Point Cloud Library (PCL)  1.11.0-dev
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 
49 namespace pcl
50 {
51 
52 template <typename Scalar, typename Roots> inline void
53 computeRoots2 (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 
67 template <typename Matrix, typename Roots> inline void
68 computeRoots (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  const 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 
132 template <typename Matrix, typename Vector> inline void
133 eigen22 (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 
171 template <typename Matrix, typename Vector> inline void
172 eigen22 (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 
225 template <typename Matrix, typename Vector> inline void
226 computeCorrespondingEigenVector (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 
256 namespace detail
257 {
258 
259 template <typename Vector, typename Scalar>
260 struct EigenVector {
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  */
272 template <typename Vector, typename Matrix> static EigenVector<Vector, typename Matrix::Scalar>
273 getLargest3x3Eigenvector (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 EigenVector<Vector, Scalar> {crossProduct.row (index) / length,
289  length};
290 }
291 
292 } // namespace detail
293 
294 
295 template <typename Matrix, typename Vector> inline void
296 eigen33 (const Matrix& mat, typename Matrix::Scalar& eigenvalue, Vector& eigenvector)
297 {
298  using Scalar = typename Matrix::Scalar;
299  // Scale the matrix so its entries are in [-1,1]. The scaling is applied
300  // only when at least one matrix entry has magnitude larger than 1.
301 
302  Scalar scale = mat.cwiseAbs ().maxCoeff ();
303  if (scale <= std::numeric_limits < Scalar > ::min ())
304  scale = Scalar (1.0);
305 
306  Matrix scaledMat = mat / scale;
307 
308  Vector eigenvalues;
309  computeRoots (scaledMat, eigenvalues);
310 
311  eigenvalue = eigenvalues (0) * scale;
312 
313  scaledMat.diagonal ().array () -= eigenvalues (0);
314 
315  eigenvector = detail::getLargest3x3Eigenvector<Vector> (scaledMat).vector;
316 }
317 
318 
319 template <typename Matrix, typename Vector> inline void
320 eigen33 (const Matrix& mat, Vector& evals)
321 {
322  using Scalar = typename Matrix::Scalar;
323  Scalar scale = mat.cwiseAbs ().maxCoeff ();
324  if (scale <= std::numeric_limits < Scalar > ::min ())
325  scale = Scalar (1.0);
326 
327  Matrix scaledMat = mat / scale;
328  computeRoots (scaledMat, evals);
329  evals *= scale;
330 }
331 
332 
333 template <typename Matrix, typename Vector> inline void
334 eigen33 (const Matrix& mat, Matrix& evecs, Vector& evals)
335 {
336  using Scalar = typename Matrix::Scalar;
337  // Scale the matrix so its entries are in [-1,1]. The scaling is applied
338  // only when at least one matrix entry has magnitude larger than 1.
339 
340  Scalar scale = mat.cwiseAbs ().maxCoeff ();
341  if (scale <= std::numeric_limits < Scalar > ::min ())
342  scale = Scalar (1.0);
343 
344  Matrix scaledMat = mat / scale;
345 
346  // Compute the eigenvalues
347  computeRoots (scaledMat, evals);
348 
349  if ( (evals (2) - evals (0)) <= Eigen::NumTraits < Scalar > ::epsilon ())
350  {
351  // all three equal
352  evecs.setIdentity ();
353  }
354  else if ( (evals (1) - evals (0)) <= Eigen::NumTraits < Scalar > ::epsilon ())
355  {
356  // first and second equal
357  Matrix tmp;
358  tmp = scaledMat;
359  tmp.diagonal ().array () -= evals (2);
360 
