Ww = OO L5 
S9, 1 NEO 
  
ISPRS Commission III, Vol.34, Part 3A »Photogrammetric Computer Vision“, Graz, 2002 
  
  
Figure 1: The trifocal plane 
e The orientation can be based upon homologous points in the 
same way as on (infinitely long) homologous straight lines. 
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 5.2) 
possible. 
3.2 Derivation of the Trifocal Tensor 
The trifocal tensor can be introduced intuitively based on homol- 
ogous straight lines (Hartley and Zisserman, 2000). Given are a 
straight line l in the first and a straight line 1” in the second im- 
age (cf. Figure 2). The planes x" = Pl” and x” = P"'l" con- 
structed from these lines intersect in the 3D straight line L, the image 
of which in the third camera is in general the straight line I". 
  
Figure 2: Trifocal tensor from three intersecting lines 
For the fundamental matrix as well as for the trifocal tensor the 
projection matrices can always be transformed in a way that the 
projection matrix of the first image is P^ — [l|0]. The other 
two projection matrices can then be written as P” = [Ajas] and 
P"' = [B|b4]. Based on this the intersection of the three lines in 3D 
space can be defined algebraically by requiring that the 4 x 3 ma- 
trix M = [x’, =”, ='"] has rank 2. Points on the line of intersection 
may be represented as X — o X4 4- 8 X2 with X; and X» linearly 
independent. These points are incident to all three planes and thus 
x X = m 7X = x" TX =0. This implies MX = 0 and because 
of M'X, — 0 and M'X; — 0 M has a 2D null-space. 
Since the rank of M is 2, there is a linear dependence among the 
columns. If we denote 
I AT’ BI" 
M- (mi, m», ms] = 0 all” bly” , 
4 4 
the linear relation may be written m; = am: + Sms. Because 
the lower left element of M is 0 it follows that @ = k(bil”) and 
B — —k(all") for some scalar k. Applying this to the upper three 
vectors we obtain up to a homogeneous scale factor 
r= (Ir AT = (a BT” = (I. ATI" 113 (@Ta,)B'r” 
The i-th coordinate of I’ can thus be written 
I = 1" (b;a yr" T. i" (a. by" bd a j by" i LT (a4 bL" 
By denoting T; — ai bl — 84 bl the incidence relation can finally be 
written as 
C= inn (3) 
T is a bilinear transformation and defines the trifocal tensor which is 
usually written as 77 * Itisa 3x3 x 3 cube made up of 27 elements. 
e 73 * has 18 DOF at maximum. Le., not all cubes are trifocal 
tensors. The number 18 is obtained by subtracting from the 
33 parameters of the three projection matrices the 15 parame- 
ters for a projective transformation (homogenous 4 x 4 matrix) 
of space. For the solution either the conditions have to be en- 
forced or the trifocal tensor has to be minimally parameterized. 
Both lead to non-linear equations. Practical investigations have 
shown, that it is important to use the conditions, because the 
solution is not stable otherwise. 
e There are direct relations among the coefficients of the trifocal 
tensor, the fundamental matrices of the three image pairs, and 
the three projection matrices. They can be employed to deter- 
mine from the trifocal tensor the fundamental matrices and after 
the choice of a coordinate system also the projection matrices. 
4 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. 
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