point_pluecker_line_to_hom_mat3dT_point_pluecker_line_to_hom_mat3dPointPlueckerLineToHomMat3dPointPlueckerLineToHomMat3dpoint_pluecker_line_to_hom_mat3d (Operator)

Name

point_pluecker_line_to_hom_mat3dT_point_pluecker_line_to_hom_mat3dPointPlueckerLineToHomMat3dPointPlueckerLineToHomMat3dpoint_pluecker_line_to_hom_mat3d — Approximate a 3D affine transformation from 3D point-to-line correspondences.

Signature

Description

point_pluecker_line_to_hom_mat3dpoint_pluecker_line_to_hom_mat3dPointPlueckerLineToHomMat3dPointPlueckerLineToHomMat3dpoint_pluecker_line_to_hom_mat3d approximates a 3D affine transformation from 3D point-to-line correspondences and returns it as the homogeneous transformation matrix HomMat3DHomMat3DHomMat3DhomMat3Dhom_mat_3d. The input tuples must be of same length and determine at least 3 point-to-line correspondences. The corresponding points and lines must be at the same index positions in the tuples. The points are given by (PointXPointXPointXpointXpoint_x, PointYPointYPointYpointYpoint_y, PointZPointZPointZpointZpoint_z). The lines are given by their Plücker coordinates (LineDirectionXLineDirectionXLineDirectionXlineDirectionXline_direction_x, LineDirectionYLineDirectionYLineDirectionYlineDirectionYline_direction_y, LineDirectionZLineDirectionZLineDirectionZlineDirectionZline_direction_z) and (LineMomentXLineMomentXLineMomentXlineMomentXline_moment_x, LineMomentYLineMomentYLineMomentYlineMomentYline_moment_y, LineMomentZLineMomentZLineMomentZlineMomentZline_moment_z) For the definition of Plücker coordinates, see “Solution Guide III-C - 3D Vision”.

The type of the 3D transformation to compute is specified with TransformationTypeTransformationTypeTransformationTypetransformationTypetransformation_type. For TransformationTypeTransformationTypeTransformationTypetransformationTypetransformation_type = 'rigid'"rigid""rigid""rigid""rigid", a rigid 3D transformation (a rotation and a translation) is computed. This is the only transformation type that is supported at the moment.

Note that for the minimum number of 3 correspondences, up to 8 valid solutions are possible. point_pluecker_line_to_hom_mat3dpoint_pluecker_line_to_hom_mat3dPointPlueckerLineToHomMat3dPointPlueckerLineToHomMat3dpoint_pluecker_line_to_hom_mat3d returns an arbitrarily chosen valid solution in this case. If a unique solution is desired, at least 4 correspondences should be specified. Furthermore, note that no well-defined solution exists if all lines are parallel. In this case, point_pluecker_line_to_hom_mat3dpoint_pluecker_line_to_hom_mat3dPointPlueckerLineToHomMat3dPointPlueckerLineToHomMat3dpoint_pluecker_line_to_hom_mat3d returns an error.

The returned transformation minimizes the sum of the squared orthogonal distances of the points transformed with HomMat3DHomMat3DHomMat3DhomMat3Dhom_mat_3d from their corresponding lines (see distance_point_pluecker_linedistance_point_pluecker_lineDistancePointPlueckerLineDistancePointPlueckerLinedistance_point_pluecker_line).

HomMat3DHomMat3DHomMat3DhomMat3Dhom_mat_3d can be used directly with operators that transform 3D data using affine transformations, e.g., affine_trans_point_3daffine_trans_point_3dAffineTransPoint3dAffineTransPoint3daffine_trans_point_3d. To transform lines given in Plücker coordinates, the operator dual_quat_trans_line_3ddual_quat_trans_line_3dDualQuatTransLine3dDualQuatTransLine3ddual_quat_trans_line_3d can be used. Here, HomMat3DHomMat3DHomMat3DhomMat3Dhom_mat_3d must be converted into a pose using hom_mat3d_to_posehom_mat3d_to_poseHomMat3dToPoseHomMat3dToPosehom_mat3d_to_pose, the pose must be inverted using pose_invertpose_invertPoseInvertPoseInvertpose_invert and then converted into a dual quaternion using pose_to_dual_quatpose_to_dual_quatPoseToDualQuatPoseToDualQuatpose_to_dual_quat.

Execution Information

• Multithreading type: reentrant (runs in parallel with non-exclusive operators).
• Multithreading scope: global (may be called from any thread).
• Processed without parallelization.

Parameters

Example (HDevelop)

* In this example, we assume that correspondences between points
* (PX, PY, PZ) and Plücker lines (LX, LY, LZ) - (MX, MY, MZ) have
* already been computed.
point_pluecker_line_to_hom_mat3d ('rigid', PX, PY, PZ, LX, LY, LZ, \
                                  MX, MY, MZ, HomMat3D)
hom_mat3d_to_pose (HomMat3D, Pose)
* Compute the residual errors in 3D space, i.e., the distance of the
* points (PX, PY, PZ) transformed by HomMat3D from the lines.
affine_trans_point_3d (HomMat3D, PX, PY, PZ, TPX, TPY, TPZ)
distance_point_pluecker_line (TPX, TPY, TPZ, LX, LY, LZ, \
                              MX, MY, MZ, D)
* The same distance can also be computed by transforming the lines
* instead of the points.  Here, the inverse pose must be used.
pose_invert (Pose, PoseInv)
pose_to_dual_quat (PoseInv, DualQuat)
dual_quat_trans_line_3d (DualQuat, 'moment', LX, LY, LZ, MX, MY, MZ, \
                         TLX, TLY, TLZ, TMX, TMY, TMZ)
distance_point_pluecker_line (PX, PY, PZ, TLX, TLY, TLZ, \
                              TMX, TMY, TMZ, D)

Possible Predecessors

get_line_of_sightget_line_of_sightGetLineOfSightGetLineOfSightget_line_of_sight, points_to_pluecker_linepoints_to_pluecker_linePointsToPlueckerLinePointsToPlueckerLinepoints_to_pluecker_line, point_direction_to_pluecker_linepoint_direction_to_pluecker_linePointDirectionToPlueckerLinePointDirectionToPlueckerLinepoint_direction_to_pluecker_line

Possible Successors

hom_mat3d_to_posehom_mat3d_to_poseHomMat3dToPoseHomMat3dToPosehom_mat3d_to_pose, affine_trans_point_3daffine_trans_point_3dAffineTransPoint3dAffineTransPoint3daffine_trans_point_3d, pose_invertpose_invertPoseInvertPoseInvertpose_invert, pose_to_dual_quatpose_to_dual_quatPoseToDualQuatPoseToDualQuatpose_to_dual_quat, dual_quat_trans_line_3ddual_quat_trans_line_3dDualQuatTransLine3dDualQuatTransLine3ddual_quat_trans_line_3d, distance_point_pluecker_linedistance_point_pluecker_lineDistancePointPlueckerLineDistancePointPlueckerLinedistance_point_pluecker_line

See also

vector_to_hom_mat3dvector_to_hom_mat3dVectorToHomMat3dVectorToHomMat3dvector_to_hom_mat3d

Module

Foundation