Projective invariants

In case of projective invariants the relation between the two representations (Euclidean object database vs. output of the projective reconstruction) can be described with a 3D projective transformation (collineation).

The number of parameters which describes the used entities are as follows.

• 3D point can be described with a 4-vector determined up to a scale. The degree of freedom is 3.

• 3D line can be described with a 6-vector determined up to a scale and a constraint (Plücker). The degree of freedom is 4.

• 3D projective transformation can be described with a 4x4 matrix determined up to a scale. The degree of freedom is 15.

Using these values the minimum number of entities to determine the invariant(s) is

• 6 points yield 6×3−15+0=3 independent invariants

• 4 points and a line yield (4×3+4)−15+0=1 independent invariant

• 2 points and 3 lines yield (2×3+3×4)−15+0=3 independent invariants

• 3 points and 2 lines yield (3×3+2×4)−15+0=2 independent invariants

• 4 lines yield 4×4−15+1=2 independent invariants

The basic element of the projective invariants is the cross ratio and its generalizations for higher dimensions (see Appendix C).

In the following, using the different geometric configurations to calculate invariants, it is supposed that the ele-ments are in general positions. Apart from the trivial degenerate cases the nontrivial configurations will be determined.

An invariant could be undetermined, if one or more determinants are zero. This means for example coincident point(s) and/or line(s). All of these cases are eliminated from further investigation.

Invariants of 6 points

As shown in (4.1) and also in [84], the number of independent solutions is 3. Using the ratio of product of determinants, a possible combination of independent invariants are

I1 = |Q1Q2Q3Q5| |Q1Q2Q4Q6|

|Q₁Q₂Q₃Q₆| |Q₁Q₂Q₄Q₅| I2 = |Q1Q2Q3Q5| |Q1Q3Q4Q6|

|Q₁Q₂Q₃Q₆| |Q₁Q₃Q₄Q₅| I3 = |Q1Q2Q3Q5| |Q2Q3Q4Q6|

|Q1Q2Q3Q6| |Q2Q3Q4Q5|

There are many ways to create a geometric configuration to represent the situation from which it is possible to calculate the cross ratio. Taking two points Q₁and Q₂as the axis, and using the remaining points Q_i,i=3,4,5,6, four planes (pencil of planes) can be formed. The cross ratio of these planes can be determined as the cross ratio of points created as the intersection of these planes with an arbitrary line not intersecting the axis.

I₁ = {Q₁Q₂Q₃,Q₁Q₂Q₄; Q₁Q₂Q₅,Q₁Q₂Q₆} I2 = {Q1Q3Q2,Q1Q3Q4; Q1Q3Q5,Q1Q3Q6} I3 = {Q2Q3Q1,Q2Q3Q4; Q2Q3Q5,Q2Q3Q6}

where{}denotes the cross ratio. The first 2 points define the axis of the pencil of planes. A further point and the axis define a plane, hence four points yields four planes. Intersecting the planes by a new line the intersection points can be used to define the cross ratio.

Invariant of 4 points and a line

Let Q_L,i,i=1,2 denote two arbitrary distinct points on the line L. In this case the invariant in the determinant form is:

I = |QL,1QL,2Q1Q3| |QL,1QL,2Q2Q4|

|QL,1QL,2Q₁Q₄| |QL,1QL,2Q₂Q₃|

The geometrical situation is similar to the 6 point case, but the axis of the pencil of planes is the line. Therefore cross-ratio can be defined, as

I = {QL,1QL,2Q1,QL,1QL,2Q2; QL,1QL,2Q3,QL,1QL,2Q4} Invariants of 3 points and 2 lines

Let the two lines denoted by L and K, and QL,i,QK,i,i=1,2 are two points on these lines, respectively. As shown in (4.1), there must be two independent invariants for this configuration.

I₁ = |QL,1QL,2Q1Q2| |QK,1QK,2Q1Q3|

|Q_L,1Q_L,2Q₁Q₃| |Q_K,1Q_K,2Q₁Q₂| I2 = |QL,1QL,2Q1Q2| |QK,1QK,2Q2Q3|

|Q_L,1Q_L,2Q₂Q₃| |Q_K,1Q_K,2Q₁Q₂|

A possible geometric configuration to determine the cross ratio is the three planes formed by L and points Q_i,i=1,2,3 (three planes), and the fourth plane generated by the three points. Using the line K to cut through these planes, the intersection of the line and the planes gives four points on the line defining the cross ratio. The other invariant can be determined by interchanging the role of the lines.

