_cells_.radiusSearch(x_trans_pt, resolution_, neighborhood, distances);
431 for (
const auto& cell : neighborhood) {
433 const auto& x_pt = (*input_)[idx];
434 const Eigen::Vector3d x = x_pt.getVector3fMap().template cast<double>();
437 const Eigen::Vector3d x_trans =
438 x_trans_pt.getVector3fMap().template cast<double>() - cell->getMean();
441 const Eigen::Matrix3d c_inv = cell->getInverseCov();
445 computePointDerivatives(x);
448 updateHessian(hessian, x_trans, c_inv);
453 template <
typename Po
intSource,
typename Po
intTarget>
456 Eigen::Matrix<double, 6, 6>& hessian,
457 const Eigen::Vector3d& x_trans,
458 const Eigen::Matrix3d& c_inv)
const
462 gauss_d2_ * std::exp(-gauss_d2_ * x_trans.dot(c_inv * x_trans) / 2);
465 if (e_x_cov_x > 1 || e_x_cov_x < 0 || std::isnan(e_x_cov_x)) {
470 e_x_cov_x *= gauss_d1_;
472 for (
int i = 0; i < 6; i++) {
475 const Eigen::Vector3d cov_dxd_pi = c_inv * point_jacobian_.col(i);
477 for (Eigen::Index j = 0; j < hessian.cols(); j++) {
480 e_x_cov_x * (-gauss_d2_ * x_trans.dot(cov_dxd_pi) *
481 x_trans.dot(c_inv * point_jacobian_.col(j)) +
482 x_trans.dot(c_inv * point_hessian_.block<3, 1>(3 * i, j)) +
483 point_jacobian_.col(j).dot(cov_dxd_pi));
488 template <
typename Po
intSource,
typename Po
intTarget>
511 if (g_t * (a_l - a_t) > 0) {
519 if (g_t * (a_l - a_t) < 0) {
533 template <
typename Po
intSource,
typename Po
intTarget>
546 if (a_t == a_l && a_t == a_u) {
551 enum class EndpointsCondition { Case1, Case2, Case3, Case4 };
552 EndpointsCondition condition;
555 condition = EndpointsCondition::Case4;
557 else if (f_t > f_l) {
558 condition = EndpointsCondition::Case1;
560 else if (g_t * g_l < 0) {
561 condition = EndpointsCondition::Case2;
563 else if (std::fabs(g_t) <= std::fabs(g_l)) {
564 condition = EndpointsCondition::Case3;
567 condition = EndpointsCondition::Case4;
571 case EndpointsCondition::Case1: {
574 const double z = 3 * (f_t - f_l) / (a_t - a_l) - g_t - g_l;
575 const double w = std::sqrt(z * z - g_t * g_l);
577 const double a_c = a_l + (a_t - a_l) * (w - g_l - z) / (g_t - g_l + 2 * w);
582 a_l - 0.5 * (a_l - a_t) * g_l / (g_l - (f_l - f_t) / (a_l - a_t));
584 if (std::fabs(a_c - a_l) < std::fabs(a_q - a_l)) {
587 return 0.5 * (a_q + a_c);
590 case EndpointsCondition::Case2: {
593 const double z = 3 * (f_t - f_l) / (a_t - a_l) - g_t - g_l;
594 const double w = std::sqrt(z * z - g_t * g_l);
596 const double a_c = a_l + (a_t - a_l) * (w - g_l - z) / (g_t - g_l + 2 * w);
600 const double a_s = a_l - (a_l - a_t) / (g_l - g_t) * g_l;
602 if (std::fabs(a_c - a_t) >= std::fabs(a_s - a_t)) {
608 case EndpointsCondition::Case3: {
611 const double z = 3 * (f_t - f_l) / (a_t - a_l) - g_t - g_l;
612 const double w = std::sqrt(z * z - g_t * g_l);
613 const double a_c = a_l + (a_t - a_l) * (w - g_l - z) / (g_t - g_l + 2 * w);
617 const double a_s = a_l - (a_l - a_t) / (g_l - g_t) * g_l;
