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Camillo Ressl 
J, is a homography-matrix from image y, to image y; due to the plane o, whose position and attitude is determined by 
image Vo; in detail o, goes through O» with ny, 2 R;C;'-e, and du = -Va, Similarly if we keep the k-index fixed as k 
212 [1,2,3), we get the following matrix K, (e, being the t" column of E44): 
K,=v,-¢ B-v{.A (4.6) 
K, is a homography-matrix from image y, to image wy» due to the plane B, whose position and attitude is determined by 
image Vs; in detail D, goes through Oz with ng, = RC; "e, and dg, = -v5,". 
Besides their geometrical properties (discussed earlier in section 2.3) these two groups of homography-matrices are of 
interest because of two things: Firstly, with the equations of section 2.3 it is possible to compute the missing epipoles: 
v il. = . — 71 — 
V1, =K; * Va = Vz 73, Yız and Vi;7J, : V3 =>  V347K, v, (4.7) 
Because of the inversion in these computations at least one regular J,- and K,-matrix is needed. Since the determination 
whether a matrix is singular or not using noisy data is an ill-posed problem, a good way to overcome this is to choose 
that Jj- and K,-matrix which has the best conditioning number because it is very unlikely (though possible) that all three 
Jc or Ky matrices are singular. 
Secondly, using these J- and K,-matrices and the computed epipoles we can determine the fundamental matrices 
between the views V, and y» and between y, and y; with equation (2.6). With these so derived fundamental matrices 
and the given IOR-matrices C,, C, C; we are able to compute the rotation matrices R; and R; (by setting R, 2 L, i.e. by 
working with the relative orientation of successive images); [Brandstätter 1991]. In case of unknown IOR-matrices it is 
possible to compute a common IOR for all images (C, = C, = C3); [Niini 1994]. But for this purpose a fundamental 
matrix between y» and y; is necessary, which can be computed using a homography-matrix H between these images. A 
rank-2 homography H»; resp. H3; from image w» to image w^; resp. from image y; to image V» can be computed by: 
4 1 341.1 
H,-J,K,'-— -v,-e, resp. Hy zkK,J,!— v.e with s, t € (1,2,3] (4.8) 
Vai Vor 
Whether Hy; or Hs; is to be chosen depends on which homography J, or K, is the best conditioned one. An IOR- and 
XOR-computation for a given tensor can be found in [Ressl 1997]. 
6 FUTURE WORK 
Within this paper an introduction to the relative orientation of three images in a linear way by means of the trifocal 
tensor was given. This trifocal tensor is of interest because of three reasons: Firstly, the theoretical and geometrical 
background of the underlying relations (some of them have been reported in this paper); secondly, as a means for 
computing approximate values for the XOR and the common IOR of three images (i.e. image orientation and 
calibration); and thirdly, the trifocal tensor constitutes a first suitable means to detect blunders ‘on a small scale’ in 
advance, on the contrary to gross-error-detection in the whole image set by means of data-snooping. 
In [Torr, Zisserman 1997] a method called RANSAC (random sample consensus) is used for the blunder-detection. In 
the course of that out of a large set of n point correspondences (order 100) a subset of six homologous points is selected 
and the trifocal tensor underlying this subset is computed uniquely. Afterwards the number of outliers is computed; i.e. 
those of the remaining (1-6) point-correspondences not being involved in the tensor’s computation having an error 
above a certain threshold. After that the whole procedure is repeated for several (~500) other subsets of six points out of 
the given point correspondences. This is done to make sure (~95% probability) that there is at least one subset 
containing only good data points. Out of the resulting multitude of trifocal tensors the one having least outliers is 
chosen as an approximation for the following least-squares-adjustment including all the correspondences considered as 
inliers yielding an improved estimation. 
Of further interest is the topic of dangerous surfaces. In [Shashua, Maybank 1996] it is shown, that if the trifocal tensor 
is computed by means of >6 homologous point-triples no such surfaces exist, but 10 certain singular positions for the 
points in space (including the projection centers). The arrangement of these dangerous points is not given, except that 
these points arise as the intersection (base points) of a linear system of cubic surfaces. Also not given is their ‘sphere of 
influence’, i.e. how far away points must lie to allow a solution. In this connection it must be mentioned that the 
  
International Archives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part B3. Amsterdam 2000. 775