Point Cloud Library (PCL) 1.14.0
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eigen.hpp
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38
39#pragma once
40
41#include <pcl/common/eigen.h>
42#include <pcl/console/print.h>
43
44#include <array>
45#include <algorithm>
46#include <cmath>
47
48
49namespace pcl
50{
51
52template <typename Scalar, typename Roots> inline void
53computeRoots2 (const Scalar& b, const Scalar& c, Roots& roots)
54{
55 roots (0) = Scalar (0);
56 Scalar d = Scalar (b * b - 4.0 * c);
57 if (d < 0.0) // no real roots ! THIS SHOULD NOT HAPPEN!
58 d = 0.0;
59
60 Scalar sd = std::sqrt (d);
61
62 roots (2) = 0.5f * (b + sd);
63 roots (1) = 0.5f * (b - sd);
64}
65
66
67template <typename Matrix, typename Roots> inline void
68computeRoots (const Matrix& m, Roots& roots)
69{
70 using Scalar = typename Matrix::Scalar;
71
72 // The characteristic equation is x^3 - c2*x^2 + c1*x - c0 = 0. The
73 // eigenvalues are the roots to this equation, all guaranteed to be
74 // real-valued, because the matrix is symmetric.
75 Scalar c0 = m (0, 0) * m (1, 1) * m (2, 2)
76 + Scalar (2) * m (0, 1) * m (0, 2) * m (1, 2)
77 - m (0, 0) * m (1, 2) * m (1, 2)
78 - m (1, 1) * m (0, 2) * m (0, 2)
79 - m (2, 2) * m (0, 1) * m (0, 1);
80 Scalar c1 = m (0, 0) * m (1, 1) -
81 m (0, 1) * m (0, 1) +
82 m (0, 0) * m (2, 2) -
83 m (0, 2) * m (0, 2) +
84 m (1, 1) * m (2, 2) -
85 m (1, 2) * m (1, 2);
86 Scalar c2 = m (0, 0) + m (1, 1) + m (2, 2);
87
88 if (std::abs (c0) < Eigen::NumTraits < Scalar > ::epsilon ()) // one root is 0 -> quadratic equation
89 computeRoots2 (c2, c1, roots);
90 else
91 {
92 constexpr Scalar s_inv3 = Scalar(1.0 / 3.0);
93 const Scalar s_sqrt3 = std::sqrt (Scalar (3.0));
94 // Construct the parameters used in classifying the roots of the equation
95 // and in solving the equation for the roots in closed form.
96 Scalar c2_over_3 = c2 * s_inv3;
97 Scalar a_over_3 = (c1 - c2 * c2_over_3) * s_inv3;
98 if (a_over_3 > Scalar (0))
99 a_over_3 = Scalar (0);
100
101 Scalar half_b = Scalar (0.5) * (c0 + c2_over_3 * (Scalar (2) * c2_over_3 * c2_over_3 - c1));
102
103 Scalar q = half_b * half_b + a_over_3 * a_over_3 * a_over_3;
104 if (q > Scalar (0))
105 q = Scalar (0);
106
107 // Compute the eigenvalues by solving for the roots of the polynomial.
108 Scalar rho = std::sqrt (-a_over_3);
109 Scalar theta = std::atan2 (std::sqrt (-q), half_b) * s_inv3;
110 Scalar cos_theta = std::cos (theta);
111 Scalar sin_theta = std::sin (theta);
112 roots (0) = c2_over_3 + Scalar (2) * rho * cos_theta;
113 roots (1) = c2_over_3 - rho * (cos_theta + s_sqrt3 * sin_theta);
114 roots (2) = c2_over_3 - rho * (cos_theta - s_sqrt3 * sin_theta);
115
116 // Sort in increasing order.
117 if (roots (0) >= roots (1))
118 std::swap (roots (0), roots (1));
119 if (roots (1) >= roots (2))
120 {
121 std::swap (roots (1), roots (2));
122 if (roots (0) >= roots (1))
123 std::swap (roots (0), roots (1));
124 }
125
126 if (roots (0) <= 0) // eigenval for symmetric positive semi-definite matrix can not be negative! Set it to 0
127 computeRoots2 (c2, c1, roots);
128 }
129}
130
131
132template <typename Matrix, typename Vector> inline void
133eigen22 (const Matrix& mat, typename Matrix::Scalar& eigenvalue, Vector& eigenvector)
134{
135 // if diagonal matrix, the eigenvalues are the diagonal elements
136 // and the eigenvectors are not unique, thus set to Identity
137 if (std::abs (mat.coeff (1)) <= std::numeric_limits<typename Matrix::Scalar>::min ())
138 {
139 if (mat.coeff (0) < mat.coeff (2))
140 {
141 eigenvalue = mat.coeff (0);
142 eigenvector[0] = 1.0;
143 eigenvector[1] = 0.0;
144 }
145 else
146 {
147 eigenvalue = mat.coeff (2);
148 eigenvector[0] = 0.0;
149 eigenvector[1] = 1.0;
150 }
151 return;
152 }
153
154 // 0.5 to optimize further calculations
155 typename Matrix::Scalar trace = static_cast<typename Matrix::Scalar> (0.5) * (mat.coeff (0) + mat.coeff (3));
156 typename Matrix::Scalar determinant = mat.coeff (0) * mat.coeff (3) - mat.coeff (1) * mat.coeff (1);
157
158 typename Matrix::Scalar temp = trace * trace - determinant;
159
160 if (temp < 0)
161 temp = 0;
162
163 eigenvalue = trace - std::sqrt (temp);
164
165 eigenvector[0] = -mat.coeff (1);
166 eigenvector[1] = mat.coeff (0) - eigenvalue;
167 eigenvector.normalize ();
168}
169
170
171template <typename Matrix, typename Vector> inline void
172eigen22 (const Matrix& mat, Matrix& eigenvectors, Vector& eigenvalues)
173{
174 // if diagonal matrix, the eigenvalues are the diagonal elements
175 // and the eigenvectors are not unique, thus set to Identity
