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In the previous section we were able to combine rotation and scaling into a single composite operation by matrix multiplication. Unfortunately, translation cannot yet be included in the composite operator since we do it by addition rather than by multiplication.
Suppose we wish to rotate the image by about the point . Our rotation matrix always rotates about the origin, so we must combine three transformations to accomplish this:
For step (i), we have and
For step (ii),
For step (iii), we can from step (i):
In this example we were unable to find a single matrix operator to do the entire job. The total transformation took the form
This is called an affine transformation because it involves both multiplication by and addition of a constant matrix. This is in contrast to the more desirable linear transformation , which involves only multiplication by .
We will now move toward a modified representation of the image and the operators by rewriting the last equation as
where in the example we had and .
The matrix looks like
and the points are called homogeneous coordinates . We can modify Equation 5 so that the new point matrix is also in homogeneous coordinates:
In the new representation, each point in the image has a third coordinate, which is always a 1. The homogeneous transformation is a matrix,
which is capable of translation, rotation, and scaling all by matrix multi- plication. Thus, using homogeneous coordinates, we can build compositetransformations that include translation.
In homogeneous coordinates, we have
The composite transformation to triple the size of an image and then move it 2 units to the left is
On the other hand, the composite transformation to move an image 2 units to the left and then triple its size is
In the latter case, the distance of the translation is also tripled.
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