361  evecs.col (2) = detail::getLargest3x3Eigenvector<Vector> (tmp).vector;
362  evecs.col (1) = evecs.col (2).unitOrthogonal ();
363  evecs.col (0) = evecs.col (1).cross (evecs.col (2));
364  }
365  else if ( (evals (2) - evals (1)) <= Eigen::NumTraits < Scalar > ::epsilon ())
366  {
367  // second and third equal
368  Matrix tmp;
369  tmp = scaledMat;
370  tmp.diagonal ().array () -= evals (0);
371 
372  evecs.col (0) = detail::getLargest3x3Eigenvector<Vector> (tmp).vector;
373  evecs.col (1) = evecs.col (0).unitOrthogonal ();
374  evecs.col (2) = evecs.col (0).cross (evecs.col (1));
375  }
376  else
377  {
378  std::array<Scalar, 3> eigenVecLen;
379 
380  for (int i = 0; i < 3; ++i)
381  {
382  Matrix tmp = scaledMat;
383  tmp.diagonal ().array () -= evals (i);
384  const auto vec_len = detail::getLargest3x3Eigenvector<Vector> (tmp);
385  evecs.col (i) = vec_len.vector;
386  eigenVecLen[i] = vec_len.length;
387  }
388 
389  // @TODO: might be redundant or over-complicated as per @SergioRAgostinho
390  // see: https://github.com/PointCloudLibrary/pcl/pull/3441#discussion_r341024181
391  const auto minmax_it = std::minmax_element (eigenVecLen.cbegin (), eigenVecLen.cend ());
392  int min_idx = std::distance (eigenVecLen.cbegin (), minmax_it.first);
393  int max_idx = std::distance (eigenVecLen.cbegin (), minmax_it.second);
394  int mid_idx = 3 - min_idx - max_idx;
395 
396  evecs.col (min_idx) = evecs.col ( (min_idx + 1) % 3).cross (evecs.col ( (min_idx + 2) % 3)).normalized ();
397  evecs.col (mid_idx) = evecs.col ( (mid_idx + 1) % 3).cross (evecs.col ( (mid_idx + 2) % 3)).normalized ();
398  }
399  // Rescale back to the original size.
400  evals *= scale;
401 }
402 
403 
404 template <typename Matrix> inline typename Matrix::Scalar
405 invert2x2 (const Matrix& matrix, Matrix& inverse)
406 {
407  using Scalar = typename Matrix::Scalar;
408  Scalar det = matrix.coeff (0) * matrix.coeff (3) - matrix.coeff (1) * matrix.coeff (2);
409 
410  if (det != 0)
411  {
412  //Scalar inv_det = Scalar (1.0) / det;
413  inverse.coeffRef (0) = matrix.coeff (3);
414  inverse.coeffRef (1) = -matrix.coeff (1);
415  inverse.coeffRef (2) = -matrix.coeff (2);
416  inverse.coeffRef (3) = matrix.coeff (0);
417  inverse /= det;
418  }
419  return det;
420 }
421 
422 
423 template <typename Matrix> inline typename Matrix::Scalar
424 invert3x3SymMatrix (const Matrix& matrix, Matrix& inverse)
425 {
426  using Scalar = typename Matrix::Scalar;
427  // elements
428  // a b c
429  // b d e
430  // c e f
431  //| a b c |-1 | fd-ee ce-bf be-cd |
432  //| b d e | = 1/det * | ce-bf af-cc bc-ae |
433  //| c e f | | be-cd bc-ae ad-bb |
434 
435  //det = a(fd-ee) + b(ec-fb) + c(eb-dc)
436 
437  Scalar fd_ee = matrix.coeff (4) * matrix.coeff (8) - matrix.coeff (7) * matrix.coeff (5);
438  Scalar ce_bf = matrix.coeff (2) * matrix.coeff (5) - matrix.coeff (1) * matrix.coeff (8);
439  Scalar be_cd = matrix.coeff (1) * matrix.coeff (5) - matrix.coeff (2) * matrix.coeff (4);
440 
441  Scalar det = matrix.coeff (0) * fd_ee + matrix.coeff (1) * ce_bf + matrix.coeff (2) * be_cd;
442 
443  if (det != 0)
444  {
445  //Scalar inv_det = Scalar (1.0) / det;
446  inverse.coeffRef (0) = fd_ee;
447  inverse.coeffRef (1) = inverse.coeffRef (3) = ce_bf;
448  inverse.coeffRef (2) = inverse.coeffRef (6) = be_cd;
449  inverse.coeffRef (4) = (matrix.coeff (0) * matrix.coeff (8) - matrix.coeff (2) * matrix.coeff (2));
450  inverse.coeffRef (5) = inverse.coeffRef (7) = (matrix.coeff (1) * matrix.coeff (2) - matrix.coeff (0) * matrix.coeff (5));