176 if (std::abs (mat.coeff (1)) <= std::numeric_limits<typename Matrix::Scalar>::min ())
177 {
178 if (mat.coeff (0) < mat.coeff (3))
179 {
180 eigenvalues.coeffRef (0) = mat.coeff (0);
181 eigenvalues.coeffRef (1) = mat.coeff (3);
182 eigenvectors.coeffRef (0) = 1.0;
183 eigenvectors.coeffRef (1) = 0.0;
184 eigenvectors.coeffRef (2) = 0.0;
185 eigenvectors.coeffRef (3) = 1.0;
186 }
187 else
188 {
189 eigenvalues.coeffRef (0) = mat.coeff (3);
190 eigenvalues.coeffRef (1) = mat.coeff (0);
191 eigenvectors.coeffRef (0) = 0.0;
192 eigenvectors.coeffRef (1) = 1.0;
193 eigenvectors.coeffRef (2) = 1.0;
194 eigenvectors.coeffRef (3) = 0.0;
195 }
196 return;
197 }
198
199 // 0.5 to optimize further calculations
200 typename Matrix::Scalar trace = static_cast<typename Matrix::Scalar> (0.5) * (mat.coeff (0) + mat.coeff (3));
201 typename Matrix::Scalar determinant = mat.coeff (0) * mat.coeff (3) - mat.coeff (1) * mat.coeff (1);
202
203 typename Matrix::Scalar temp = trace * trace - determinant;
204
205 if (temp < 0)
206 temp = 0;
207 else
208 temp = std::sqrt (temp);
209
210 eigenvalues.coeffRef (0) = trace - temp;
211 eigenvalues.coeffRef (1) = trace + temp;
212
213 // either this is in a row or column depending on RowMajor or ColumnMajor
214 eigenvectors.coeffRef (0) = -mat.coeff (1);
215 eigenvectors.coeffRef (2) = mat.coeff (0) - eigenvalues.coeff (0);
216 typename Matrix::Scalar norm = static_cast<typename Matrix::Scalar> (1.0)
217 / static_cast<typename Matrix::Scalar> (std::sqrt (eigenvectors.coeffRef (0) * eigenvectors.coeffRef (0) + eigenvectors.coeffRef (2) * eigenvectors.coeffRef (2)));
218 eigenvectors.coeffRef (0) *= norm;
219 eigenvectors.coeffRef (2) *= norm;
220 eigenvectors.coeffRef (1) = eigenvectors.coeffRef (2);
221 eigenvectors.coeffRef (3) = -eigenvectors.coeffRef (0);
222}
223
224
225template <typename Matrix, typename Vector> inline void
226computeCorrespondingEigenVector (const Matrix& mat, const typename Matrix::Scalar& eigenvalue, Vector& eigenvector)
227{
228 using Scalar = typename Matrix::Scalar;
229 // Scale the matrix so its entries are in [-1,1]. The scaling is applied
230 // only when at least one matrix entry has magnitude larger than 1.
231
232 Scalar scale = mat.cwiseAbs ().maxCoeff ();
233 if (scale <= std::numeric_limits < Scalar > ::min ())
234 scale = Scalar (1.0);
235
236 Matrix scaledMat = mat / scale;
237
238 scaledMat.diagonal ().array () -= eigenvalue / scale;
239
240 Vector vec1 = scaledMat.row (0).cross (scaledMat.row (1));
241 Vector vec2 = scaledMat.row (0).cross (scaledMat.row (2));
242 Vector vec3 = scaledMat.row (1).cross (scaledMat.row (2));
243
244 Scalar len1 = vec1.squaredNorm ();
245 Scalar len2 = vec2.squaredNorm ();
246 Scalar len3 = vec3.squaredNorm ();
247
248 if (len1 >= len2 && len1 >= len3)
249 eigenvector = vec1 / std::sqrt (len1);
250 else if (len2 >= len1 && len2 >= len3)
251 eigenvector = vec2 / std::sqrt (len2);
252 else
253 eigenvector = vec3 / std::sqrt (len3);
254}
255
256namespace detail
257{
258
259template <typename Vector, typename Scalar>
261 Vector vector;
262 Scalar length;
263}; // struct EigenVector
264
265/**
266 * @brief returns the unit vector along the largest eigen value as well as the
267 * length of the largest eigenvector
268 * @tparam Vector Requested result type, needs to be explicitly provided and has
269 * to be implicitly constructible from ConstRowExpr
270 * @tparam Matrix deduced input type providing similar in API as Eigen::Matrix
271 */
272template <typename Vector, typename Matrix> static EigenVector<Vector, typename Matrix::Scalar>
273getLargest3x3Eigenvector (const Matrix scaledMatrix)
274{
275 using Scalar = typename Matrix::Scalar;
276 using Index = typename Matrix::Index;
277
278 Matrix crossProduct;
279 crossProduct << scaledMatrix.row (0).cross (scaledMatrix.row (1)),
280 scaledMatrix.row (0).cross (scaledMatrix.row (2)),
281 scaledMatrix.row (1).cross (scaledMatrix.row (2));
282
283 // expression template, no evaluation here
284 const auto len = crossProduct.rowwise ().norm ();
285
286 Index index;
287 const Scalar length = len.maxCoeff (&index); // <- first evaluation
288 return {crossProduct.row (index) / length, length};
289}
290
291} // namespace detail
292
293
294template <typename Matrix, typename Vector> inline void
295eigen33 (const Matrix& mat, typename Matrix::Scalar& eigenvalue, Vector& eigenvector)
296{
297 using Scalar = typename Matrix::Scalar;
298 // Scale the matrix so its entries are in [-1,1]. The scaling is applied
299 // only when at least one matrix entry has magnitude larger than 1.
