Projective geometry describes a group based on central (conic) projections. Con-fining ourselves to an image’s two dimensions, each projection can be visualised as a central projection from an arbitrary plane Π0 onto a second plane π, compare Figure 2.1. The totality of all those projections from one plane onto another forms the projective group [138].
Since any dimensional plane in 3D can be transferred into any other
two-Projective Transformations 19
PSfrag replacements π Π0
Figure 2.1: A central projection from one plane onto another.
dimensional plane by rotation and translation1, we can think of any plane Π0 as a rotated and translated version of the special plane Π formed by the points X = (X, Y,0)T. Any point X on Π is transformed into a new point X0 on an arbitrary plane Π0 with
X0 =RX+t, (2.1)
where R∈IR3×3 is the matrix of rotation and t∈ IR3 the vector of translation.
Since the third coordinate of X was chosen to be 0, the rigid transformations be-tween Π and Π0 (translation and rotation) can be combined into a single 3× 3 transformation matrix, namely
This makes the nonlinear nature of projection in Euclidean coordinates apparent.
Equation (2.3) does not yet describe the group of 2D projective transformations; in particular the rij are not general, since they are columns of a rotation matrix with
1Possibly by an infinite amount.
20 Projective Transformations only 3 degrees of freedom [103]. Repeated application of Equations (2.2) and (2.3) leads to the form of a general projective transformation:
x = X0
This transformation has 8 degrees of freedom (DOF), despite having 9 parameters pij — any one parameter pij 6= 0 can arbitrarily be set to pij = 1 by multiplying both numerator and denominator with 1/pij. Such a transformation, and equally any projective transformation from a space of dimensionality n into a space of the same dimensionality n, is sometimes called a homography.
2.2.1 Homogeneous Coordinates
Equation (2.4) can be expressed by a single, linear matrix transformation such that
if the convention is adopted that x
This 3-vector representation of a point is known as homogeneous coordinates. Its main advantage is the fact that, using homogeneous coordinates, a projection can be expressed by a single matrix multiplication, which hides the nonlinearity inherent in projection and is therefore handy for computational purposes. For this reason homogeneous coordinates will be used throughout the remainder of this thesis, unless otherwise stated.
In homogeneous coordinates any finite two-dimensional point x = (x, y)T can be expressed as the triplet X = (X, Y, Z)T with Z 6= 0. The conversion between the
2.2.2 The Euclidean Group 21 From Equations (2.8) and (2.9) it is clear that the homogeneous representation X is only defined up to an arbitrary scale factor k 6= 0; only the ratio of homogeneous coordinates is significant. We also see from Equation (2.9) that in the limit Z →0 a point at infinity can be expressed quite naturally as X = (X, Y,0)T, compare also Section 2.4.2. Any non-singular matrix P ∈ IR3×3 forms a valid projective transformation with eight degrees of freedom (see above).
The group of projective transformations discussed above contains several subgroups.
These are discussed in the next sections, going from the more special to the more general.
2.2.2 The Euclidean Group
Equation (2.6) describes a Euclidean transform if Peucl =k
It is easy to show that all orthogonal matrices describe either rotations (det(R) = 1) or reflections (det(R) =−1). The usual parameterisations for a rotation or reflection are
The Euclidean transformation therefore has 3 degrees of freedom (the angle of ro-tation α and the vector of translation t = (tx, ty)T), and it is easy to see that all transformations of this type form a group. Compare Figure 2.2(a) on Page 23 for examples of all possible Euclidean transformations.
2.2.3 The Similarity Group
The similarity group is a generalisation of the Euclidean group through the addition of a uniform scale-factors to the matrix of rotation or reflectionR. Equation (2.10) becomes
22 Projective Transformations Consequently, a similarity transformation has 4 degrees of freedom. It is again easy to see that all similarity transformations form a group. Figure 2.2(b) on Page 23 gives examples of similarity transformations.
2.2.4 The Affine Group
The affine group is derived from the similarity group through the inclusion of anisotropic scaling and skew. This introduces two additional degrees of freedom, resulting in 6 degrees of freedom altogether. An affine transformation has the ma-trix
where ax and ay describe skew in x-direction (i. e. parallel to the x-axis) and y-direction respectively. For ax =−ay this also describes a rotation around the origin and isotropic scaling; the effect of skew can conversely be created by a suitable combination of rotations and anisotropic scaling. Figure 2.2(c) on Page 23 gives examples of affine transformations, in particular skew in y-direction.
2.2.5 The Projective Group
The projective group finally can be derived from the affine group by introducing so-called perspective skew in thex- andy-direction. This has also been called projective shear or chirp and keystoning. This is simply the full matrix in Equation (2.5), or
Pproj =PaffPproj skew (2.17)
where the projective skew alone can be parametrised as Pproj skew =
if bx and by describe projective skew in x-direction (i. e. symmetric around the x-axis) and y-direction respectively. An example of projective shear in one or both directions can be seen in Figure 2.2(d).
Figure 2.2 and Table 2.1 give an overview over the projective group and its subgroups as well as some invariant features.
2.2.5 The Projective Group 23
Figure 2.2: Visual effects of different group-actions: (a) Euclidean, (b) Simi-larity, (c) Affine, (d) Projective.
Group DOF Matrix Invariant properties
Projective
cross-ratio (ratio of ratios of collinear lengths)
ratio of lengths of collinear or parallel segments (e. g. midpoints) ratio of areas
Table 2.1: Common subgroups of the projective group and their geometric properties. Groups lower in the table inherit from groups higher in the table (but the converse is of course not true). See also [103, introduction].
24 Camera Models
X
0 Z
f
Y
PSfrag replacements
(x, y)
(X, Y, Z)
Figure 2.3: Generic camera Model.