Camillo Ressl 
  
convention of summation certain mathematical relations can be expressed in a very efficient way. This convention Says 
that a sum is made of all the seme indices appearing ds co- and contra-variant. So, for example the scalar Product 
s(x,y) = x..y of two vectors x — (x! x? xy andy = (y! y. y 9 can be written in a shorter way as: s(x,y) — xy. The 
product A-B = C of two matrices À and B can be written as A} By = =C x The contra-variant indices relate to the rows 
and the co-variant ones to the columns. As it can be seen the index responsible for the summation disappears (i resp. j in 
the examples above). Such indices are termed saturated (or dummy) while the remaining ones are termed free indices, 
Using this indexation and the convention of summation the following lente sion of matrices may be simplified 
AEB+CED=F = A’; EB, +! j B.D", = Br (AS, BY wi, ,;D' 'm) = F'n. The indexation and the convention 
of summation are everything of tensor calculus that is needed for the following: 
3 AHISTORICAL REVIEW 
The existence of dependencies among 3 images of an object has been known in photogrammetry since, e.g., [Rinner, 
Burkhardt 1972] (trilinear image-fields resp. trilinear relations). The term ‘trilinear’ means, in this case, that à 
homologous line in the first and second image is related to a line in the third image. These trilinear relations are used, 
e.g., to transfer image contents (image-pair of maximum image-content) or to create an object's ground plan by means 
of two images. Despite those and other interesting properties of these trilinear relations, they remained quite unused in 
photogrammetry, maybe due to the lack of a compact mathematical formulization. 
In computer vision the first to discover redundancies within the contents of 3 images were [Spetsakis, Aloimonos 1990] 
They found three relations between homologous points and one relation between homologous lines in three images 
expressed in terms of 27 coefficients, arranged in three 3x3-matrices. [Shashua 1995] showed that as a matter of fact 
these 27 coefficients and one homologous triple of points in three views form together nine linear equations (four of 
them being independent), which he called ‘trilinearities’, since they consist of products of three image coordinates and 
one of the 27 coefficients. Furthermore [Hartley 1994] showed that the same 27 coefficients and a homologous triple of 
lines actually create two equations. He also proposed for this set of 3x3x3 coefficients the term ‘frifocal tensor, 
nevertheless the tensor is sometimes also referenced as ‘trilinear tensor’. 
4 THE TRIFOCAL TENSOR AND ITS OBSERVATION EQUATIONS 
Given are three images Vi, V», Vs. In view of a relative orientation the first image's projection center O, is set to 0. So 
the central projection of these three images is given by: 
Pi =C; Ry [Es eH 
p,-C; RT E,,-0, 1" A=C; R; R,C, v47-C; R; 0, (4.1) 
P5 =C; Rj 1£,,0,H*] B-C, Rj ‘R,-C, MET z-C RT :O; 
If the images Ay, As, A; of a 3d-straight-line m (not lying in the epipolar-plane of O, and O3) are given then it can be 
shown that the following relation holds: 
AT A a) B 42) 
So using (4.2) the mapping of m in image y, can be computed using the images of m in y» and v and the orientation 
elements A, B, v;, v1. If the products and sums in (4.2) are replaced by tensorial-expressions we get the following 
three-lines-relation (the indices i, j, k running from 1 to 3 are put in brackets for better distinction): 
7 2124 JE) 7 (3,1 JD 3 (K)_ 9 (y! i) “ yo (n 7 7 7 jk 43 
Mai) Ak) Vi 4 770 Ai t^p V Mk)” Bey = sd Vs) Bi A) = Mi) Ay Mk) T, ( 3 
a air 
T 
7 
T;* is the trifocal tensor which is not unique, because it depends on the choice of the 1* image (this is expressed in (43) 
in such a way that only A; can be predicted by A, and À; — no way for a Prediction of A; by A; and A; or of A3 by M and 
À»). If one multiplies this three-lines-relation (4.3) by p,=(x, y, D) =p," — a point in image y, lying on A, — one gels 
  
772 International Archives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part B3. Amsterdam 2000. 
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