mz*(sin(^)* sin(0 cos(^)* sin(to)* cos mz*(cos(^)* sin(0 sin(^)* sin(to)* cos mz* cos(tt) cos(0 0
Note that in one formulation, the x-axis rescaling factor mx appears in the first row of the transformation matrix, whereas in the other it appears in the first column.
The situation is similar in two dimensions, where a five-parameter model can be constructed analogously to the three-dimensional nine-parameter model. The two added dimensions allow for anisotropic rescaling either before or after rotations are applied. Figure 8 illustrates the elementary anisotropic rescaling transformations in two dimensions.
Two numerical examples of the three-dimensional case are required. The first uses a sequence of transformations in which an image is first rescaled by factors 1.5 along the x-axis, 2.0 along the y-axis and 2.5 along the z-axis, then rotated around the x-axis by 7°, then rotated around the y-axis by 11°; then rotated around the z-axis by 13°, and finally translated 2 units along x, 3 units along y, and 5 units along z:
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