16 Projective geometry
The concepts presented in the following two chapters concentrates on concepts of pro-jective geometry. This chapter and the next one introduce most of the geometric concepts used in the rest of the thesis. This chapter focuses on projective geometry and introduces concepts as points, lines an planes in two or three dimensions. A lot of attention goes to the analysis of geometry in projective, affine, metric and Euclidean layers. Projective geome-try is used for its simplicity in formalism, additional structure and properties that can then be introduced were needed through this hierarchy of geometric strata. This section was in-spired by the introductions on projective geometry found in Faugeras’ book [5]. A detailed description on the subject can be found in the recent book by Hartley and Zisserman [8].
3.1 Projective Geometry
A point in projective n-space Pn is given by a (n + 1)-vector of coordinates x = [x1...xn+1]T. At least one of these entries of the vector should differ from zero. These coordinates are called homogeneous coordinates. In this thesis the coordinate vector and the point itself will be denoted with the same symbol. Two points denoted by (n + 1)-vectors x and y are equal if and only if there exists a nonzero scalar λ such that x = λy.
This will be indicated by x∼ y.
A collineation is a mapping between projective spaces. A collineation from Pm to Pn can be mathematically denoted by a (m + 1) × (n + 1) matrix H, where points are transformed linearly: x0 ∼ Hx. Matrices H and λH with a nonzero scalar λ represent the same collineation.
A projective basis is the extension of a coordinate system to projective geometry. A projective basis is a set of n+ 2 points such that no n + 1 of them are linearly dependent.
The set el = [0, · · · , 1, · · · , 0]T, ∀l, 1≤l≤n + 1, where 1 is in the lth position and en+2 = [1, 1, · · · , 1]T is the standard projective basis. A projective point ofPncan be described as a linear combination of any n+ 1 points of the standard basis. For example:
m =
n+1
X
l=1
λlel
3.1 Projective Geometry 17
It can be shown [4] that any projective basis can be transformed into a unique collineation of the standard projective basis. Similarly, if two sets of points m1, ..., mn+2and m01, ..., m0n+2 both form a projective basis, then there exists a uniquely resolved collineation T such that ml0 ∼ T ml, ∀l, 1≤l≤n + 2. This collineation T describes the different combination of projective basis. Notice that T is invertible.
3.1.1 The Projective Plane
The projective plane is the projective space P2. A point inP2 is represented by a 3-vector m= [x, y, z]T. A line l is also represented by a 3-vector. A point m is located on a line l if and only if
lTm= 0 (3.1)
This equation, however, can also be described as the expression that ”the line l passes through the point m” or ”the point m in on the line l”. This symmetry in the equation shows that there is no formal difference between points and lines in the projective plane. This is known as the principle of duality. A line l passing through two points m1 and m2 is given by their vector product m1 × m2. This can also be written as
The dual formulation gives the intersection of two lines. All the lines passing through a specific point form a pencil of lines. If two lines l1 and l2 are distinct elements of the pencil, all the other lines can be obtained through the following equation:
l∼ λ1l1+ λ2l2 (3.3)
for some scalars λ1and λ2. Note that the ratio λλ1
2 is important.
18 Projective geometry
3.1.2 The Projective 3D Space
A projective 3D space typically means the dimension of the projective space is 3, where is the projective space P3. An element in P3 is represented by a 4-entry vector M = [X, Y, Z, W ]T. In P3 the duality of an element is a plane, which is also denoted as a 4-entry vector. A point M lies on a planeΠ can be denoted mathematically as:
ΠTM = 0 (3.4)
A line can be written into a linear combination of two points as:
λ1M1+ λ2M2
or can be produced by the intersection of two planesΠ1 ∩ Π2.
3.1.3 Projective Transformations
We can denote a transformation between the images as a homography of P2 → P2 , which can be represented by a3 × 3-matrix H. With the same properties of matrices, H and λH represent the same homography for all nonzero scalars λ. A point is transformed as follows:
m7→ m0 ∼ Hm (3.5)
The corresponding transformation of a line can be obtained by transforming the points which are on the line and then finding the line defined by these points:
l0Tm0 = lTH−1Hm= lTm= 0 (3.6) From the previous equation it is easy to derive a transformation equation for a line (H−T = (H−1)T = (HT)−1):
l7→ l0 ∼ H−Tl (3.7)
3.2 Analysis of 3D Geometry 19
Similar reasons can be considered inP3 gives the following equations for transforma-tions of points and planes in 3D space:
M 7→ M0 ∼ T M, (3.8)
Π 7→ Π0 ∼ T−TΠ (3.9)
where T is a4 × 4-matrix.
3.2 Analysis of 3D Geometry
Usually we define the real world as a Euclidean 3D space. But in some particular cases it is not sufficient to use the full Euclidean structure of 3D space. Euclidean 3D space is only suitable for less structured and thus simpler projective geometry. Intermediate layers are formed by the affine and metric geometry. These structures can be thought of as different geometric layers which can be overlaid on the world for different transformations.
The most complicated is Euclidean, then metric, next affine and finally projective structure.