451  inverse.coeffRef (8) = (matrix.coeff (0) * matrix.coeff (4) - matrix.coeff (1) * matrix.coeff (1));
452  inverse /= det;
453  }
454  return det;
455 }
456 
457 
458 template <typename Matrix> inline typename Matrix::Scalar
459 invert3x3Matrix (const Matrix& matrix, Matrix& inverse)
460 {
461  using Scalar = typename Matrix::Scalar;
462 
463  //| a b c |-1 | ie-hf hc-ib fb-ec |
464  //| d e f | = 1/det * | gf-id ia-gc dc-fa |
465  //| g h i | | hd-ge gb-ha ea-db |
466  //det = a(ie-hf) + d(hc-ib) + g(fb-ec)
467 
468  Scalar ie_hf = matrix.coeff (8) * matrix.coeff (4) - matrix.coeff (7) * matrix.coeff (5);
469  Scalar hc_ib = matrix.coeff (7) * matrix.coeff (2) - matrix.coeff (8) * matrix.coeff (1);
470  Scalar fb_ec = matrix.coeff (5) * matrix.coeff (1) - matrix.coeff (4) * matrix.coeff (2);
471  Scalar det = matrix.coeff (0) * (ie_hf) + matrix.coeff (3) * (hc_ib) + matrix.coeff (6) * (fb_ec);
472 
473  if (det != 0)
474  {
475  inverse.coeffRef (0) = ie_hf;
476  inverse.coeffRef (1) = hc_ib;
477  inverse.coeffRef (2) = fb_ec;
478  inverse.coeffRef (3) = matrix.coeff (6) * matrix.coeff (5) - matrix.coeff (8) * matrix.coeff (3);
479  inverse.coeffRef (4) = matrix.coeff (8) * matrix.coeff (0) - matrix.coeff (6) * matrix.coeff (2);
480  inverse.coeffRef (5) = matrix.coeff (3) * matrix.coeff (2) - matrix.coeff (5) * matrix.coeff (0);
481  inverse.coeffRef (6) = matrix.coeff (7) * matrix.coeff (3) - matrix.coeff (6) * matrix.coeff (4);
482  inverse.coeffRef (7) = matrix.coeff (6) * matrix.coeff (1) - matrix.coeff (7) * matrix.coeff (0);
483  inverse.coeffRef (8) = matrix.coeff (4) * matrix.coeff (0) - matrix.coeff (3) * matrix.coeff (1);
484 
485  inverse /= det;
486  }
487  return det;
488 }
489 
490 
491 template <typename Matrix> inline typename Matrix::Scalar
492 determinant3x3Matrix (const Matrix& matrix)
493 {
494  // result is independent of Row/Col Major storage!
495  return matrix.coeff (0) * (matrix.coeff (4) * matrix.coeff (8) - matrix.coeff (5) * matrix.coeff (7)) +
496  matrix.coeff (1) * (matrix.coeff (5) * matrix.coeff (6) - matrix.coeff (3) * matrix.coeff (8)) +
497  matrix.coeff (2) * (matrix.coeff (3) * matrix.coeff (7) - matrix.coeff (4) * matrix.coeff (6)) ;
498 }
499 
500 
501 void
502 getTransFromUnitVectorsZY (const Eigen::Vector3f& z_axis,
503  const Eigen::Vector3f& y_direction,
504  Eigen::Affine3f& transformation)
505 {
506  Eigen::Vector3f tmp0 = (y_direction.cross(z_axis)).normalized();
507  Eigen::Vector3f tmp1 = (z_axis.cross(tmp0)).normalized();
508  Eigen::Vector3f tmp2 = z_axis.normalized();
509 
510  transformation(0,0)=tmp0[0]; transformation(0,1)=tmp0[1]; transformation(0,2)=tmp0[2]; transformation(0,3)=0.0f;
511  transformation(1,0)=tmp1[0]; transformation(1,1)=tmp1[1]; transformation(1,2)=tmp1[2]; transformation(1,3)=0.0f;
512  transformation(2,0)=tmp2[0]; transformation(2,1)=tmp2[1]; transformation(2,2)=tmp2[2]; transformation(2,3)=0.0f;
513  transformation(3,0)=0.0f; transformation(3,1)=0.0f; transformation(3,2)=0.0f; transformation(3,3)=1.0f;
514 }
515 
516 
517 Eigen::Affine3f
518 getTransFromUnitVectorsZY (const Eigen::Vector3f& z_axis,
519  const Eigen::Vector3f& y_direction)
520 {
521  Eigen::Affine3f transformation;
522  getTransFromUnitVectorsZY (z_axis, y_direction, transformation);
523  return (transformation);
524 }
525 
526 
527 void
528 getTransFromUnitVectorsXY (const Eigen::Vector3f& x_axis,
529  const Eigen::Vector3f& y_direction,
530  Eigen::Affine3f& transformation)
531 {
