hom_mat3d_project — Project an affine 3D transformation matrix to a 2D projective
transformation matrix.
hom_mat3d_project( : : HomMat3D, PrincipalPointRow, PrincipalPointCol, Focus : HomMat2D)
hom_mat3d_project 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_mat2d,
hom_vector_to_proj_hom_mat2d, and
proj_match_points_ransac).
Matrices for rotation, scale, and translation can be constructed
using the operators hom_mat3d_identity,
hom_mat3d_scale, hom_mat3d_rotate,
hom_mat3d_translate and pose_to_hom_mat3d.
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_image,
the first row of HomMat2D contains the transformation of
the row axis and the second row contains the transformation of the
column axis of the image.
The point (PrincipalPointRow, PrincipalPointCol)
is the principal point of the projection and the point
(PrincipalPointRow, PrincipalPointCol, 0)
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 Focus pixels
away form the camera. As a result, using the identity matrix as the
input matrix HomMat3D leads to a matrix HomMat2D
which also represents the identity in 2D.
Consequently, the parameter Focus 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 ,
(r,c)=(PrincipalPointRow,PrincipalPointCol)
and f=Focus.
Then the projective transformation matrix is calculated as follows: First, a 3x4 projection matrix is calculated as:
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:
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_image:
The overall transformation can be written as:
HomMat3D (input_control) hom_mat3d → (real)
3×4 3D transformation matrix.
PrincipalPointRow (input_control) point.y → (real / integer)
Row coordinate of the principal point.
Default value: 256
Suggested values: 16, 32, 64, 128, 240, 256, 512
PrincipalPointCol (input_control) point.x → (real / integer)
Column coordinate of the principal point.
Default value: 256
Suggested values: 16, 32, 64, 128, 256, 320, 512
Focus (input_control) number → (real / integer)
Focal length in pixels.
Default value: 256
Suggested values: 1, 2, 5, 256, 32768
HomMat2D (output_control) hom_mat2d → (real)
Homogeneous projective transformation matrix.
hom_mat3d_identity,
hom_mat3d_rotate,
hom_mat3d_translate,
hom_mat3d_scale
projective_trans_image,
projective_trans_point_2d,
projective_trans_region,
projective_trans_contour_xld,
hom_mat2d_invert
Foundation