Camillo Ressl
To compute a tensor that meets with these constraints but avoiding to explicitly include them in the computation ope
possibility is to introduce a new parameterisation for the tensor having exactly 18 coefficients. One such method js
presented by [Torr, Zisserman 1997]: By choosing Es;5 for the IOR-matrix C, — which makes no difference for the
projective relations — it is possible to compute the tensor by six homologous point triples across three views. Of the 34
image-coordinates of these triples, convenient 18 coordinates are kept fixed. In this way a consistent and minimal
parameterisation of the tensor is achieved. The unknowns themselves are obtained as (up to 3) solutions of a cubi,
equation. Due to this fixing of erroneous observations in the images one may entertain suspicion that errors in the
calculated tensor may be induced, furthermore no correct minimisation of the measurement-errors of all observations i;
possible. And as it is shown by the results in [Torr, Zisserman 1997] the standard deviation depends on the choice of the
6 points resp. the fixed 18 coordinates.
Another consistent and minimal parameterisation is proposed by [Papadopoulo, Faugeras 1998]. There 18 special
quantities are selected as unknowns, parameterising the epipoles and some parts of the tensor. The solution for th;
selected unknowns is unique and is achieved by minimising measurement-error. This method depends on approximate
values; e.g. obtained by the eigen-value algorithm mentioned. Furthermore that parameterisation itself is not unique; ie,
depending on the configuration of the three images other 18 quantities have to be chosen.
The photogrammetric standard case of known IOR is not dealt with in the papers of the computer vision community,
Using calibrated images even 15 constraints must hold among the 27 elements of the tensor. Their form is still not
known, either.
5 THE HOMOGRAPHIES AND CORRELATIONS OF THE TENSOR
One can imagine the trifocal tensor T;" formed as a 3x3x3 cube of numbers and the cube's edges related to the
indices i, j, k. If we keep one index fixed we slice a 3x3 matrix out of the tensor. Since we have three indices we gd
three different kinds of matrices — different also in their geometrical meaning. If we keep the i-index fixed as i =1=
(1,2,3], we get the following matrix I, (e, being the t" column of E5,4):
I, =v, -e-B"-A-e] -v], (4.1)
I, describes a linear mapping (correlation) of the lines À4 in image y; to the points p» in the image y». Such a point p; in
WV» is the image-point of the intersecting-point of the projection-plane due to À; and the projection-ray Ry-Cy-e, of image
w,. Therefore, all mapped image-points p; lie on one common line 1, in image y» — the epipolar-line of this particular
projection-ray. The line 1, can be computed by:
L'1,=0 orby 1,~[valcA-e~Fure (42)
Since 1, is the left kernel of I, it is also interesting to look at the right kernel r, of I, which is the epipolar-line in V of
that particular projection-ray:
lr,z0 orby rm,-[va]- Be - Fise, (43)
If all three r, and 1, are computed and arranged as the rows of the matrices R, and L, then the epipols v», and v3, can be
computed quite easily:
L..v,-0 and R,v,-90 (44)
The columns of I, are linear-combinations of the two vectors v3, and A-e, , so the rank of I, will be 2 in general. Thes
three L-matrices are the basic element for [Papadopoulo, Faugeras 1998] to find their consistent and minimal
parameterisation. The 12 constraints they found are: rank(Rx) = 2, rank(Lx) = 2 and det 27 |, Fo V, (which are 10
constraints).
If we keep the j-index fixed as j = 1 = {1,2,3}, we get the following matrix J, (e, being the t" column of E44):
J, =v} -B-v, eA (45
774 International Archives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part B3. Amsterdam 2000.
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