532  Eigen::Vector3f tmp2 = (x_axis.cross(y_direction)).normalized();
533  Eigen::Vector3f tmp1 = (tmp2.cross(x_axis)).normalized();
534  Eigen::Vector3f tmp0 = x_axis.normalized();
535 
536  transformation(0,0)=tmp0[0]; transformation(0,1)=tmp0[1]; transformation(0,2)=tmp0[2]; transformation(0,3)=0.0f;
537  transformation(1,0)=tmp1[0]; transformation(1,1)=tmp1[1]; transformation(1,2)=tmp1[2]; transformation(1,3)=0.0f;
538  transformation(2,0)=tmp2[0]; transformation(2,1)=tmp2[1]; transformation(2,2)=tmp2[2]; transformation(2,3)=0.0f;
539  transformation(3,0)=0.0f; transformation(3,1)=0.0f; transformation(3,2)=0.0f; transformation(3,3)=1.0f;
540 }
541 
542 
543 Eigen::Affine3f
544 getTransFromUnitVectorsXY (const Eigen::Vector3f& x_axis,
545  const Eigen::Vector3f& y_direction)
546 {
547  Eigen::Affine3f transformation;
548  getTransFromUnitVectorsXY (x_axis, y_direction, transformation);
549  return (transformation);
550 }
551 
552 
553 void
554 getTransformationFromTwoUnitVectors (const Eigen::Vector3f& y_direction,
555  const Eigen::Vector3f& z_axis,
556  Eigen::Affine3f& transformation)
557 {
558  getTransFromUnitVectorsZY (z_axis, y_direction, transformation);
559 }
560 
561 
562 Eigen::Affine3f
563 getTransformationFromTwoUnitVectors (const Eigen::Vector3f& y_direction,
564  const Eigen::Vector3f& z_axis)
565 {
566  Eigen::Affine3f transformation;
567  getTransformationFromTwoUnitVectors (y_direction, z_axis, transformation);
568  return (transformation);
569 }
570 
571 
572 void
573 getTransformationFromTwoUnitVectorsAndOrigin (const Eigen::Vector3f& y_direction,
574  const Eigen::Vector3f& z_axis,
575  const Eigen::Vector3f& origin,
576  Eigen::Affine3f& transformation)
577 {
578  getTransformationFromTwoUnitVectors(y_direction, z_axis, transformation);
579  Eigen::Vector3f translation = transformation*origin;
580  transformation(0,3)=-translation[0]; transformation(1,3)=-translation[1]; transformation(2,3)=-translation[2];
581 }
582 
583 
584 template <typename Scalar> void
585 getEulerAngles (const Eigen::Transform<Scalar, 3, Eigen::Affine> &t, Scalar &roll, Scalar &pitch, Scalar &yaw)
586 {
587  roll = std::atan2 (t (2, 1), t (2, 2));
588  pitch = asin (-t (2, 0));
589  yaw = std::atan2 (t (1, 0), t (0, 0));
590 }
591 
592 
593 template <typename Scalar> void
594 getTranslationAndEulerAngles (const Eigen::Transform<Scalar, 3, Eigen::Affine> &t,
595  Scalar &x, Scalar &y, Scalar &z,
596  Scalar &roll, Scalar &pitch, Scalar &yaw)
597 {
598  x = t (0, 3);
599  y = t (1, 3);
600  z = t (2, 3);
601  roll = std::atan2 (t (2, 1), t (2, 2));
602  pitch = asin (-t (2, 0));
603  yaw = std::atan2 (t (1, 0), t (0, 0));
604 }
605 
606 
607 template <typename Scalar> void
608 getTransformation (Scalar x, Scalar y, Scalar z,
609  Scalar roll, Scalar pitch, Scalar yaw,
610  Eigen::Transform<Scalar, 3, Eigen::Affine> &t)
611 {
612  Scalar A = std::cos (yaw), B = sin (yaw), C = std::cos (pitch), D = sin (pitch),
613  E = std::cos (roll), F = sin (roll), DE = D*E, DF = D*F;
614 
615  t (0, 0) = A*C; t (0, 1) = A*DF - B*E; t (0, 2) = B*F + A*DE; t (0, 3) = x;
616  t (1, 0) = B*C; t (1, 1) = A*E + B*DF; t (1, 2) = B*DE - A*F; t (1, 3) = y;
617  t (2, 0) = -D; t (2, 1) = C*F; t (2, 2) = C*E; t (2, 3) = z;
618  t (3, 0) = 0; t (3, 1) = 0; t (3, 2) = 0; t (3, 3) = 1;
619 }
620 
621 
622 template <typename Derived> void
623 saveBinary (const Eigen::MatrixBase<Derived>& matrix, std::ostream& file)
624 {
625  std::uint32_t rows = static_cast<std::uint32_t> (matrix.rows ()), cols = static_cast<std::uint32_t> (matrix.cols ());