300
301 Scalar scale = mat.cwiseAbs ().maxCoeff ();
302 if (scale <= std::numeric_limits < Scalar > ::min ())
303 scale = Scalar (1.0);
304
305 Matrix scaledMat = mat / scale;
306
307 Vector eigenvalues;
308 computeRoots (scaledMat, eigenvalues);
309
310 eigenvalue = eigenvalues (0) * scale;
311
312 scaledMat.diagonal ().array () -= eigenvalues (0);
313
314 eigenvector = detail::getLargest3x3Eigenvector<Vector> (scaledMat).vector;
315}
316
317
318template <typename Matrix, typename Vector> inline void
319eigen33 (const Matrix& mat, Vector& evals)
320{
321 using Scalar = typename Matrix::Scalar;
322 Scalar scale = mat.cwiseAbs ().maxCoeff ();
323 if (scale <= std::numeric_limits < Scalar > ::min ())
324 scale = Scalar (1.0);
325
326 Matrix scaledMat = mat / scale;
327 computeRoots (scaledMat, evals);
328 evals *= scale;
329}
330
331
332template <typename Matrix, typename Vector> inline void
333eigen33 (const Matrix& mat, Matrix& evecs, Vector& evals)
334{
335 using Scalar = typename Matrix::Scalar;
336 // Scale the matrix so its entries are in [-1,1]. The scaling is applied
337 // only when at least one matrix entry has magnitude larger than 1.
338
339 Scalar scale = mat.cwiseAbs ().maxCoeff ();
340 if (scale <= std::numeric_limits < Scalar > ::min ())
341 scale = Scalar (1.0);
342
343 Matrix scaledMat = mat / scale;
344
345 // Compute the eigenvalues
346 computeRoots (scaledMat, evals);
347
348 if ( (evals (2) - evals (0)) <= Eigen::NumTraits < Scalar > ::epsilon ())
349 {
350 // all three equal
351 evecs.setIdentity ();
352 }
353 else if ( (evals (1) - evals (0)) <= Eigen::NumTraits < Scalar > ::epsilon ())
354 {
355 // first and second equal
356 Matrix tmp;
357 tmp = scaledMat;
358 tmp.diagonal ().array () -= evals (2);
359
360 evecs.col (2) = detail::getLargest3x3Eigenvector<Vector> (tmp).vector;
361 evecs.col (1) = evecs.col (2).unitOrthogonal ();
362 evecs.col (0) = evecs.col (1).cross (evecs.col (2));
363 }
364 else if ( (evals (2) - evals (1)) <= Eigen::NumTraits < Scalar > ::epsilon ())
365 {
366 // second and third equal
367 Matrix tmp;
368 tmp = scaledMat;
369 tmp.diagonal ().array () -= evals (0);
370
371 evecs.col (0) = detail::getLargest3x3Eigenvector<Vector> (tmp).vector;
372 evecs.col (1) = evecs.col (0).unitOrthogonal ();
373 evecs.col (2) = evecs.col (0).cross (evecs.col (1));
374 }
375 else
376 {
377 std::array<Scalar, 3> eigenVecLen;
378
379 for (int i = 0; i < 3; ++i)
380 {
381 Matrix tmp = scaledMat;
382 tmp.diagonal ().array () -= evals (i);
383 const auto vec_len = detail::getLargest3x3Eigenvector<Vector> (tmp);
384 evecs.col (i) = vec_len.vector;
385 eigenVecLen[i] = vec_len.length;
386 }
387
388 // @TODO: might be redundant or over-complicated as per @SergioRAgostinho
389 // see: https://github.com/PointCloudLibrary/pcl/pull/3441#discussion_r341024181
390 const auto minmax_it = std::minmax_element (eigenVecLen.cbegin (), eigenVecLen.cend ());
391 int min_idx = std::distance (eigenVecLen.cbegin (), minmax_it.first);
392 int max_idx = std::distance (eigenVecLen.cbegin (), minmax_it.second);
393 int mid_idx = 3 - min_idx - max_idx;
394
395 evecs.col (min_idx) = evecs.col ( (min_idx + 1) % 3).cross (evecs.col ( (min_idx + 2) % 3)).normalized ();
396 evecs.col (mid_idx) = evecs.col ( (mid_idx + 1) % 3).cross (evecs.col ( (mid_idx + 2) % 3)).normalized ();
397 }
398 // Rescale back to the original size.
399 evals *= scale;
400}
401
402
403template <typename Matrix> inline typename Matrix::Scalar
404invert2x2 (const Matrix& matrix, Matrix& inverse)
405{
406 using Scalar = typename Matrix::Scalar;
407 Scalar det = matrix.coeff (0) * matrix.coeff (3) - matrix.coeff (1) * matrix.coeff (2);
408
409 if (det != 0)
410 {
411 //Scalar inv_det = Scalar (1.0) / det;
412 inverse.coeffRef (0) = matrix.coeff (3);
413 inverse.coeffRef (1) = -matrix.coeff (1);
414 inverse.coeffRef (2) = -matrix.coeff (2);
415 inverse.coeffRef (3) = matrix.coeff (0);
416 inverse /= det;
417 }
418 return det;
419}
420
421
422template <typename Matrix> inline typename Matrix::Scalar
423invert3x3SymMatrix (const Matrix& matrix, Matrix& inverse)
424{
425 using Scalar = typename Matrix::Scalar;
426 // elements
427 // a b c
428 // b d e
429 // c e f
430 //| a b c |-1 | fd-ee ce-bf be-cd |
431 //| b d e | = 1/det * | ce-bf af-cc bc-ae |
432 //| c e f | | be-cd bc-ae ad-bb |
433
434 //det = a(fd-ee) + b(ec-fb) + c(eb-dc)
435
436 Scalar fd_ee = matrix.coeff (4) * matrix.coeff (8) - matrix.coeff (7) * matrix.coeff (5);
437 Scalar ce_bf = matrix.coeff (2) * matrix.coeff (5) - matrix.coeff (1) * matrix.coeff (8);
438 Scalar be_cd = matrix.coeff (1) * matrix.coeff (5) - matrix.coeff (2) * matrix.coeff (4);
439
440 Scalar det = matrix.coeff (0) * fd_ee + matrix.coeff (1) * ce_bf + matrix.coeff (2) * be_cd;
441
442 if (det != 0)
443 {
444 //Scalar inv_det = Scalar (1.0) / det;
445 inverse.coeffRef (0) = fd_ee;
446 inverse.coeffRef (1) = inverse.coeffRef (3) = ce_bf;
