## Perspective Transformations

The most general linear transformation is the perspective transformation. Lines that were parallel before perspective transformation can intersect after transformation. This transformation is not generally useful for tomographic imaging data, but is relevant for radiologic images where radiation from a point source interacts with an object to produce a projected image on a plane. Likewise, it is relevant for photographs where the light collected has all passed through the focal point of the lens. The perspective transformation also rationalizes the extra constant row in the matrix formulation of affine transformations. Figure 9 illustrates a two-dimensional perspective image.

To explain how perspective is incorporated into a matrix framework, it is easiest to consider the case of a one-dimensional image. One-dimensional transformations have only two parameters, scaling and displacement, parameterized as follows:

The matrix V describes the orientation of the special reference frame with respect to the coordinate axes. Rescaling within this reference frame is performed by the matrix product

By analogy with two-dimensional and three-dimensional transformations, this is expressed in matrix formulation using a two-by-two matrix: FIGURE 9 Perspective distortions in a two-dimensional image. Note that after transformation, the horizontal lines will converge to a point somewhere to the right of the figure and the vertical lines will converge to a point somewhere below the figure. Small rotations and skews are also present in the transformed image. Perspective distortions are the most general linear transformations and are not considered affine transformations because parallel lines do not remain parallel after transformation.

FIGURE 9 Perspective distortions in a two-dimensional image. Note that after transformation, the horizontal lines will converge to a point somewhere to the right of the figure and the vertical lines will converge to a point somewhere below the figure. Small rotations and skews are also present in the transformed image. Perspective distortions are the most general linear transformations and are not considered affine transformations because parallel lines do not remain parallel after transformation. x'
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