Approximate a rigid affine transformation from point correspondences.

vector_to_rigid approximates a rigid affine transformation, i.e., a transformation consisting of a rotation and a translation, from at least two point correspondences and returns it as the homogeneous transformation matrix HomMat2D. The matrix consists of 2 components: a rotation matrix R and a translation vector t (also see hom_mat2d_rotate and hom_mat2d_translate):

                              / 1 0 tx \   / R00 R01 0 \
  HomMat2D  =  |  R   t |  =  | 0 1 ty | * | R10 R11 0 |  =  H(t) * H(R)
               | 0 0  1 |     \ 0 0 1  /   \  0   0  1 /
                         

The point correspondences are passed in the tuples (Px, Py) and (Qx,Qy), where corresponding points must be at the same index positions in the tuples. The transformation is always overdetermined. Therefore, the returned transformation is the transformation that minimizes the distances between the original points (Px,Py) and the transformed points (Qx,Qy), as described in the following equation (points as homogeneous vectors):

             || / Qx[i] \                / Px[i] \ ||^2
 sum of all  || | Qy[i] |  -  HomMat2D * | Py[i] | ||  =  minimum
             || \  1    /                \  1    / ||

HomMat2D can be used directly with operators that transform data using affine transformations, e.g., affine_trans_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.

vector_to_rigid is reentrant and processed without parallelization.

affine_trans_image, affine_trans_region, affine_trans_contour_xld, affine_trans_polygon_xld, affine_trans_point_2d

vector_to_hom_mat2d, vector_to_similarity

dvf_to_hom_mat2d

Foundation