447 inverse.coeffRef (2) = inverse.coeffRef (6) = be_cd;
448 inverse.coeffRef (4) = (matrix.coeff (0) * matrix.coeff (8) - matrix.coeff (2) * matrix.coeff (2));
449 inverse.coeffRef (5) = inverse.coeffRef (7) = (matrix.coeff (1) * matrix.coeff (2) - matrix.coeff (0) * matrix.coeff (5));
450 inverse.coeffRef (8) = (matrix.coeff (0) * matrix.coeff (4) - matrix.coeff (1) * matrix.coeff (1));
451 inverse /= det;
452 }
453 return det;
454}
455
456
457template <typename Matrix> inline typename Matrix::Scalar
458invert3x3Matrix (const Matrix& matrix, Matrix& inverse)
459{
460 using Scalar = typename Matrix::Scalar;
461
462 //| a b c |-1 | ie-hf hc-ib fb-ec |
463 //| d e f | = 1/det * | gf-id ia-gc dc-fa |
464 //| g h i | | hd-ge gb-ha ea-db |
465 //det = a(ie-hf) + d(hc-ib) + g(fb-ec)
466
467 Scalar ie_hf = matrix.coeff (8) * matrix.coeff (4) - matrix.coeff (7) * matrix.coeff (5);
468 Scalar hc_ib = matrix.coeff (7) * matrix.coeff (2) - matrix.coeff (8) * matrix.coeff (1);
469 Scalar fb_ec = matrix.coeff (5) * matrix.coeff (1) - matrix.coeff (4) * matrix.coeff (2);
470 Scalar det = matrix.coeff (0) * (ie_hf) + matrix.coeff (3) * (hc_ib) + matrix.coeff (6) * (fb_ec);
471
472 if (det != 0)
473 {
474 inverse.coeffRef (0) = ie_hf;
475 inverse.coeffRef (1) = hc_ib;
476 inverse.coeffRef (2) = fb_ec;
477 inverse.coeffRef (3) = matrix.coeff (6) * matrix.coeff (5) - matrix.coeff (8) * matrix.coeff (3);
478 inverse.coeffRef (4) = matrix.coeff (8) * matrix.coeff (0) - matrix.coeff (6) * matrix.coeff (2);
479 inverse.coeffRef (5) = matrix.coeff (3) * matrix.coeff (2) - matrix.coeff (5) * matrix.coeff (0);
480 inverse.coeffRef (6) = matrix.coeff (7) * matrix.coeff (3) - matrix.coeff (6) * matrix.coeff (4);
481 inverse.coeffRef (7) = matrix.coeff (6) * matrix.coeff (1) - matrix.coeff (7) * matrix.coeff (0);
482 inverse.coeffRef (8) = matrix.coeff (4) * matrix.coeff (0) - matrix.coeff (3) * matrix.coeff (1);
483
484 inverse /= det;
485 }
486 return det;
487}
488
489
490template <typename Matrix> inline typename Matrix::Scalar
491determinant3x3Matrix (const Matrix& matrix)
492{
493 // result is independent of Row/Col Major storage!
494 return matrix.coeff (0) * (matrix.coeff (4) * matrix.coeff (8) - matrix.coeff (5) * matrix.coeff (7)) +
495 matrix.coeff (1) * (matrix.coeff (5) * matrix.coeff (6) - matrix.coeff (3) * matrix.coeff (8)) +
496 matrix.coeff (2) * (matrix.coeff (3) * matrix.coeff (7) - matrix.coeff (4) * matrix.coeff (6)) ;
497}
498
499
500void
501getTransFromUnitVectorsZY (const Eigen::Vector3f& z_axis,
502 const Eigen::Vector3f& y_direction,
503 Eigen::Affine3f& transformation)
504{
505 Eigen::Vector3f tmp0 = (y_direction.cross(z_axis)).normalized();
506 Eigen::Vector3f tmp1 = (z_axis.cross(tmp0)).normalized();
507 Eigen::Vector3f tmp2 = z_axis.normalized();
508
509 transformation(0,0)=tmp0[0]; transformation(0,1)=tmp0[1]; transformation(0,2)=tmp0[2]; transformation(0,3)=0.0f;
510 transformation(1,0)=tmp1[0]; transformation(1,1)=tmp1[1]; transformation(1,2)=tmp1[2]; transformation(1,3)=0.0f;
511 transformation(2,0)=tmp2[0]; transformation(2,1)=tmp2[1]; transformation(2,2)=tmp2[2]; transformation(2,3)=0.0f;
512 transformation(3,0)=0.0f; transformation(3,1)=0.0f; transformation(3,2)=0.0f; transformation(3,3)=1.0f;
513}
514
515
516Eigen::Affine3f
517getTransFromUnitVectorsZY (const Eigen::Vector3f& z_axis,
518 const Eigen::Vector3f& y_direction)
519{
520 Eigen::Affine3f transformation;
521 getTransFromUnitVectorsZY (z_axis, y_direction, transformation);
522 return (transformation);
523}
524
525
526void
527getTransFromUnitVectorsXY (const Eigen::Vector3f& x_axis,
528 const Eigen::Vector3f& y_direction,
529 Eigen::Affine3f& transformation)
530{
531 Eigen::Vector3f tmp2 = (x_axis.cross(y_direction)).normalized();
532 Eigen::Vector3f tmp1 = (tmp2.cross(x_axis)).normalized();
533 Eigen::Vector3f tmp0 = x_axis.normalized();
534
535 transformation(0,0)=tmp0[0]; transformation(0,1)=tmp0[1]; transformation(0,2)=tmp0[2]; transformation(0,3)=0.0f;
536 transformation(1,0)=tmp1[0]; transformation(1,1)=tmp1[1]; transformation(1,2)=tmp1[2]; transformation(1,3)=0.0f;
537 transformation(2,0)=tmp2[0]; transformation(2,1)=tmp2[1]; transformation(2,2)=tmp2[2]; transformation(2,3)=0.0f;
538 transformation(3,0)=0.0f; transformation(3,1)=0.0f; transformation(3,2)=0.0f; transformation(3,3)=1.0f;
539}
540
541
542Eigen::Affine3f
543getTransFromUnitVectorsXY (const Eigen::Vector3f& x_axis,
544 const Eigen::Vector3f& y_direction)
545{
546 Eigen::Affine3f transformation;
547 getTransFromUnitVectorsXY (x_axis, y_direction, transformation);
548 return (transformation);
549}
550
551
552void
553getTransformationFromTwoUnitVectors (const Eigen::Vector3f& y_direction,
554 const Eigen::Vector3f& z_axis,
555 Eigen::Affine3f& transformation)
556{
557 getTransFromUnitVectorsZY (z_axis, y_direction, transformation);
558}
559
560
561Eigen::Affine3f