626  file.write (reinterpret_cast<char*> (&rows), sizeof (rows));
627  file.write (reinterpret_cast<char*> (&cols), sizeof (cols));
628  for (std::uint32_t i = 0; i < rows; ++i)
629  for (std::uint32_t j = 0; j < cols; ++j)
630  {
631  typename Derived::Scalar tmp = matrix(i,j);
632  file.write (reinterpret_cast<const char*> (&tmp), sizeof (tmp));
633  }
634 }
635 
636 
637 template <typename Derived> void
638 loadBinary (Eigen::MatrixBase<Derived> const & matrix_, std::istream& file)
639 {
640  Eigen::MatrixBase<Derived> &matrix = const_cast<Eigen::MatrixBase<Derived> &> (matrix_);
641 
642  std::uint32_t rows, cols;
643  file.read (reinterpret_cast<char*> (&rows), sizeof (rows));
644  file.read (reinterpret_cast<char*> (&cols), sizeof (cols));
645  if (matrix.rows () != static_cast<int>(rows) || matrix.cols () != static_cast<int>(cols))
646  matrix.derived().resize(rows, cols);
647 
648  for (std::uint32_t i = 0; i < rows; ++i)
649  for (std::uint32_t j = 0; j < cols; ++j)
650  {
651  typename Derived::Scalar tmp;
652  file.read (reinterpret_cast<char*> (&tmp), sizeof (tmp));
653  matrix (i, j) = tmp;
654  }
655 }
656 
657 
658 template <typename Derived, typename OtherDerived>
659 typename Eigen::internal::umeyama_transform_matrix_type<Derived, OtherDerived>::type
660 umeyama (const Eigen::MatrixBase<Derived>& src, const Eigen::MatrixBase<OtherDerived>& dst, bool with_scaling)
661 {
662 #if EIGEN_VERSION_AT_LEAST (3, 3, 0)
663  return Eigen::umeyama (src, dst, with_scaling);
664 #else
665  using TransformationMatrixType = typename Eigen::internal::umeyama_transform_matrix_type<Derived, OtherDerived>::type;
666  using Scalar = typename Eigen::internal::traits<TransformationMatrixType>::Scalar;
667  using RealScalar = typename Eigen::NumTraits<Scalar>::Real;
668  using Index = typename Derived::Index;
669 
670  static_assert (!Eigen::NumTraits<Scalar>::IsComplex, "Numeric type must be real.");
671  static_assert ((Eigen::internal::is_same<Scalar, typename Eigen::internal::traits<OtherDerived>::Scalar>::value),
672  "You mixed different numeric types. You need to use the cast method of matrixbase to cast numeric types explicitly.");
673 
674  enum { Dimension = PCL_EIGEN_SIZE_MIN_PREFER_DYNAMIC (Derived::RowsAtCompileTime, OtherDerived::RowsAtCompileTime) };
675 
676  using VectorType = Eigen::Matrix<Scalar, Dimension, 1>;
677  using MatrixType = Eigen::Matrix<Scalar, Dimension, Dimension>;
678  using RowMajorMatrixType = typename Eigen::internal::plain_matrix_type_row_major<Derived>::type;
679 
680  const Index m = src.rows (); // dimension
681  const Index n = src.cols (); // number of measurements
682 
683  // required for demeaning ...
684  const RealScalar one_over_n = 1 / static_cast<RealScalar> (n);
685 
686  // computation of mean
687  const VectorType src_mean = src.rowwise ().sum () * one_over_n;
688  const VectorType dst_mean = dst.rowwise ().sum () * one_over_n;
689 
690  // demeaning of src and dst points
691  const RowMajorMatrixType src_demean = src.colwise () - src_mean;
692  const RowMajorMatrixType dst_demean = dst.colwise () - dst_mean;
693 
694  // Eq. (36)-(37)
695  const Scalar src_var = src_demean.rowwise ().squaredNorm ().sum () * one_over_n;
696 
697  // Eq. (38)
698  const MatrixType sigma (one_over_n * dst_demean * src_demean.transpose ());
699 
700  Eigen::JacobiSVD<MatrixType> svd (sigma, Eigen::ComputeFullU | Eigen::ComputeFullV);
701 
702  // Initialize the resulting transformation with an identity matrix...
