hom_vector_to_proj_hom_mat2d — Compute a homogeneous transformation matrix using given point correspondences.
hom_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
/ Px \ / Qx \ HomMat2D * | Py | = | Qy |. \ Pw / \ Qw /
If fewer than 4 pairs of points (Px,Py,Pw), (Qx,Qy,Qw) 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, hom_vector_to_proj_hom_mat2d seeks to minimize the transformation error. To achieve such a minimization, two different algorithms are available. The algorithm to use can be chosen using the parameter Method. For conventional geometric problems Method='normalized_dlt' usually yields better results. However, if one of the coordinates Qw or Pw equals 0, Method='dlt' must be chosen.
In contrast to vector_to_proj_hom_mat2d, hom_vector_to_proj_hom_mat2d uses homogeneous coordinates for the points, and hence points at infinity (Pw = 0 or Qw = 0) can be used to determine the transformation. If finite points are used, typically Pw and Qw are set to 1. In this case, vector_to_proj_hom_mat2d can also be used. vector_to_proj_hom_mat2d has the advantage that one additional optimization method can be used and that the covariances of the points can be taken into account. 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.
Input points 1 (x coordinate).
Input points 1 (y coordinate).
Input points 1 (w coordinate).
Input points 2 (x coordinate).
Input points 2 (y coordinate).
Input points 2 (w coordinate).
Default value: 'normalized_dlt'
List of values: 'dlt', 'normalized_dlt'
Homogeneous 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
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.