hom_mat3d_projectHomMat3dProjectHomMat3dProjecthom_mat3d_projectT_hom_mat3d_project🔗

Short description🔗

hom_mat3d_projectHomMat3dProjectHomMat3dProjecthom_mat3d_projectT_hom_mat3d_project — Project an affine 3D transformation matrix to a 2D projective transformation matrix.

Signature🔗

hom_mat3d_project( hom_mat3d HomMat3D, point.y PrincipalPointRow, point.x PrincipalPointCol, number Focus, out hom_mat2d HomMat2D )

Description🔗

hom_mat3d_projectHomMat3dProject calculates a homogeneous projection matrix from a homogeneous 3x4 transformation matrix describing an affine transformation in 3D. The result can be used to project a plane, in particular a plane containing an image. The projection matrix defines a projective transformation between two (two-dimensional) planes.

This can be used to create perspective distortions, which occur in a projection of a plane rotated around an axis other than the z axis. Usually, however, projective transformations are determined from point correspondences (see vector_to_proj_hom_mat2dVectorToProjHomMat2d, hom_vector_to_proj_hom_mat2dHomVectorToProjHomMat2d, and proj_match_points_ransacProjMatchPointsRansac).

Matrices for rotation, scale, and translation can be constructed using the operators hom_mat3d_identityHomMat3dIdentity, hom_mat3d_scaleHomMat3dScale, hom_mat3d_rotateHomMat3dRotate, hom_mat3d_translateHomMat3dTranslate and pose_to_hom_mat3dPoseToHomMat3d.

Note that for 3D transformations the x-axis represents the column axis while the y-axis represents the row axis (see also Calibration) while in projective_trans_imageProjectiveTransImage, the first row of HomMat2DhomMat2Dhom_mat_2d contains the transformation of the row axis and the second row contains the transformation of the column axis of the image.

The point (PrincipalPointRowprincipalPointRowprincipal_point_row, PrincipalPointColprincipalPointColprincipal_point_col) is the principal point of the projection and the point (PrincipalPointRowprincipalPointRowprincipal_point_row, PrincipalPointColprincipalPointColprincipal_point_col, 00) can thus be interpreted as the position of the camera in a virtual three-dimensional space. The direction of view is along the positive z-axis.

In this virtual space the plane containing the input image as well as the image plane are located at z=Focus, which is Focusfocusfocus pixels away form the camera. As a result, using the identity matrix as the input matrix HomMat3DhomMat3Dhom_mat_3d leads to a matrix HomMat2DhomMat2Dhom_mat_2d which also represents the identity in 2D.

Consequently, the parameter Focusfocusfocus is the “focal distance” of the virtual camera used and its unit is pixels. Its value influences the degree of perspective distortions. The same input matrix at a bigger focal distance results in weaker distortions than at a low focal distance.

Let H be the affine 3D matrix with elements \(h_{ij}\), (r,c)=(PrincipalPointRow,PrincipalPointCol) and f=Focus.

Then the projective transformation matrix is calculated as follows: First, a 3x4 projection matrix is calculated as:

\[\begin{eqnarray*} Q = \left(\begin{array}{ccc} f & 0 & c \\ 0 & f & r \\ 0 & 0 & 1 \end{array} \right) \\ \cdot \\ \left(\begin{array}{cccc} 1 & 0 & 0 & -c \\ 0 & 1 & 0 & -r \\ 0 & 0 & 1 & 0 \end{array} \right) \\ \cdot \\ \left(\begin{array}{cccc} h_{11} & h_{12} & h_{13} & h_{14} \\ h_{21} & h_{22} & h_{23} & h_{24} \\ h_{31} & h_{32} & h_{33} & h_{34} \\ 0 & 0 & 0 & 1 \end{array} \right) \end{eqnarray*}\]