562getTransformationFromTwoUnitVectors (const Eigen::Vector3f& y_direction,
563 const Eigen::Vector3f& z_axis)
564{
565 Eigen::Affine3f transformation;
566 getTransformationFromTwoUnitVectors (y_direction, z_axis, transformation);
567 return (transformation);
568}
569
570
571void
572getTransformationFromTwoUnitVectorsAndOrigin (const Eigen::Vector3f& y_direction,
573 const Eigen::Vector3f& z_axis,
574 const Eigen::Vector3f& origin,
575 Eigen::Affine3f& transformation)
576{
577 getTransformationFromTwoUnitVectors(y_direction, z_axis, transformation);
578 Eigen::Vector3f translation = transformation*origin;
579 transformation(0,3)=-translation[0]; transformation(1,3)=-translation[1]; transformation(2,3)=-translation[2];
580}
581
582
583template <typename Scalar> void
584getEulerAngles (const Eigen::Transform<Scalar, 3, Eigen::Affine> &t, Scalar &roll, Scalar &pitch, Scalar &yaw)
585{
586 roll = std::atan2 (t (2, 1), t (2, 2));
587 pitch = asin (-t (2, 0));
588 yaw = std::atan2 (t (1, 0), t (0, 0));
589}
590
591
592template <typename Scalar> void
593getTranslationAndEulerAngles (const Eigen::Transform<Scalar, 3, Eigen::Affine> &t,
594 Scalar &x, Scalar &y, Scalar &z,
595 Scalar &roll, Scalar &pitch, Scalar &yaw)
596{
597 x = t (0, 3);
598 y = t (1, 3);
599 z = t (2, 3);
600 roll = std::atan2 (t (2, 1), t (2, 2));
601 pitch = asin (-t (2, 0));
602 yaw = std::atan2 (t (1, 0), t (0, 0));
603}
604
605
606template <typename Scalar> void
607getTransformation (Scalar x, Scalar y, Scalar z,
608 Scalar roll, Scalar pitch, Scalar yaw,
609 Eigen::Transform<Scalar, 3, Eigen::Affine> &t)
610{
611 Scalar A = std::cos (yaw), B = sin (yaw), C = std::cos (pitch), D = sin (pitch),
612 E = std::cos (roll), F = sin (roll), DE = D*E, DF = D*F;
613
614 t (0, 0) = A*C; t (0, 1) = A*DF - B*E; t (0, 2) = B*F + A*DE; t (0, 3) = x;
615 t (1, 0) = B*C; t (1, 1) = A*E + B*DF; t (1, 2) = B*DE - A*F; t (1, 3) = y;
616 t (2, 0) = -D; t (2, 1) = C*F; t (2, 2) = C*E; t (2, 3) = z;
617 t (3, 0) = 0; t (3, 1) = 0; t (3, 2) = 0; t (3, 3) = 1;
618}
619
620
621template <typename Derived> void
622saveBinary (const Eigen::MatrixBase<Derived>& matrix, std::ostream& file)
623{
624 std::uint32_t rows = static_cast<std::uint32_t> (matrix.rows ()), cols = static_cast<std::uint32_t> (matrix.cols ());
625 file.write (reinterpret_cast<char*> (&rows), sizeof (rows));
626 file.write (reinterpret_cast<char*> (&cols), sizeof (cols));
627 for (std::uint32_t i = 0; i < rows; ++i)
628 for (std::uint32_t j = 0; j < cols; ++j)
629 {
630 typename Derived::Scalar tmp = matrix(i,j);
631 file.write (reinterpret_cast<const char*> (&tmp), sizeof (tmp));
632 }
633}
634
635
636template <typename Derived> void
637loadBinary (Eigen::MatrixBase<Derived> const & matrix_, std::istream& file)
638{
639 Eigen::MatrixBase<Derived> &matrix = const_cast<Eigen::MatrixBase<Derived> &> (matrix_);
640
641 std::uint32_t rows, cols;
642 file.read (reinterpret_cast<char*> (&rows), sizeof (rows));
643 file.read (reinterpret_cast<char*> (&cols), sizeof (cols));
644 if (matrix.rows () != static_cast<int>(rows) || matrix.cols () != static_cast<int>(cols))
645 matrix.derived().resize(rows, cols);
646
647 for (std::uint32_t i = 0; i < rows; ++i)
648 for (std::uint32_t j = 0; j < cols; ++j)
649 {
650 typename Derived::Scalar tmp;
651 file.read (reinterpret_cast<char*> (&tmp), sizeof (tmp));
652 matrix (i, j) = tmp;
653 }
654}
655
656
657template <typename Derived, typename OtherDerived>
658typename Eigen::internal::umeyama_transform_matrix_type<Derived, OtherDerived>::type
659umeyama (const Eigen::MatrixBase<Derived>& src, const Eigen::MatrixBase<OtherDerived>& dst, bool with_scaling)
660{
661#if EIGEN_VERSION_AT_LEAST (3, 3, 0)
662 return Eigen::umeyama (src, dst, with_scaling);
663#else
664 using TransformationMatrixType = typename Eigen::internal::umeyama_transform_matrix_type<Derived, OtherDerived>::type;
665 using Scalar = typename Eigen::internal::traits<TransformationMatrixType>::Scalar;
666 using RealScalar = typename Eigen::NumTraits<Scalar>::Real;
667 using Index = typename Derived::Index;
668
669 static_assert (!Eigen::NumTraits<Scalar>::IsComplex, "Numeric type must be real.");
670 static_assert ((Eigen::internal::is_same<Scalar, typename Eigen::internal::traits<OtherDerived>::Scalar>::value),
671 "You mixed different numeric types. You need to use the cast method of matrixbase to cast numeric types explicitly.");
672
673 enum { Dimension = PCL_EIGEN_SIZE_MIN_PREFER_DYNAMIC (Derived::RowsAtCompileTime, OtherDerived::RowsAtCompileTime) };
674
675 using VectorType = Eigen::Matrix<Scalar, Dimension, 1>;
676 using MatrixType = Eigen::Matrix<Scalar, Dimension, Dimension>;
677 using RowMajorMatrixType = typename Eigen::internal::plain_matrix_type_row_major<Derived>::type;
678
679 const Index m = src.rows (); // dimension
680 const Index n = src.cols (); // number of measurements
681
682 // required for demeaning ...