Another possible geometric interpretation is to define a plane from the three points. Taking the intersection points of the lines with this plane yields five coplanar point. It was shown in [72], that this configuration has two independent invariants.

Invariants of 2 points and 3 lines

Let Li,i=1,2,3 and Qj,j=1,2 be the three lines and two points, respectively. Pairing every line with every point, they form 6 planes. Using the principle of duality between 3D points and planes, this case could be traced back to the case of six points.

Geometrically, four planes could be defined from a pair of a line and a point. For example, let the four planes:

(L1,Q1), (L1,Q2), (L2,Q1), and (L2,Q2). The remaining line L3 intersects these planes and the four intersection points on the line determine the cross ratio. The other two invariants could be calculated using lines 1,3 and 2,3 in plane definition.

Invariants of 4 lines

Let Li,i=1,2,3,4 be the four lines. As stated earlier, this configuration has 4×4−15+1=2 projective invariants.

This means, that there exists an isotropy subgroup of any collineation of 3D projective space, that leaves the four lines in place [47]. In order to determine the form of such collineations and the geometric construction of the invariants, the transversals of four lines in general position (no two of them intersect) must be defined.

As stated in [91], there exists a uniquely defined double ruled quadratic surface, which contains three given lines, Li,i=1,2,3. Let the equation of the quadric be:

Ξ =AX²+BY²+CZ²+DW²+EXY+FXZ+GXW+HYZ+IYW+JZW =0

Let the points of a line written into the form Qi = µiMi+νiNi,i = 1,2,3, where Mi, Niare the base points of the line L_i. The lines are contained in the quadric, if∀(µ, ν),Ξ(Q_i)=0. The coefficients of the termsµ², µν, ν² must be vanished, this gives three contraints for a given line. A quadric is defined by 10 coefficients (A, . . . ,J) but up to a non-zero scale factor (yields 9 parameters) therefore three lines uniquely determine the quadric.

A quadric (surface) is double ruled if through every one of its points there are two distinct lines that lie on it. This means, that there are two distinct sets of lines, that rule the quadric. The lines in one of the set never intersect an other line in the same set, but intersect every line of the other set. Because the three lines do not intersect, they belong to the same set.

Let the intersections of the quadric with the remaining line L₄be QI,i,i=1,2. Choosing two lines from the other ruling set that pass through the QI,ipoints, these lines will intersect every four original lines. The intersection points on these lines determine two, independent cross ratios.

As noted above, it should exist an isotropy subgroup of collineations of one dimension, that leave the four lines fixed. Because each transversal intersects the four lines in four points (and the lines are fixed), therefore the two transversals must be also held fixed by the collineations. Choosing the coordinate system in such a way, that the equations of the transversals are X=Y =0 and Z =W =0 [47], the required isotropy subgroup of collineations can be represented with TG =hα, α, β, βidiagonal matrix. Every collineation is defined up to a nonzero scaling factor, therefore dim(TG)=1.

Algebraically the invariants can be written as:

I₁ = |Q1,1Q1,2Q2,1Q2,2| |Q3,1Q3,2Q4,1Q4,2|

|Q1,1Q1,2Q3,1Q3,2| |Q2,1Q2,2Q4,1Q4,2| I₂ = |Q1,1Q1,2Q2,1Q2,2| |Q3,1Q3,2Q4,1Q4,2|

|Q1,1Q1,2Q4,1Q4,2| |Q2,1Q2,2Q3,1Q3,2| (4.2) where Qi,jdenotes the j^′th point on the line Li.

Degenerate cases

Examining the invariants in (4.2), it can be seen, that an invariant could be undetermined, if one or more determinants in the denominator are zero. This means, that the four points in the given determinant are coplanar, hence the lines

involved in that determinant are coplanar (parallel or intersecting). If only two lines are coplanar, then there is a geometric configuration, that yields one invariant. The remaining two lines intersect the plane formed by the coplanar lines in two points. Taking the intersection of the line determined by these points with the coplanar lines yields four collinear points, that can be used to calculate a cross ratio.

The coplanarity of lines (namely points of lines) could be tested as the determinant composed from the four points vanishes. In practice this means that when the absolute value of the determinant is bellow a limit, those cases are eliminated.

The other, more restricted cases (three or four lines are coplanar) are completly useless.