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36 2 Image formation

Transformation Matrix # DoF Preserves Icon

translation I t 3 orientation
3×4

rigid (Euclidean) R t 6 lengths
3×4

similarity sR t 7 angles
3×4

affine A 12 parallelism
3×4

˜

projective H 15 straight lines
4×4
Table 2.2 Hierarchy of 3D coordinate transformations. Each transformation also preserves the properties listed
in the rows below it, i.e., similarity preserves not only angles but also parallelism and straight lines. The 3 × 4
T
matrices are extended with a fourth [0 1] row to form a full 4 × 4 matrix for homogeneous coordinate transfor-
mations. The mnemonic icons are drawn in 2D but are meant to suggest transformations occurring in a full 3D
cube.

the deformation is linear in the motion parameters, it does not generally preserve straight
lines (only lines parallel to the square axes). However, it is often quite useful, e.g., in the
interpolation of sparse grids using splines (Section 8.3).

2.1.3 3D transformations

The set of three-dimensional coordinate transformations is very similar to that available for
2D transformations and is summarized in Table 2.2. As in 2D, these transformations form a
nested set of groups. Hartley and Zisserman (2004, Section 2.4) give a more detailed descrip-
tion of this hierarchy.

Translation. 3D translations can be written as x = x + t or

x = I t ¯ x (2.23)
where I is the (3 × 3) identity matrix and 0 is the zero vector.

Rotation + translation. Also known as 3D rigid body motion or the 3D Euclidean trans-
formation, it can be written as x = Rx + t or

x = R t ¯ x (2.24)

T
where R is a 3 × 3 orthonormal rotation matrix with RR = I and |R| =1. Note that
sometimes it is more convenient to describe a rigid motion using

x = R(x − c)= Rx − Rc, (2.25)
where c is the center of rotation (often the camera center).
Compactly parameterizing a 3D rotation is a non-trivial task, which we describe in more
detail below.