683 const RealScalar one_over_n = 1 / static_cast<RealScalar> (n);
684
685 // computation of mean
686 const VectorType src_mean = src.rowwise ().sum () * one_over_n;
687 const VectorType dst_mean = dst.rowwise ().sum () * one_over_n;
688
689 // demeaning of src and dst points
690 const RowMajorMatrixType src_demean = src.colwise () - src_mean;
691 const RowMajorMatrixType dst_demean = dst.colwise () - dst_mean;
692
693 // Eq. (36)-(37)
694 const Scalar src_var = src_demean.rowwise ().squaredNorm ().sum () * one_over_n;
695
696 // Eq. (38)
697 const MatrixType sigma (one_over_n * dst_demean * src_demean.transpose ());
698
699 Eigen::JacobiSVD<MatrixType> svd (sigma, Eigen::ComputeFullU | Eigen::ComputeFullV);
700
701 // Initialize the resulting transformation with an identity matrix...
702 TransformationMatrixType Rt = TransformationMatrixType::Identity (m + 1, m + 1);
703
704 // Eq. (39)
705 VectorType S = VectorType::Ones (m);
706
707 if ( svd.matrixU ().determinant () * svd.matrixV ().determinant () < 0 )
708 S (m - 1) = -1;
709
710 // Eq. (40) and (43)
711 Rt.block (0,0,m,m).noalias () = svd.matrixU () * S.asDiagonal () * svd.matrixV ().transpose ();
712
713 if (with_scaling)
714 {
715 // Eq. (42)
716 const Scalar c = Scalar (1)/ src_var * svd.singularValues ().dot (S);
717
718 // Eq. (41)
719 Rt.col (m).head (m) = dst_mean;
720 Rt.col (m).head (m).noalias () -= c * Rt.topLeftCorner (m, m) * src_mean;
721 Rt.block (0, 0, m, m) *= c;
722 }
723 else
724 {
725 Rt.col (m).head (m) = dst_mean;
726 Rt.col (m).head (m).noalias () -= Rt.topLeftCorner (m, m) * src_mean;
727 }
728
729 return (Rt);
730#endif
731}
732
733
734template <typename Scalar> bool
735transformLine (const Eigen::Matrix<Scalar, Eigen::Dynamic, 1> &line_in,
736 Eigen::Matrix<Scalar, Eigen::Dynamic, 1> &line_out,
737 const Eigen::Transform<Scalar, 3, Eigen::Affine> &transformation)
738{
739 if (line_in.innerSize () != 6 || line_out.innerSize () != 6)
740 {
741 PCL_DEBUG ("transformLine: lines size != 6\n");
742 return (false);
743 }
744
745 Eigen::Matrix<Scalar, 3, 1> point, vector;
746 point << line_in.template head<3> ();
747 vector << line_out.template tail<3> ();
748
749 pcl::transformPoint (point, point, transformation);
750 pcl::transformVector (vector, vector, transformation);
751 line_out << point, vector;
752 return (true);
753}
754
755
756template <typename Scalar> void
757transformPlane (const Eigen::Matrix<Scalar, 4, 1> &plane_in,
758 Eigen::Matrix<Scalar, 4, 1> &plane_out,
759 const Eigen::Transform<Scalar, 3, Eigen::Affine> &transformation)
760{
761 Eigen::Hyperplane < Scalar, 3 > plane;
762 plane.coeffs () << plane_in;
763 plane.transform (transformation);
764 plane_out << plane.coeffs ();
765
766 // Versions prior to 3.3.2 don't normalize the result
767 #if !EIGEN_VERSION_AT_LEAST (3, 3, 2)
768 plane_out /= plane_out.template head<3> ().norm ();
769 #endif
770}
771
772
773template <typename Scalar> void
776 const Eigen::Transform<Scalar, 3, Eigen::Affine> &transformation)
777{
778 std::vector<Scalar> values (plane_in->values.begin (), plane_in->values.end ());
779 Eigen::Matrix < Scalar, 4, 1 > v_plane_in (values.data ());
780 pcl::transformPlane (v_plane_in, v_plane_in, transformation);
781 plane_out->values.resize (4);
782 std::copy_n(v_plane_in.data (), 4, plane_out->values.begin ());
783}
784
785
786template <typename Scalar> bool
787checkCoordinateSystem (const Eigen::Matrix<Scalar, Eigen::Dynamic, 1> &line_x,
788 const Eigen::Matrix<Scalar, Eigen::Dynamic, 1> &line_y,
789 const Scalar norm_limit,
790 const Scalar dot_limit)
791{
792 if (line_x.innerSize () != 6 || line_y.innerSize () != 6)
793 {
794 PCL_DEBUG ("checkCoordinateSystem: lines size != 6\n");
795 return (false);
796 }
797
798 if (line_x.template head<3> () != line_y.template head<3> ())
799 {
800 PCL_DEBUG ("checkCoorZdinateSystem: vector origins are different !\n");
801 return (false);
802 }
803
804 // Make a copy of vector directions
805 // X^Y = Z | Y^Z = X | Z^X = Y
806 Eigen::Matrix<Scalar, 3, 1> v_line_x (line_x.template tail<3> ()),
807 v_line_y (line_y.template tail<3> ()),
808 v_line_z (v_line_x.cross (v_line_y));
809
810 // Check vectors norms
811 if (v_line_x.norm () < 1 - norm_limit || v_line_x.norm () > 1 + norm_limit)
812 {
813 PCL_DEBUG ("checkCoordinateSystem: line_x norm %d != 1\n", v_line_x.norm ());
814 return (false);
815 }
816
817 if (v_line_y.norm () < 1 - norm_limit || v_line_y.norm () > 1 + norm_limit)
