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34 2 Image formation

Scaled rotation. Also known as the similarity transform, this transformation can be ex-

pressed as x = sRx + t where s is an arbitrary scale factor. It can also be written as

a
−bt x
x = sR t ¯ x = ¯ x, (2.18)

b a t y
2
2
where we no longer require that a + b =1. The similarity transform preserves angles
between lines.
Affine. The affine transformation is written as x = A¯x, where A is an arbitrary 2 × 3

matrix, i.e.,

a 00 a 01 a 02

x = ¯ x. (2.19)
a 10 a 11 a 12
Parallel lines remain parallel under affine transformations.
Projective. This transformation, also known as a perspective transform or homography,
operates on homogeneous coordinates,
˜

˜ x = H ˜x, (2.20)
˜
˜
where H is an arbitrary 3 × 3 matrix. Note that H is homogeneous, i.e., it is only defined
˜
up to a scale, and that two H matrices that differ only by scale are equivalent. The resulting

homogeneous coordinate ˜x must be normalized in order to obtain an inhomogeneous result
x, i.e.,
h 00 x + h 01 y + h 02 h 10 x + h 11 y + h 12
x = and y = . (2.21)

h 20 x + h 21 y + h 22 h 20 x + h 21 y + h 22
Perspective transformations preserve straight lines (i.e., they remain straight after the trans-
formation).
Hierarchy of 2D transformations. The preceding set of transformations are illustrated
in Figure 2.4 and summarized in Table 2.1. The easiest way to think of them is as a set
of (potentially restricted) 3 × 3 matrices operating on 2D homogeneous coordinate vectors.
Hartley and Zisserman (2004) contains a more detailed description of the hierarchy of 2D
planar transformations.
The above transformations form a nested set of groups, i.e., they are closed under com-
position and have an inverse that is a member of the same group. (This will be important
later when applying these transformations to images in Section 3.6.) Each (simpler) group is
a subset of the more complex group below it.

Co-vectors. While the above transformations can be used to transform points in a 2D
plane, can they also be used directly to transform a line equation? Consider the homogeneous
˜ ˜
equation l · ˜x =0. If we transform x = Hx, we obtain

˜ ˜ ˜ ˜ ˜ T ˜ (2.22)
T
T
l · ˜x = l H ˜x =(H l ) ˜x = l · ˜x =0,
˜ ˜ −T ˜
i.e., l = H l. Thus, the action of a projective transformation on a co-vector such as a 2D
line or 3D normal can be represented by the transposed inverse of the matrix, which is equiv-
˜
alent to the adjoint of H, since projective transformation matrices are homogeneous. Jim