The concept of stratification is closely related to the groups of transformations acting on geometric entities and leaving some properties of configurations of these elements in-variant. Attached to the projective stratum is the set of projective transformations, attached to the affine stratum is the set of affine transformations, attached to the metric stratum is the set of similarities and attached to the Euclidean stratum is the set of Euclidean transforma-tions. It is important to notice that these groups are subgroups of each other, e.g. the metric group is a subgroup of the affine group and both are subgroups of the projective group.
An important aspect related to these groups are their invariants. An invariant is a prop-erty of a derivation of geometric entities that is not altered by any transformation belonging to a specific group. Invariants therefore can guild us what measurements we can do consid-ering a specific stratum of geometry. These invariants are often related to geometric entities which stay unchanged after applying the transformations to a specific group. These
geomet-20 Projective geometry
ric entities with invariants related play an important role in part of this thesis.Recovering them allows us to upgrade the structure of the geometry to a higher level of the geometric stratification.
In the following sections, different strata of geometry are discussed. The associated groups of transformations, their invariants and the corresponding invariant structures are presented.
3.2.1 Projective Stratum
The simplest stratum is the projective stratum. It is the less structured one and has the least number of invariants and the largest group of transformations related to it. The group of projective transformations or collineations is composed with the most general group of linear transformations.
A projective transformation of 3D space can be denoted by a 4 × 4-matrix, where the matrix is invertible:
This transformation matrix is only defined up to a nonzero scale factor and has therefore 15 degrees of freedom.
Relations of collinearity, incidence and tangency are projectively invariant. The cross-ratio is an invariant property under projective transformations as well. It is defined as follows: Assume that the four points M1, M2, M3 and M4 are collinear. Then they can be
3.2 Analysis of 3D Geometry 21
The cross-ratio does not depend on the choice of the reference points M and M0 and is invariant under the group of projective transformations of P3. It can be derived that a similar cross-ratio invariant for four line intersecting in a point or four planes intersecting in a line.
We can regard cross-ratio as the coordinate of the fourth point is the linear combination of the first three points, since three points form a basis for a projective line inP1. Similarly, two invariants can be found for five coplanar points, three invariants for six coplanar points, all in general position.
3.2.2 Affine Stratum
The affine stratum has more structure than the projective one, but less structure than the metric or the Euclidean strata. Differs from projective stratum, the affine stratum identifies a special plane, which called the plane at infinity.
To define this plane at infinity, we have W = 0 and thus Π∞ = [0, 0, 0, 1]T. We can consider that the projective space contains the affine space under the one-to-one mapping:
A3 → P3: [X, Y, Z]T 7→ [X, Y, Z, 1]T. The plane W = 0 in P3 can be thought as con-taining the limit points forkMk = ∞. The Affine transformation is usually denoted as the following:
The affine transformation can be rewritten in the matrix form: M0 ∼ TAM with:
TA∼
22 Projective geometry
Therefore, the affine transformation has 12 independent degrees of freedom. All in-variants under the projective stratum also exsist under the affine stratum. For the more restrictive affine group, parallelism is added as a new invariant property. Lines or planes having their intersection at infinity are called parallel. Another new invariant property for affine group is the ratio of lengths along some direction.
3.2.3 Metric Stratum
The metric stratum resembles in the group of similarities. This stratum differs from the Euclidean stratum only up to a scale factor. The metric transformations correspond to Euclidean transformations complemented with a scaling. When no absolute measure-ment is available, reconstruction in the metric coordinate is the highest level of geometric structure that 3D reconstruction from images can achieve.
A metric tranformation can be represented as the following:
with rij the coefficients of an orthonormal matrix, which is usually denoted by R such that RTR= RRT = I and thus R−1 = RT. Recall that R is a rotation matrix if and only if RRT = I and det(R) = 1. In homogeneous coordinates, Equation 3.14 can be rewritten as M0 = TMM , with
A metric transformation therefore has 7 independent degrees of freedom, 3 for trans-lation, 3 for orientation and 1 for scale. In metric stratum there are two important new invariants properties: relative lengths and angles.
3.2 Analysis of 3D Geometry 23
3.2.4 Euclidean Stratum
The only difference between Euclidean stratum and metric stratum is absolute length.
Therefore, the Euclidean transformation has 6 independent degrees of freedom, 3 for trans-lation and 3 for rotation. A Euclidean transformation has the following matrix form:
TE ∼
with rij the coefficients of an orthonormal matrix, as described previously. If det(R) = 1 then, this transformation is simply the same as a rigid-body transformation in space.
3.2.5 Comparison of the Different Strata
In this chapter some concepts of projective geometry were introduced. Based on these concepts, some methods can be invented by doing the inverse of the projection process and obtain 3D reconstructions of the observed scenes, where is the main objective of this thesis.
We can list a table in order to compare different strata described previously:
24 Projective geometry
ambiguity DOF transformation in matrix form invariants
projective 15 TP =
Table 3.1: Comparison of Different Geometric Strata. ”Number of degrees of freedom, trans-formation in matrix form and invariants corresponding to different geometric strata.”
3.2 Analysis of 3D Geometry 25
Figure 3.1: Shapes which are equivalent to a cube under different geometric transforms.
26 Projective geometry