2D/3D Geometric Transformations. CS485/685 Computer Vision Dr. George Bebis. 2D Translation. Moves a point to a new location by adding translation amounts to the coordinates of the point. or. or. 2D Translation (cont’d).
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CS485/685 Computer Vision
Dr. George Bebis
(θ>0: counterclockwise rotation)
(x, y) (xw, yw, w)
line in the space of
* Translate P1 to origin
* Perform scaling and rotation
* Translate to P2
translation + rotation
rotation + translation
(only 4 multiplications and 4 additions)
upper 2x2 submatrix is ortonormal
affine transformed object
G. Bebis, M. Georgiopoulos, N. da Vitoria Lobo, and M. Shah, " Recognition by learning
affine transformations", Pattern Recognition, Vol. 32, No. 10, pp. 1783-1799, 1999.
projective (8 parameters)
affine (6 parameters)
(x, y,z) (xw, yw, zw, w)
(1) Recover the translation T and
rotation R from RxRyRzto xyz.
that aligns RxRyRz with xyz
(2) Apply T and Ron P3 to compute
its coordinates in the RxRyRz system.
Thus, the rotation matrix R
is given by: