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e For all points x" in the second image which lie on the straight 
line 
ie FX 
holds lx" — 0 and, therefore, the coplanarity condition. Le., 
1” is the epipolar line of x" in the second image. It can be used 
to predict the geometrical location of x" in the second image 
in the form of a straight line. The computation can be based 
solely on F. There is no need to know the parameters of the 
orientation of the two cameras. 
e The epipoles e; 2 and e»,1 in the first and the second image, 
respectively, are mathematically the left and right eigenvectors 
corresponding to the eigenvalue O of the fundamental matrix: 
el oF = Qand Fez; = 0. 
e The bilinear form is linear in the coefficients of the fundamen- 
tal matrix. This allows for a direct determination from homol- 
ogous points. 
e The 3 x 3 fundamental matrix has 9 elements and is singular 
with rank 2. Because it is additionally homogenous, i.e., a mul- 
tiplication with a scalar # 0 does not change the result, it has 
only 7 DOF. The condition |F| = 0 has to be enforced, which 
is cubic in the parameters of F. 
If calibration data is available, the fundamental matrix reduces to the 
essential matrix E and its bilinear form "x TE"x” = 0 with the 
reduced image coordinates "x’ = K’~'x’ and "x" = K"~'x". The 
essential matrix can be obtained from the fundamental matrix via 
E — K"'FK' with the calibration matrices K' and K" of the first 
and the second image, respectively. From the point of view of the 
geometrical spaces, the fundamental matrix describes a projective 
space and the essential matrix a "similar" space. 
For three images we use the trifocal tensor. Its concept and its linear 
computation were presented in (Hartley, 1997). In photogrammetry 
(Fórstner, 2000, Ressel, 2000, Theiss et al., 2000) have described 
the trifocal tensor and reported about experiments. The orientation 
based on image triplets has some advantages over the orientation 
based on image pairs: 
e The orientation can be based upon homologous points in the 
same way as on (infinitely long) homologous straight lines. 
e The conditions for the homologous points as well as for the 
homologous straight lines are linear in the observed homoge- 
nous coordinates. They are additionally linear in the elements 
of three 3 x 3-matrices or more precisely of the 3 x 3 x 3-tensor 
constructed from these matrices. 
e Practical experience shows that the local geometry of an image 
strip or an image sequence can be much more precisely and, 
what is more important, also much more robustly determined 
from image triplets and their conditions than from only weakly 
overdetermined pairs. This is true for the trifocal tensor but also 
for bundle triangulation. Opposed to the latter, the trifocal ten- 
sor has like the fundamental matrix its strength in its linearity. 
Linearity equals speed and this makes the determination of ap- 
proximate solutions, e.g., based on RANSAC (cf. Section 4.1) 
possible. 
The trifocal tensor can be introduced intuitively based on homol- 
ogous straight lines (Hartley and Zisserman, 2000). Given are a 
straight line 1’ in the first and a straight line 1” in the second im- 
age. The planes ©’ = PI” and x” PTY" constructed from 
these lines intersect in the 3D straight line L, the image of which in 
the third camera is in general the straight line 1”. 
rU = PTT Y" 
T is a bilinear transformation and defines the trifocal tensor which 
is usually written as qu It is a 3 x 3 x 3 cube made up of 27 
elements. To has 18 DOF at maximum. Le., not all cubes are 
trifocal tensors. The number 18 is, e.g., obtained by subtracting from 
the 33 parameters of the three projection matrices the 15 parameters 
for a projective transformation (homogenous 4 x 4 matrix) of space. 
3 POINT TRANSFER WITH THE TRIFOCAL TENSOR 
Based on the trifocal tensor a prediction of points and straight lines 
in the third image is feasible without determining the point or the 
straight line in space. Le., the trifocal tensor describes relations be- 
tween measurements in the images without the need to reconstruct 
3D geometry explicitly. In principle this corresponds to the epipolar 
line for the image pair, but opposed to it the result is unique. 
For the general case the prediction for points could be done by in- 
tersecting the epipolar lines in the third image corresponding to the 
homologous points in the first and the second image, respectively. 
This is true only if the epipolar lines are not parallel, which is the 
case if a point lies on the trifocal plane, or if the projection centers 
are collinear. The latter is often valid or at least nearly valid, e.g., for 
aerial images from one flight strip. 
The restriction of the preceding paragraph does not hold if we em- 
ploy the trifocal tensor: Given two homologous points x' and x" 
in the first and the second image one chooses a line 1” through x”. 
Then, the point x" can be computed by transferring x' from the 
first to the third view via the homography defined by 1/7; 5 ie, 
ak — a. 
This transfer via the homography implies the intersection of the 
plane defined by the projection center of the second camera O" and 
the line 1” with the ray defined by the projection center of the first 
camera O' and x' (cf. Fig. 1). This intersection is not defined if 1” 
is taken to be the epipolar line corresponding to x'. The plane de- 
fined by the epipolar line and O" comprises the point X which is 
projected to x’, x", and x” as well as the projection center O" of the 
first camera. In this plane lies also the ray defined by O' and x". 
On the other hand, if one takes the line perpendicular to the epipolar 
line through x", also the projection plane becomes in one direction 
perpendicular to the ray defined by x' and O' and thus the intersec- 
tion geometry becomes optimal. In (Hartley and Zisserman, 2000) 
it is recommended to use optimal triangulation to compute a pair x 
and x” which satisfies x""F,2x" = 0. We do not do this because 
we are focusing on speed rather than on optimum quality. 
Thus, our basic algorithm looks as follows: 
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