ISPRS Commission III, Vol.34, Part 3A ,,Photogrammetric Computer Vision*, Graz, 2002 
  
1 S aj cH 010 
x= 1 0 t+m y 0 e, Q0 0 
0 0 1 g- 0.140 
With the projection matrix (K is the matrix K multiplied by an arbi- 
trary scalar # 0) we finally end up with 
P=KR(I|-Xo) and x =PX. (2) 
With P image coordinates can be predicted linearly from object co- 
ordinates. Due to its homogeneity (multiplication with an arbitrary 
scalar # 0 does not change the projection) the 3 x 4 matrix P has 
only 11 degrees of freedom (DOF). 
2.3 Fundamental and Essential Matrix 
The fundamental matrix describes the (projective) relative orienta- 
tion of the image pair. We assume that we have n homologous points 
X; in the first image and x’ in the second image. The projection ma- 
trices are P, — K'R'(I|0) and P? — K"R"(I| — T). 
Assuming R' — 1, this corresponds to the method of relative ori- 
entation of successive photographs To simplify the presentation, we 
transform the observed image points in the system of an ideal camera 
with projection matrix (/|0). We obtain the reduced image coordi- 
nates marked with the superscript index "^" *x' 2 R/^!K'-!x' and 
"x" = R'"-!K"-1x". The condition, that the rays should inter- 
sect, implies the coplanarity of x', x", and T. Presented in the same 
coordinate system we obtain 
k k 
X ("Tx trs. =0 
0 —T3 T» 
with Sr = S(T) = T3 0 —T1 
—T "T 0 
S(T) is a skew-symmetric matrix for the vector T with rank 2, which 
allows the vector product V — T x U to be written as a matrix vector 
multiplication: V — Sr. U — —Sy T. Putting things together we end 
up with 
ICY RS, RAR 5 x" zs. 
This relation is linear in the image coordinates of both images, i.e., 
bilinear. With the 3 x 3 fundamental matrix the coplanarity equation 
as condition for the homology of image points can be represented in 
a simple and elegant way: 
F= (KY RS, RIK" ie, TRY m0. 
The latter equation has some important properties: 
e Because it refers to the original measured data, there is no need 
for a reduction of the image coordinates. The reduction is con- 
tained in the fundamental matrix. The bilinear form is linear 
in the coefficients of the fundamental matrix. This allows for a 
direct determination from homologous points. 
(+ P)O) 3 
e For all points x" in the second image which lie on the straight 
line 
= Fly’ 
? 
holds Ix" — 0 and, therefore, the coplanarity condition. Ie. 
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. 
The 3 x 3 fundamental matrix has 9 elements. As S7 is of rank 2, the 
fundamental matrix is singular with rank 2. Because it is additionally 
homogenous, 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 
E = Sr R^! "X E x” => 0 
’ 
with reduced image coordinates "x' — K'^!x' and "x" — K^ 1x". 
The essential matrix can be obtained from the fundamental matrix 
from E — K"'FK. 
3 TRIFOCAL TENSOR 
The idea of the trifocal tensor and its linear computation was pre- 
sented for the first time in (Hartley, 1994, Shashua, 1994). The com- 
putation of a consistent tensor was described in (Torr and Zisserman, 
1997). (Faugeras and Papadopoulo, 1997) deals with constraints on 
the trifocal tensor. In photogrammetry (Férstner, 2000, Ressel, 2000, 
Theiss et al., 2000) have described the trifocal tensor and reported 
about experiments. 
3.1 Trifocal Geometry from the Image Pair 
When extending the image pair by another image we basically as- 
sume that all three projection centers are different. When they are 
additionally not collinear, they form the trifocal plane. This plane 
intersects the image planes in the three trifocal lines ty, to, and ts, 
which comprise the epipoles e; ; (cf. Figure 1). It is important to 
note that the three fundamental matrices F12, F23, and F3, are not in- 
dependent. They have to comply with the following three conditions 
arising from the coplanarity of all projection centers and epipoles: 
T T T 
e2,3F126e1,3 = e3.1F23e2,1 = ei ?F31e3,2 =0. 
To see why this is true, observe that, e.g., the epipolar line of the 
epipole e;,5 in the second image is represented by F;2e1,3. €1,5 is 
the image point of O"" in the first image just as e2,3 in the second 
image. From the latter follows the first condition. 
The orientation based on image triplets has some general advantages 
over the orientation based on image pairs: 
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3.2 
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