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os 1990].
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rientation
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Camillo Ressl
Pi Ag, 707 pi A. uy TH (4.4)
1
This is one single point-line-line-equation. If instead of the homologous lines A, and A; the homologous points
p y, ly and p.-(x, v, ly are known — so (pj p» p3) forming a homologous point-triple — one can use
these image-points to create fictitious lines in each of the two images (V2, V3). With h = (2,3) these three lines are the
line pt, =0 1 -y, )" going through py, parallel to the x-axis of image w,, the line Hy, =1 0 -x, y going through p,
parallel to the y-axis of image i, and the line p,, =(y, —x, 0) connecting the origin of the image-system of image wy
and the image-point py. Using these three lines one has 3x3 = 9 possibilities to choose A, resp Az out of the set
(la ya Ho) resp. (Hs Hy3 Ho) to form equation (4.4). These nine equations are the nine trilinearities reported in
[Shashua 1995]. Since only four of them are linearly independent one may erase Mon and combine the other two to the
10 —,
matrices L, - 01 | Then, using again tensorial-expressions we can write four independent trilinearities (three-
h
points-relations) in the following form, which are valid as long as no image-point {p, ps} is sited on the line of infinity:
(7) yp
pL
2( jJ)
qu p)* 290m l,m-(1,2)
The fourth possible relation within three images would be a point-point-line-relation. Given the homologous point p;
and the homologous line À; resp. À5 then the homologous point p; resp. p3 can be computed by:
(D (7) AI (K) (i) TA
py ~p" Ay TL” resp. py -p Ax jT, (4.6)
The trifocal tensor includes all the projective geometric constraints inherent in three views, so it plays the same role for
three views as the fundamental matrix plays for two. If one has homologous points or lines in two views, the location of
the homologous partner in the third view can be computed by means of the tensor using equations (4.6) resp. (4.2). This
‘tensorial-transfer’ of points works even in the case when the epipolar-transfer of points by means of fundamental
matrices fails. The tensorial-transfer of lines, however, fails for object-lines lying in the epipolar-plane of O; and O,.
Although the tensor depends on the XOR and IOR of the three views, it can be computed from image correspondences
alone. So if one wants to compute the trifocal tensor for three given images, it holds: Every homologous triple of points
resp. every homologous triple of lines gives four resp. two independent equations expressed linearly in the elements of
the tensor; i.e. the equations (4.5) resp. (4.3). The trifocal tensor allows the use of homologous lines for the relative
orientation for the first time; this was not possible with the fundamental matrix.
Since the tensor is made of 3x3x3 = 27 elements it may be computed given a sufficient number of point-triples (= 7) or
line-triples (= 13) or combinations using a least-squares-adjustment by minimising the homogenous equations’ right
side. Since these equations (4.5) resp. (4.3) are homogenous the trifocal tensor is determined only up to scale (same
property for the fundamental matrix) and so this scale has to be chosen; e.g. by IIT;^l| 2 1, which leads to an eigen-value
problem [Hartley 1994]. For such a linear solution no approximate tensor is needed. One must be aware, however, of
the fact that such a linear solution is obtained by minimising non-meaningful quantities whose minimization is justified
neither geometrically nor stochastically, since not all of the point observations are treated the same way, i.e. the
observations’ weights become dependent on the point location in the images (i.e. minimising ‘algebraic-error’ instead
of ‘measurement-error’).
Besides the fact of minimising algebraic-error, this linear solution has also the disadvantage that it is totally over-
parameterised because it is computed with 26 unknowns although the relative orientation in the case of three calibrated
images has only 11 DOF and in the case of three uncalibrated images 18 DOF. So we see that the tensor’s elements
should meet with 16 resp. 9 constraints (one of these constraints is the tensor’s scale which has to be fixed in advance).
So, this unconstrained linear solution minimising algebraic-error will have a strong bias depending on the image-noise
(and the image-distortion, which are not included in the TFT-modelling). By minimising measurement-error without
considering the mentioned constraints the situation gets slightly better. According to [Torr, Zisserman 1997] these
constraints have been investigated but are not as yet thoroughly understood. In [Papadopoulo, Faugeras 1998] 12
constraints can be found; i.e. some of them should be dependent. All of these constraints are non-linear and are partly
expressed in terms of the epipoles.
International Archives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part B3. Amsterdam 2000. 773