Since the image of a plane containing points \((x, y, f, 1)^T\) is to be calculated the last two columns of Q can be joined:

\[\begin{eqnarray*} R = \left(\begin{array}{ccc} r_{11} & r_{12} & r_{13} \\ r_{21} & r_{22} & r_{23} \\ r_{31} & r_{32} & r_{33} \end{array} \right) \\ = \\ \left(\begin{array}{ccc} q_{11} & q_{12} & f \cdot q_{13} + q_{14} \\ q_{21} & q_{22} & f \cdot r_{23} + q_{24} \\ q_{31} & q_{32} & f \cdot r_{33} + q_{34} \end{array} \right) \\ = Q \cdot \\ \left(\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & f \\ 0 & 0 & 1 \end{array} \right) \end{eqnarray*}\]

Finally, the columns and rows of R are swapped in a way that the first row of P contains the transformation of the row coordinates and the second row contains the transformation of the column coordinates so that P can be used directly in projective_trans_imageProjectiveTransImage:

\[\begin{eqnarray*} P = \left(\begin{array}{ccc} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{array} \right) \\ \cdot R \cdot \\ \left(\begin{array}{ccc} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{array} \right) \end{eqnarray*}\]

The overall transformation can be written as:

\[\begin{eqnarray*} P = \left(\begin{array}{ccc} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{array} \right) \\ \cdot \\ \left(\begin{array}{ccc} f & 0 & c \\ 0 & f & r \\ 0 & 0 & 1 \end{array} \right) \\ \cdot \\ \left(\begin{array}{cccc} 1 & 0 & 0 & -c \\ 0 & 1 & 0 & -r \\ 0 & 0 & 1 & 0 \end{array} \right) \\ \cdot H \cdot \\ \left(\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & f \\ 0 & 0 & 1 \end{array} \right) \cdot \\ \left(\begin{array}{ccc} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{array} \right) \end{eqnarray*}\]

Execution information🔗

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🔗

HomMat3DhomMat3Dhom_mat_3d (input_control) hom_mat3d → (real)HTuple (double)HHomMat3D, HTuple (double)Sequence[float]Htuple (double)

3×4 3D transformation matrix.

PrincipalPointRowprincipalPointRowprincipal_point_row (input_control) point.y → (real / integer)HTuple (double / Hlong)HTuple (double / int / long)Union[float, int]Htuple (double / Hlong)

Row coordinate of the principal point.

Default: 256256
Suggested values: 16, 32, 64, 128, 240, 256, 51216, 32, 64, 128, 240, 256, 512

PrincipalPointColprincipalPointColprincipal_point_col (input_control) point.x → (real / integer)HTuple (double / Hlong)HTuple (double / int / long)Union[float, int]Htuple (double / Hlong)

Column coordinate of the principal point.

Default: 256256
Suggested values: 16, 32, 64, 128, 256, 320, 51216, 32, 64, 128, 256, 320, 512

Focusfocusfocus (input_control) number → (real / integer)HTuple (double / Hlong)HTuple (double / int / long)Union[float, int]Htuple (double / Hlong)

Focal length in pixels.

Default: 256256
Suggested values: 1, 2, 5, 256, 327681, 2, 5, 256, 32768

HomMat2DhomMat2Dhom_mat_2d (output_control) hom_mat2d → (real)HTuple (double)HHomMat2D, HTuple (double)Sequence[float]Htuple (double)

Homogeneous projective transformation matrix.

Combinations with other operators🔗

Combinations

Possible predecessors

hom_mat3d_identityHomMat3dIdentity, hom_mat3d_rotateHomMat3dRotate, hom_mat3d_translateHomMat3dTranslate, hom_mat3d_scaleHomMat3dScale

Possible successors

projective_trans_imageProjectiveTransImage, projective_trans_point_2dProjectiveTransPoint2d, projective_trans_regionProjectiveTransRegion, projective_trans_contour_xldProjectiveTransContourXld, hom_mat2d_invertHomMat2dInvert

Module🔗

Foundation