703  TransformationMatrixType Rt = TransformationMatrixType::Identity (m + 1, m + 1);
704 
705  // Eq. (39)
706  VectorType S = VectorType::Ones (m);
707 
708  if ( svd.matrixU ().determinant () * svd.matrixV ().determinant () < 0 )
709  S (m - 1) = -1;
710 
711  // Eq. (40) and (43)
712  Rt.block (0,0,m,m).noalias () = svd.matrixU () * S.asDiagonal () * svd.matrixV ().transpose ();
713 
714  if (with_scaling)
715  {
716  // Eq. (42)
717  const Scalar c = Scalar (1)/ src_var * svd.singularValues ().dot (S);
718 
719  // Eq. (41)
720  Rt.col (m).head (m) = dst_mean;
721  Rt.col (m).head (m).noalias () -= c * Rt.topLeftCorner (m, m) * src_mean;
722  Rt.block (0, 0, m, m) *= c;
723  }
724  else
725  {
726  Rt.col (m).head (m) = dst_mean;
727  Rt.col (m).head (m).noalias () -= Rt.topLeftCorner (m, m) * src_mean;
728  }
729 
730  return (Rt);
731 #endif
732 }
733 
734 
735 template <typename Scalar> bool
736 transformLine (const Eigen::Matrix<Scalar, Eigen::Dynamic, 1> &line_in,
737  Eigen::Matrix<Scalar, Eigen::Dynamic, 1> &line_out,
738  const Eigen::Transform<Scalar, 3, Eigen::Affine> &transformation)
739 {
740  if (line_in.innerSize () != 6 || line_out.innerSize () != 6)
741  {
742  PCL_DEBUG ("transformLine: lines size != 6\n");
743  return (false);
744  }
745 
746  Eigen::Matrix<Scalar, 3, 1> point, vector;
747  point << line_in.template head<3> ();
748  vector << line_out.template tail<3> ();
749 
750  pcl::transformPoint (point, point, transformation);
751  pcl::transformVector (vector, vector, transformation);
752  line_out << point, vector;
753  return (true);
754 }
755 
756 
757 template <typename Scalar> void
758 transformPlane (const Eigen::Matrix<Scalar, 4, 1> &plane_in,
759  Eigen::Matrix<Scalar, 4, 1> &plane_out,
760  const Eigen::Transform<Scalar, 3, Eigen::Affine> &transformation)
761 {
762  Eigen::Hyperplane < Scalar, 3 > plane;
763  plane.coeffs () << plane_in;
764  plane.transform (transformation);
765  plane_out << plane.coeffs ();
766 
767  // Versions prior to 3.3.2 don't normalize the result
768  #if !EIGEN_VERSION_AT_LEAST (3, 3, 2)
769  plane_out /= plane_out.template head<3> ().norm ();
770  #endif
771 }
772 
773 
774 template <typename Scalar> void
776  pcl::ModelCoefficients::Ptr plane_out,
777  const Eigen::Transform<Scalar, 3, Eigen::Affine> &transformation)
778 {
779  std::vector<Scalar> values (plane_in->values.begin (), plane_in->values.end ());
780  Eigen::Matrix < Scalar, 4, 1 > v_plane_in (values.data ());
781  pcl::transformPlane (v_plane_in, v_plane_in, transformation);
782  plane_out->values.resize (4);
783  std::copy_n(v_plane_in.data (), 4, plane_in->values.begin ());
784 }
785 
786 
787 template <typename Scalar> bool
788 checkCoordinateSystem (const Eigen::Matrix<Scalar, Eigen::Dynamic, 1> &line_x,
789  const Eigen::Matrix<Scalar, Eigen::Dynamic, 1> &line_y,
790  const Scalar norm_limit,
791  const Scalar dot_limit)
792 {
793  if (line_x.innerSize () != 6 || line_y.innerSize () != 6)
794  {
795  PCL_DEBUG ("checkCoordinateSystem: lines size != 6\n");
796  return (false);
797  }
798 
799  if (line_x.template head<3> () != line_y.template head<3> ())
800  {
801  PCL_DEBUG ("checkCoorZdinateSystem: vector origins are different !\n");
802  return (false);
803  }
804 
805  // Make a copy of vector directions
806  // X^Y = Z | Y^Z = X | Z^X = Y
807  Eigen::Matrix<Scalar, 3, 1> v_line_x (line_x.template tail<3> ()),
808  v_line_y (line_y.template tail<3> ()),
809  v_line_z (v_line_x.cross (v_line_y));
810 
811  // Check vectors norms
812  if (v_line_x.norm () < 1 - norm_limit || v_line_x.norm () > 1 + norm_limit)
813  {
814  PCL_DEBUG ("checkCoordinateSystem: line_x norm %d != 1\n", v_line_x.norm ());
815  return (false);
816  }
817 
