vector_to_proj_hom_mat2d — Compute a projective transformation matrix using given point
correspondences.
vector_to_proj_hom_mat2d( : : Px, Py, Qx, Qy, Method, CovXX1, CovYY1, CovXY1, CovXX2, CovYY2, CovXY2 : HomMat2D, Covariance)
vector_to_proj_hom_mat2d determines the homogeneous
projective transformation matrix HomMat2D that optimally
fulfills the following equations given by at least 4 point
correspondences
If fewer than 4 pairs of points (Px,Py),
(Qx,Qy) are given, there exists no unique
solution, if exactly 4 pairs are supplied the matrix
HomMat2D transforms them in exactly the desired way, and if
there are more than 4 point pairs given,
vector_to_proj_hom_mat2d seeks to minimize the
transformation error. To achieve such a minimization, several
different algorithms are available. The algorithm to use can be
chosen using the parameter Method.
Method='dlt' uses a fast and simple, but also
rather inaccurate error estimation algorithm while
Method='normalized_dlt' offers a good compromise
between speed and accuracy. Finally,
Method='gold_standard' performs a mathematically
optimal but slower optimization.
If 'gold_standard' is used and the input points have been
obtained from an operator like points_foerstner, which
provides a covariance matrix for each of the points, which specifies
the accuracy of the points, this can be taken into account by using
the input parameters CovYY1, CovXX1,
CovXY1 for the points in the first image and
CovYY2, CovXX2, CovXY2 for the points in
the second image. The covariances are symmetric 2×2
matrices. CovXX1/CovXX2 and
CovYY1/CovYY2 are a list of diagonal entries while
CovXY1/CovXY2 contains the non-diagonal entries
which appear twice in a symmetric matrix. If a different
Method than 'gold_standard' is used or the
covariances are unknown the covariance parameters can be left
empty.
In contrast to hom_vector_to_proj_hom_mat2d, points at
infinity cannot be used to determine the transformation in
vector_to_proj_hom_mat2d. If this is necessary,
hom_vector_to_proj_hom_mat2d must be used. If the
correspondence between the points has not been determined,
proj_match_points_ransac should be used to determine the
correspondence as well as the transformation.
If the points to transform are specified in standard image
coordinates, their row coordinates must be passed in
Px and their column coordinates in Py. This
is necessary to obtain a right-handed coordinate system for the
image. In particular, this assures that rotations are performed in
the correct direction. Note that the (x,y) order of the
matrices quite naturally corresponds to the usual (row,column) order
for coordinates in the image.
It should be noted that homogeneous transformation matrices refer to
a general right-handed mathematical coordinate system. If a
homogeneous transformation matrix is used to transform images,
regions, XLD contours, or any other data that has been extracted
from images, the row coordinates of the transformation must be
passed in the x coordinates, while the column coordinates must be
passed in the y coordinates. Consequently, the order of passing row
and column coordinates follows the usual order
(Row,Column). This convention is essential to
obtain a right-handed coordinate system for the transformation of
iconic data, and consequently to ensure in particular that rotations
are performed in the correct mathematical direction.
Furthermore, it should be noted that if a homogeneous transformation matrix is used to transform images, regions, XLD contours, or any other data that has been extracted from images, it is assumed that the origin of the coordinate system of the homogeneous transformation matrix lies in the upper left corner of a pixel. The image processing operators that return point coordinates, however, assume a coordinate system in which the origin lies in the center of a pixel. Therefore, to obtain a consistent homogeneous transformation matrix, 0.5 must be added to the point coordinates before computing the transformation.
Px (input_control) point.x-array → (real / integer)
Input points in image 1 (row coordinate).
Py (input_control) point.y-array → (real / integer)
Input points in image 1 (column coordinate).
Qx (input_control) point.x-array → (real)
Input points in image 2 (row coordinate).
Qy (input_control) point.y-array → (real)
Input points in image 2 (column coordinate).
Method (input_control) string → (string)
Estimation algorithm.
Default value: 'normalized_dlt'
List of values: 'dlt', 'gold_standard', 'normalized_dlt'
CovXX1 (input_control) real-array → (real)
Row coordinate variance of the points in image 1.
Default value: []
CovYY1 (input_control) real-array → (real)
Column coordinate variance of the points in image 1.
Default value: []
CovXY1 (input_control) real-array → (real)
Covariance of the points in image 1.
Default value: []
CovXX2 (input_control) real-array → (real)
Row coordinate variance of the points in image 2.
Default value: []
CovYY2 (input_control) real-array → (real)
Column coordinate variance of the points in image 2.
Default value: []
CovXY2 (input_control) real-array → (real)
Covariance of the points in image 2.
Default value: []
HomMat2D (output_control) hom_mat2d → (real)
Homogeneous projective transformation matrix.
Covariance (output_control) real-array → (real)
9×9 covariance matrix of the projective transformation matrix.
proj_match_points_ransac,
proj_match_points_ransac_guided,
points_foerstner,
points_harris
projective_trans_image,
projective_trans_image_size,
projective_trans_region,
projective_trans_contour_xld,
projective_trans_point_2d,
projective_trans_pixel
hom_vector_to_proj_hom_mat2d,
proj_match_points_ransac,
proj_match_points_ransac_guided
Richard Hartley, Andrew Zisserman: “Multiple View Geometry in
Computer Vision”; Cambridge University Press, Cambridge; 2000.
Olivier Faugeras, Quang-Tuan Luong: “The Geometry of Multiple
Images: The Laws That Govern the Formation of Multiple Images of a
Scene and Some of Their Applications”; MIT Press, Cambridge, MA;
2001.
Calibration