621 if (std::fabs(a_c - a_t) < std::fabs(a_s - a_t)) {
629 return std::min(a_t + 0.66 * (a_u - a_t), a_t_next);
631 return std::max(a_t + 0.66 * (a_u - a_t), a_t_next);
635 case EndpointsCondition::Case4: {
638 const double z = 3 * (f_t - f_u) / (a_t - a_u) - g_t - g_u;
639 const double w = std::sqrt(z * z - g_t * g_u);
641 return a_u + (a_t - a_u) * (w - g_u - z) / (g_t - g_u + 2 * w);
646 template <
typename Po
intSource,
typename Po
intTarget>
649 const Eigen::Matrix<double, 6, 1>& x,
650 Eigen::Matrix<double, 6, 1>& step_dir,
655 Eigen::Matrix<double, 6, 1>& score_gradient,
656 Eigen::Matrix<double, 6, 6>& hessian,
660 const double phi_0 = -score;
662 double d_phi_0 = -(score_gradient.dot(step_dir));
676 const int max_step_iterations = 10;
677 int step_iterations = 0;
680 const double mu = 1.e-4;
682 const double nu = 0.9;
685 double a_l = 0, a_u = 0;
689 double f_l = auxilaryFunction_PsiMT(a_l, phi_0, phi_0, d_phi_0, mu);
690 double g_l = auxilaryFunction_dPsiMT(d_phi_0, d_phi_0, mu);
692 double f_u = auxilaryFunction_PsiMT(a_u, phi_0, phi_0, d_phi_0, mu);
693 double g_u = auxilaryFunction_dPsiMT(d_phi_0, d_phi_0, mu);
697 bool interval_converged = (step_max - step_min) < 0, open_interval =
true;
699 double a_t = step_init;
700 a_t = std::min(a_t, step_max);
701 a_t = std::max(a_t, step_min);
703 Eigen::Matrix<double, 6, 1> x_t = x + step_dir * a_t;
706 convertTransform(x_t, final_transformation_);
715 score = computeDerivatives(score_gradient, hessian, trans_cloud, x_t,
true);
718 double phi_t = -score;
720 double d_phi_t = -(score_gradient.dot(step_dir));
723 double psi_t = auxilaryFunction_PsiMT(a_t, phi_t, phi_0, d_phi_0, mu);
725 double d_psi_t = auxilaryFunction_dPsiMT(d_phi_t, d_phi_0, mu);
730 while (!interval_converged && step_iterations < max_step_iterations &&
732 d_phi_t <= -nu * d_phi_0 )) {
735 a_t = trialValueSelectionMT(a_l, f_l, g_l, a_u, f_u, g_u, a_t, psi_t, d_psi_t);
738 a_t = trialValueSelectionMT(a_l, f_l, g_l, a_u, f_u, g_u, a_t, phi_t, d_phi_t);
741 a_t = std::min(a_t, step_max);
742 a_t = std::max(a_t, step_min);
744 x_t = x + step_dir * a_t;
747 convertTransform(x_t, final_transformation_);
754 score = computeDerivatives(score_gradient, hessian, trans_cloud, x_t,
false);
759 d_phi_t = -(score_gradient.dot(step_dir));
762 psi_t = auxilaryFunction_PsiMT(a_t, phi_t, phi_0, d_phi_0, mu);
764 d_psi_t = auxilaryFunction_dPsiMT(d_phi_t, d_phi_0, mu);
767 if (open_interval && (psi_t <= 0 && d_psi_t >= 0)) {
768 open_interval =
false;
771 f_l += phi_0 - mu * d_phi_0 * a_l;
775 f_u += phi_0 - mu * d_phi_0 * a_u;
782 updateIntervalMT(a_l, f_l, g_l, a_u, f_u, g_u, a_t, psi_t, d_psi_t);
788 updateIntervalMT(a_l, f_l, g_l, a_u, f_u, g_u, a_t, phi_t, d_phi_t);
797 if (step_iterations) {
798 computeHessian(hessian, trans_cloud);
806 #endif // PCL_REGISTRATION_NDT_IMPL_H_