ISPRS Commission III, Vol.34, Part 3A ,,Photogrammetric Computer Vision“, Graz, 2002
we will prove, that the first case can not occur. This prove
is outlined in the following.
First we arrange the three matrices I, as the rows of a
large matrix Z, which then has 9 rows and 3 columns.
We, however, consider the elements of Z being the column
vectors of the I, matrices. So, Z has 3 x 3 elements and
the element at row € and col n is the n°” column vector in
matrix Le, which is the image of the intersection point of
principal plane T2, with the principal ray Tie.
Now, we consider that one element (row £, col y) of Z shall
be — Vai. This may happen due to two situations: A1)
Os € rig or A2) Oi € 72,. Then, we consider that one
element (row £, col 7) of Z shall be — 0. This may happen
also due to two situations: B1) O3 — (mo, 1 rie) or B2)
Tig € 725. However, B1) implies O3 € rie (— A1)) and
B2) implies O1 € m2, (— A2)). Thus, the only possible
situations, that may return one element in Z being ~ v3;
or = 0 are the ones of Al) and A2).
These situations, however, not only return the element at
row § and col 7) of matrix Z to be ~ V3; or = 0, they further
imply: Al) returns that all elements in row £ of Z are
^: Va1; Le. the entire matrix Ig. And so all columns Ie can
be parameterized by $c (being ^ Y31 or — 0) and Ÿ31. A2)
returns that all elements in column n of Z are ~ V31; ie.
the n°” column in all three correlation matrices Iz. Again,
the parameterization of these columns is not difficult, but
what's more important: When situation A2) occurs, the
n°” column in the three correlation matrices I, will never
be used as the vectors (81,82,83) in the parameterization,
since they are not far away from Vai.
This completes the prove, that it is impossible, that one
of the three vectors {8} is v V31 or = 0, but one of the
other columns in I is different from v3; and 0. Thus, the
minimal parameterization (20) holds for any image config-
uration - provided not all three projection centers coincide.
7 SUMMARY AND FUTURE WORK
In this paper a new minimal set of constraints as well
as a new minimal parameterization for the trifocal ten-
sor (TFT) were presented. They were found using the
so-called correlation slices I, together with a new discov-
ered property of them (equ. (19)). Especially the minimal
parameterization, which is applicable for any image con-
figuration (provided not all three projection centers coin-
cide), will help to get new insights into the geometric re-
lations and properties of the TFT. With these constraints
resp. minimal parameterization it is possible to compute
the TFT with minimal (i.e. 18) DOF. Since both rely on
non-linear relations an initial solution for the TFT is re-
quired; e.g. using the well-known linear solution.
So far, the presented constraints and the minimal parame-
terization have been implemented and it will be among the
future work to investigate the advantages of each method.
The experiments so far show a benefit for the minimal
parameterized solution (equ. (20)), which can be imple-
mented rather simple and works for all practical image
configurations - as opposed to the constrained solution,
which relies on correlation slices having rank = 2 for the
constraints (15) and (19).
Also of interest are the additional constraints resp. the
minimal parameterization that arise when the interior ori-
entation of the images is known, or if it is unknown but
A- 282
common to all three images. The latter is of special in-
terest for camera calibration, which needs at least three
images taken by the same camera; e.g. [Hartley 1997].
During the future work we will also investigate, what
amount of error is induced in the resulting TFT (and
thus in the image orientation), when the constraints are
neglected and/or algebraic error is minimized instead of
measurement error. This is especially of interest when the
TFT-solution serves only as an initial start for a subse-
quent bundle-adjustment, since there already the linear
solution might be sufficient.
ACKNOWLEDGMENT
This work was supported by the Austrian Science Fund
FWF (P13901-INF).
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