818  if (v_line_y.norm () < 1 - norm_limit || v_line_y.norm () > 1 + norm_limit)
819  {
820  PCL_DEBUG ("checkCoordinateSystem: line_y norm %d != 1\n", v_line_y.norm ());
821  return (false);
822  }
823 
824  if (v_line_z.norm () < 1 - norm_limit || v_line_z.norm () > 1 + norm_limit)
825  {
826  PCL_DEBUG ("checkCoordinateSystem: line_z norm %d != 1\n", v_line_z.norm ());
827  return (false);
828  }
829 
830  // Check vectors perendicularity
831  if (std::abs (v_line_x.dot (v_line_y)) > dot_limit)
832  {
833  PCL_DEBUG ("checkCSAxis: line_x dot line_y %e = > %e\n", v_line_x.dot (v_line_y), dot_limit);
834  return (false);
835  }
836 
837  if (std::abs (v_line_x.dot (v_line_z)) > dot_limit)
838  {
839  PCL_DEBUG ("checkCSAxis: line_x dot line_z = %e > %e\n", v_line_x.dot (v_line_z), dot_limit);
840  return (false);
841  }
842 
843  if (std::abs (v_line_y.dot (v_line_z)) > dot_limit)
844  {
845  PCL_DEBUG ("checkCSAxis: line_y dot line_z = %e > %e\n", v_line_y.dot (v_line_z), dot_limit);
846  return (false);
847  }
848 
849  return (true);
850 }
851 
852 
853 template <typename Scalar> bool
854 transformBetween2CoordinateSystems (const Eigen::Matrix<Scalar, Eigen::Dynamic, 1> from_line_x,
855  const Eigen::Matrix<Scalar, Eigen::Dynamic, 1> from_line_y,
856  const Eigen::Matrix<Scalar, Eigen::Dynamic, 1> to_line_x,
857  const Eigen::Matrix<Scalar, Eigen::Dynamic, 1> to_line_y,
858  Eigen::Transform<Scalar, 3, Eigen::Affine> &transformation)
859 {
860  if (from_line_x.innerSize () != 6 || from_line_y.innerSize () != 6 || to_line_x.innerSize () != 6 || to_line_y.innerSize () != 6)
861  {
862  PCL_DEBUG ("transformBetween2CoordinateSystems: lines size != 6\n");
863  return (false);
864  }
865 
866  // Check if coordinate systems are valid
867  if (!pcl::checkCoordinateSystem (from_line_x, from_line_y) || !pcl::checkCoordinateSystem (to_line_x, to_line_y))
868  {
869  PCL_DEBUG ("transformBetween2CoordinateSystems: coordinate systems invalid !\n");
870  return (false);
871  }
872 
873  // Convert lines into Vector3 :
874  Eigen::Matrix<Scalar, 3, 1> fr0 (from_line_x.template head<3>()),
875  fr1 (from_line_x.template head<3>() + from_line_x.template tail<3>()),
876  fr2 (from_line_y.template head<3>() + from_line_y.template tail<3>()),
877 
878  to0 (to_line_x.template head<3>()),
879  to1 (to_line_x.template head<3>() + to_line_x.template tail<3>()),
880  to2 (to_line_y.template head<3>() + to_line_y.template tail<3>());
881 
882  // Code is inspired from http://stackoverflow.com/a/15277421/1816078
883  // Define matrices and points :
884  Eigen::Transform<Scalar, 3, Eigen::Affine> T2, T3 = Eigen::Transform<Scalar, 3, Eigen::Affine>::Identity ();
885  Eigen::Matrix<Scalar, 3, 1> x1, y1, z1, x2, y2, z2;
886 
887  // Axes of the coordinate system "fr"
888  x1 = (fr1 - fr0).normalized (); // the versor (unitary vector) of the (fr1-fr0) axis vector
889  y1 = (fr2 - fr0).normalized ();
890 
891  // Axes of the coordinate system "to"
892  x2 = (to1 - to0).normalized ();
893  y2 = (to2 - to0).normalized ();
894 
895  // Transform from CS1 to CS2
896  // Note: if fr0 == (0,0,0) --> CS1 == CS2 --> T2 = Identity
897  T2.linear () << x1, y1, x1.cross (y1);
898 
899  // Transform from CS1 to CS3
900  T3.linear () << x2, y2, x2.cross (y2);
901 
902  // Identity matrix = transform to CS2 to CS3
903  // Note: if CS1 == CS2 --> transformation = T3
904  transformation = Eigen::Transform<Scalar, 3, Eigen::Affine>::Identity ();
905  transformation.linear () = T3.linear () * T2.linear ().inverse ();
906  transformation.translation () = to0 - (transformation.linear () * fr0);
907  return (true);
908 }
909 
910 } // namespace pcl
911 
pcl::saveBinary
void saveBinary(const Eigen::MatrixBase< Derived > &matrix, std::ostream &file)
Write a matrix to an output stream.
