Camillo Ressl 
  
achieved by projective geometry but at the price of using more parameters than necessary, which have no easily 
comprehensible geometric meaning compared to the notion of IOR and XOR. The linear representation of the central 
projection is obtained by means of a 3x4 matrix with 11 degrees of freedom (DOF) — the ‘projection matrix’ (see 
section 2.1). Furthermore, certain indexed systems of numbers (so-called tensors) are used in computer vision, which 
describe the relative orientation of 2, 3 and 4 images in a linear manner. For five and more images no such alternative 
formulization exists. Using the so-called ‘essential matrix’ resp. ‘fundamental matrix’ one gets a linear representation of 
the relative orientation of two calibrated resp. uncalibrated images (see section 2.2). The so-called ‘trifocal tensor’ T* 
(TFT) enables a linear form of the relative orientation of three uncalibrated images. Moreover, with the so-called 
‘quadrifocal tensor’ the relative orientation of four uncalibrated images can be presented linearly; e.g. [Hartley 1998]. 
Within this paper a closer look at the trifocal tensor and the relative orientation of three images will be given. 
2 BASICS 
2.1 The central projection using homogenous coordinates 
Using homogenous coordinates it is possible to write the central projection of an image in a very compact way. If the 
XOR of y is given by the image's projection center O,, and rotation matrix R (from the image system to the global 
system of coordinates), and if the IOR of the image is given by the principal point (xo, yo), the principal distance f and 
two parameters (c B) modeling affine image deformations, then the central perspective image-point p'=(xy1)-asa 
homogenous vector — of an object point P' = (X Y Z) may be computed by the following product of matrices: 
| « -x, 100 
1 pT P 
p-C ^R [Hi C=10 5 y, E,,-|0 1 0 (2.1) 
00 -f 001 
‘= symbolizes that the left and right side in (2.1) are equal only up to scale. Using these matrices and vector p it is 
possible to write the projection ray r from the center of projection to the object point P in a compact way: r = R-Cy-p. If 
a straight line m in the object space is given, then, in general, its projection in the image will also be a straight line A, 
Due to the point-line-duality in the 2-dimensional projective space it is possible to identify the straight line Am by a 
homogenous vector Am = (a b c). (up to scale). A reasonable way to define the scale of Am Would be to set a b zl. 
In this case c would be the orthogonal Euclidean distance of the image coordinate system's origin to Am. An image 
point p is sited on A, if it holds: pA — 0. Furthermore, given the image line A,, and the elements of IOR and XOR it 
is easy to determine the normal-vector n, of the projection plane € going through O,, Am and m by: n, — RC, 
Notel: A linear mapping of points to points within the 2-dimensional projective space is termed ‘collineation’ and a 
linear mapping of points to lines within the 2-dimensional projective space is termed ‘correlation’. 
Note2: The matrix-multiplication CLR Esa, -O,,] would result in a 3x4-matrix — the projection-matrix — having ll 
DOF due to the scale ambiguity. 
2.2 The relative orientation of two images 
The relative orientation of two images y, and y» (with IORs C, and C; and with projection centers O, and O, and 
attitudes R, and R3) is obtained by the intersection of the projection rays of homologous image-points p, and p;; i.e. the 
base-vector b 2 O; - O, and the two projection rays are coplanar. Using the results of section 2.1 we can formulate this 
relation in an elegant manner using a quadratic form; e.g. [Niini 1994]: 
(R,-C,-p,)"-(bx(R,-C,-p,))}=0 — pr CL RI [b].-R,-C, p,-p; Fo p;—0 (2.2) 
Fr 
0 .—b, 5, 
The cross-product was solved by: bb} =! b, 0 -b, (23) 
-b, by 0 
This property of intersection is independent of the origin and attitude of the object-coordinate-system. As à 
consequence, the latter may be fixed, e.g., in the following way: Its origin coincides with the projection center of the 
first image and the attitude of the axes of this photograph's image-coordinate-system fixes the orientation of the object 
  
770 International Archives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part B3. Amsterdam 2000. 
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