FIGURE 7 Illustration of the numerical example of two-dimensional affine transformation. The light gray axes are centered on the coordinate (11.8514, 3.0593), which is unchanged by the transformation. These axes are initially oriented 54.9232° counterclockwise to the coordinate axes. The image is rescaled anisotropically along the gray axes, and then the image and the gray axes are rotated 24.1889° clockwise. Note that within the frame of reference defined by the gray axes, the transformation only involves anisotropic rescaling.

Note that the added parameter eases the constraints on the left upper three-by-three submatrix. For example, whereas the combinations of sines and cosines in the rigid-body model required all elements of this submatrix to have absolute values less than unity, the parameter m allows arbitrarily large absolute values, though other constraints still remain in effect.

As a numerical example, consider a sequence of transformations in which an image is first rotated around the x-axis by 7°, then rotated around the y-axis by 11°, then rotated around the z-axis by 13°, then magnified by a factor of 2 and finally translated 2 units along x, 3 units along y, and 5 units along z:


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