818 {
819 PCL_DEBUG ("checkCoordinateSystem: line_y norm %d != 1\n", v_line_y.norm ());
820 return (false);
821 }
822
823 if (v_line_z.norm () < 1 - norm_limit || v_line_z.norm () > 1 + norm_limit)
824 {
825 PCL_DEBUG ("checkCoordinateSystem: line_z norm %d != 1\n", v_line_z.norm ());
826 return (false);
827 }
828
829 // Check vectors perendicularity
830 if (std::abs (v_line_x.dot (v_line_y)) > dot_limit)
831 {
832 PCL_DEBUG ("checkCSAxis: line_x dot line_y %e = > %e\n", v_line_x.dot (v_line_y), dot_limit);
833 return (false);
834 }
835
836 if (std::abs (v_line_x.dot (v_line_z)) > dot_limit)
837 {
838 PCL_DEBUG ("checkCSAxis: line_x dot line_z = %e > %e\n", v_line_x.dot (v_line_z), dot_limit);
839 return (false);
840 }
841
842 if (std::abs (v_line_y.dot (v_line_z)) > dot_limit)
843 {
844 PCL_DEBUG ("checkCSAxis: line_y dot line_z = %e > %e\n", v_line_y.dot (v_line_z), dot_limit);
845 return (false);
846 }
847
848 return (true);
849}
850
851
852template <typename Scalar> bool
853transformBetween2CoordinateSystems (const Eigen::Matrix<Scalar, Eigen::Dynamic, 1> from_line_x,
854 const Eigen::Matrix<Scalar, Eigen::Dynamic, 1> from_line_y,
855 const Eigen::Matrix<Scalar, Eigen::Dynamic, 1> to_line_x,
856 const Eigen::Matrix<Scalar, Eigen::Dynamic, 1> to_line_y,
857 Eigen::Transform<Scalar, 3, Eigen::Affine> &transformation)
858{
859 if (from_line_x.innerSize () != 6 || from_line_y.innerSize () != 6 || to_line_x.innerSize () != 6 || to_line_y.innerSize () != 6)
860 {
861 PCL_DEBUG ("transformBetween2CoordinateSystems: lines size != 6\n");
862 return (false);
863 }
864
865 // Check if coordinate systems are valid
866 if (!pcl::checkCoordinateSystem (from_line_x, from_line_y) || !pcl::checkCoordinateSystem (to_line_x, to_line_y))
867 {
868 PCL_DEBUG ("transformBetween2CoordinateSystems: coordinate systems invalid !\n");
869 return (false);
870 }
871
872 // Convert lines into Vector3 :
873 Eigen::Matrix<Scalar, 3, 1> fr0 (from_line_x.template head<3>()),
874 fr1 (from_line_x.template head<3>() + from_line_x.template tail<3>()),
875 fr2 (from_line_y.template head<3>() + from_line_y.template tail<3>()),
876
877 to0 (to_line_x.template head<3>()),
878 to1 (to_line_x.template head<3>() + to_line_x.template tail<3>()),
879 to2 (to_line_y.template head<3>() + to_line_y.template tail<3>());
880
881 // Code is inspired from http://stackoverflow.com/a/15277421/1816078
882 // Define matrices and points :
883 Eigen::Transform<Scalar, 3, Eigen::Affine> T2, T3 = Eigen::Transform<Scalar, 3, Eigen::Affine>::Identity ();
884 Eigen::Matrix<Scalar, 3, 1> x1, y1, z1, x2, y2, z2;
885
886 // Axes of the coordinate system "fr"
887 x1 = (fr1 - fr0).normalized (); // the versor (unitary vector) of the (fr1-fr0) axis vector
888 y1 = (fr2 - fr0).normalized ();
889
890 // Axes of the coordinate system "to"
891 x2 = (to1 - to0).normalized ();
892 y2 = (to2 - to0).normalized ();
893
894 // Transform from CS1 to CS2
895 // Note: if fr0 == (0,0,0) --> CS1 == CS2 --> T2 = Identity
896 T2.linear () << x1, y1, x1.cross (y1);
897
898 // Transform from CS1 to CS3
899 T3.linear () << x2, y2, x2.cross (y2);
900
901 // Identity matrix = transform to CS2 to CS3
902 // Note: if CS1 == CS2 --> transformation = T3
903 transformation = Eigen::Transform<Scalar, 3, Eigen::Affine>::Identity ();
904 transformation.linear () = T3.linear () * T2.linear ().inverse ();
905 transformation.translation () = to0 - (transformation.linear () * fr0);
906 return (true);
907}
908
909} // namespace pcl
910
void computeCorrespondingEigenVector(const Matrix &mat, const typename Matrix::Scalar &eigenvalue, Vector &eigenvector)
determines the corresponding eigenvector to the given eigenvalue of the symmetric positive semi defin...
Definition eigen.hpp:226
void getTranslationAndEulerAngles(const Eigen::Transform< Scalar, 3, Eigen::Affine > &t, Scalar &x, Scalar &y, Scalar &z, Scalar &roll, Scalar &pitch, Scalar &yaw)
Extract x,y,z and the Euler angles (intrinsic rotations, ZYX-convention) from the given transformatio...
Definition eigen.hpp:593
void getTransformationFromTwoUnitVectorsAndOrigin(const Eigen::Vector3f &y_direction, const Eigen::Vector3f &z_axis, const Eigen::Vector3f &origin, Eigen::Affine3f &transformation)
Get the transformation that will translate origin to (0,0,0) and rotate z_axis into (0,...
Definition eigen.hpp:572
Matrix::Scalar determinant3x3Matrix(const Matrix &matrix)
Calculate the determinant of a 3x3 matrix.
Definition eigen.hpp:491
Matrix::Scalar invert3x3SymMatrix(const Matrix &matrix, Matrix &inverse)
Calculate the inverse of a 3x3 symmetric matrix.
Definition eigen.hpp:423
void loadBinary(Eigen::MatrixBase< Derived > const &matrix, std::istream &file)
Read a matrix from an input stream.
Definition eigen.hpp:637
void getTransformation(Scalar x, Scalar y, Scalar z, Scalar roll, Scalar pitch, Scalar yaw, Eigen::Transform< Scalar, 3, Eigen::Affine > &t)
Create a transformation from the given translation and Euler angles (intrinsic rotations,...
Definition eigen.hpp:607
void getEulerAngles(const Eigen::Transform< Scalar, 3, Eigen::Affine > &t, Scalar &roll, Scalar &pitch, Scalar &yaw)
Extract the Euler angles (intrinsic rotations, ZYX-convention) from the given transformation.
Definition eigen.hpp:584
void eigen22(const Matrix &mat, typename Matrix::Scalar &eigenvalue, Vector &eigenvector)
determine the smallest eigenvalue and its corresponding eigenvector
Definition eigen.hpp:133
void getTransformationFromTwoUnitVectors(const Eigen::Vector3f &y_direction, const Eigen::Vector3f &z_axis, Eigen::Affine3f &transformation)
Get the unique 3D rotation that will rotate z_axis into (0,0,1) and y_direction into a vector with x=...
Definition eigen.hpp:553
void getTransFromUnitVectorsXY(const Eigen::Vector3f &x_axis, const Eigen::Vector3f &y_direction, Eigen::Affine3f &transformation)
Get the unique 3D rotation that will rotate x_axis into (1,0,0) and y_direction into a vector with z=...
Definition eigen.hpp:527
Matrix::Scalar invert3x3Matrix(const Matrix &matrix, Matrix &inverse)
Calculate the inverse of a general 3x3 matrix.
Definition eigen.hpp:458
void eigen33(const Matrix &mat, typename Matrix::Scalar &eigenvalue, Vector &eigenvector)
determines the eigenvector and eigenvalue of the smallest eigenvalue of the symmetric positive semi d...
Definition eigen.hpp:295
void saveBinary(const Eigen::MatrixBase< Derived > &matrix, std::ostream &file)
Write a matrix to an output stream.
Definition eigen.hpp:622
Matrix::Scalar invert2x2(const Matrix &matrix, Matrix &inverse)
Calculate the inverse of a 2x2 matrix.
Definition eigen.hpp:404
void getTransFromUnitVectorsZY(const Eigen::Vector3f &z_axis, const Eigen::Vector3f &y_direction, Eigen::Affine3f &transformation)
Get the unique 3D rotation that will rotate z_axis into (0,0,1) and y_direction into a vector with x=...
Definition eigen.hpp:501
@ B
Definition norms.h:54
static EigenVector< Vector, typename Matrix::Scalar > getLargest3x3Eigenvector(const Matrix scaledMatrix)
returns the unit vector along the largest eigen value as well as the length of the largest eigenvecto...
Definition eigen.hpp:273
bool checkCoordinateSystem(const Eigen::Matrix< Scalar, Eigen::Dynamic, 1 > &line_x, const Eigen::Matrix< Scalar, Eigen::Dynamic, 1 > &line_y, const Scalar norm_limit=1e-3, const Scalar dot_limit=1e-3)
Check coordinate system integrity.
Definition eigen.hpp:787
void transformPlane(const Eigen::Matrix< Scalar, 4, 1 > &plane_in, Eigen::Matrix< Scalar, 4, 1 > &plane_out, const Eigen::Transform< Scalar, 3, Eigen::Affine > &transformation)
Transform plane vectors using an affine matrix.
Definition eigen.hpp:757
void transformPoint(const Eigen::Matrix< Scalar, 3, 1 > &point_in, Eigen::Matrix< Scalar, 3, 1 > &point_out, const Eigen::Transform< Scalar, 3, Eigen::Affine > &transformation)
Transform a point using an affine matrix.
Definition eigen.h:389
void transformVector(const Eigen::Matrix< Scalar, 3, 1 > &vector_in, Eigen::Matrix< Scalar, 3, 1 > &vector_out, const Eigen::Transform< Scalar, 3, Eigen::Affine > &transformation)
Transform a vector using an affine matrix.
Definition eigen.h:406
Eigen::internal::umeyama_transform_matrix_type< Derived, OtherDerived >::type umeyama(const Eigen::MatrixBase< Derived > &src, const Eigen::MatrixBase< OtherDerived > &dst, bool with_scaling=false)
Returns the transformation between two point sets.
Definition eigen.hpp:659
bool transformBetween2CoordinateSystems(const Eigen::Matrix< Scalar, Eigen::Dynamic, 1 > from_line_x, const Eigen::Matrix< Scalar, Eigen::Dynamic, 1 > from_line_y, const Eigen::Matrix< Scalar, Eigen::Dynamic, 1 > to_line_x, const Eigen::Matrix< Scalar, Eigen::Dynamic, 1 > to_line_y, Eigen::Transform< Scalar, 3, Eigen::Affine > &transformation)
Compute the transformation between two coordinate systems.
Definition eigen.hpp:853
bool transformLine(const Eigen::Matrix< Scalar, Eigen::Dynamic, 1 > &line_in, Eigen::Matrix< Scalar, Eigen::Dynamic, 1 > &line_out, const Eigen::Transform< Scalar, 3, Eigen::Affine > &transformation)
Transform a line using an affine matrix.
Definition eigen.hpp:735
void computeRoots2(const Scalar &b, const Scalar &c, Roots &roots)
Compute the roots of a quadratic polynom x^2 + b*x + c = 0.
Definition eigen.hpp:53
void computeRoots(const Matrix &m, Roots &roots)
computes the roots of the characteristic polynomial of the input matrix m, which are the eigenvalues
Definition eigen.hpp:68
shared_ptr< ::pcl::ModelCoefficients > Ptr
shared_ptr< const ::pcl::ModelCoefficients > ConstPtr