Definition: eigen.hpp:623
pcl
Definition: convolution.h:46
pcl::uint32_t
std::uint32_t uint32_t
Definition: types.h:58
pcl::geometry::distance
float distance(const PointT &p1, const PointT &p2)
Definition: geometry.h:60
pcl::transformVector
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:456
pcl::getTransFromUnitVectorsXY
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:528
pcl::getTransformationFromTwoUnitVectors
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:554
pcl::getTransFromUnitVectorsZY
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:502
pcl::detail::EigenVector::vector
Vector vector
Definition: eigen.hpp:261
pcl::getTransformation
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 (XYZ-convention)
Definition: eigen.hpp:608
pcl::checkCoordinateSystem
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:788
pcl::umeyama
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:660
pcl::invert2x2
Matrix::Scalar invert2x2(const Matrix &matrix, Matrix &inverse)
Calculate the inverse of a 2x2 matrix.
Definition: eigen.hpp:405
pcl::eigen33
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:296
pcl::computeRoots
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
pcl::eigen22
void eigen22(const Matrix &mat, typename Matrix::Scalar &eigenvalue, Vector &eigenvector)
determine the smallest eigenvalue and its corresponding eigenvector
Definition: eigen.hpp:133
pcl::detail::getLargest3x3Eigenvector
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
pcl::invert3x3Matrix
Matrix::Scalar invert3x3Matrix(const Matrix &matrix, Matrix &inverse)
Calculate the inverse of a general 3x3 matrix.
Definition: eigen.hpp:459
pcl::getTranslationAndEulerAngles
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 (XYZ-convention) from the given transformation.
Definition: eigen.hpp:594
pcl::getTransformationFromTwoUnitVectorsAndOrigin
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:573
pcl::invert3x3SymMatrix
Matrix::Scalar invert3x3SymMatrix(const Matrix &matrix, Matrix &inverse)
Calculate the inverse of a 3x3 symmetric matrix.
Definition: eigen.hpp:424
pcl::getEulerAngles
void getEulerAngles(const Eigen::Transform< Scalar, 3, Eigen::Affine > &t, Scalar &roll, Scalar &pitch, Scalar &yaw)
Extract the Euler angles (XYZ-convention) from the given transformation.
Definition: eigen.hpp:585
pcl::transformBetween2CoordinateSystems
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:854
pcl::transformPlane
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:758
pcl::transformLine
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:736
pcl::ModelCoefficients::Ptr
shared_ptr< ::pcl::ModelCoefficients > Ptr
Definition: ModelCoefficients.h:23
pcl::B
@ B
Definition: norms.h:54
pcl::computeCorrespondingEigenVector
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
pcl::detail::EigenVector
Definition: eigen.hpp:260
pcl::loadBinary
void loadBinary(Eigen::MatrixBase< Derived > const &matrix, std::istream &file)
Read a matrix from an input stream.
Definition: eigen.hpp:638
pcl::detail::EigenVector::length
Scalar length
Definition: eigen.hpp:262
pcl::computeRoots2
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
pcl::determinant3x3Matrix
Matrix::Scalar determinant3x3Matrix(const Matrix &matrix)
Calculate the determinant of a 3x3 matrix.
Definition: eigen.hpp:492
pcl::transformPoint
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:423