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ISPRS Commission III, Vol.34, Part 3A »Photogrammetric Computer Vision“, Graz, 2002
Figure 6: Three perspective cameras are observing a line
segment. The first camera is aligned with the world coor-
dinate system.
provides a three-view analysis of cylinders and its rela-
tionship to the three-view formulation of points (Shashua,
1995) and lines (Spetsakis and Aloimonos, 1990, Weng et
al., 1993) in terms of trilinear tensors (Hartley, 1997).
Let us first review the linear algorithm (Spetsakis and Aloi-
monos, 1990, Weng et al., 1993) to recover the motion
from three perspective views using line correspondences.
Once again let us consider that three internally calibrated
perspective cameras are observing a scene composed of
rigidly attached lines. Without loss of generality we set
the world coordinate frame as the coordinate frame of the
first camera (see Fig. 6). Furthermore, we assume that the
motion of the first camera is given by the rotation R and
the translation T' and the motion of the second camera is
given by S and U respectively.
We note that for each image, a plane (the projection plane)
is formed by the 3D line and the center of projection. For
example, the projection plane for the first camera and the
line in Fig. 6 is formed by the two points P,, P» on
the line and the center of projection O,. Let us denote
the normals of the projection planes for the cameras with
no, 1, and n» for the first, the second and the third cam-
eras respectively in their local coordinate frames. In the
world coordinate system the normals become no, Rn,
and S! n». The signed distance between the origin and the
projection planes are 0, Tn, and U^ n. We can repre-
sent these planes with four-dimensional homogeneous vec-
tors as follows:
T. RY ST ;
ES e IL, = (Fem) and Ils = (gr) :
These three planes meet at a common line. A necessary
condition for this is that the 4 x 3 matrix formed by the
vectors representing the planes has rank 2. After some al-
gebraic manipulation, this results in the following linear
system of equations:
nl En, * RU! - TS;
no x | nT Fn, | =0,where{ F der RU — TST
ni Gn, G' nut Ts
This result has earlier been obtained by Spetsakis and Aloi-
monos (Spetsakis and Aloimonos, 1990) and by Weng,
Huang and Ahuja (Weng et al., 1993). Note that these
three matrices, E, F and G, are a particular contraction
of the more general trifocal tensor described by Hartley
in (Hartley, 1997) for the projective case. When 13 or
more line correspondences are available, we can recover
the three matrices from which the actual motion parame-
ters can be extracted using the method described in (Weng
et al., 1993).
In order to study the three view-geometry of cylinders, we
first describe the axis of a cylinder in terms of the intersec-
tion of two orthogonal planes, such that one of them also
contains the origin of the coordinate system. These two
planes are defined by their normals and their distances to
the origin (n;, 0) and (n; x 1;, h;). The three-view con-
straints forces the six planes defined by the triplet of cylin-
der correspondences to all intersect in one line. This means
that the following 4 x 6 matrix is of rank two:
nlR, SÍn; h7 R, h7 S,
n) niR. Sin, hy h/f; hls,
nl R5 Sin, hl R; hls,
0. miT Un; hh + WIT. hs + hil
(33)
where h; — n; x l;, i = 0..2. Eliminating the remaining
shape parameters, 1.e., ho, h1 and h», using the fact that
the radius of the cylinder r — Ao sin(22) = hi sin(52) —
h, sin($*), we have five planes intersecting in one line.
un Rn, Stn, Rh, go a1 ho ST h, = ashg
Q0. niT Un, h7T h/t;
(34)
sin( 32) sin( 22)
where a; = Sn (2D) and as = sin The five plane, IIo,
2 , 2
Hi. Il»,
v I ya ho Shah
Ils -— ( Wir ; and IL, — hZU
intersect in one line if and only if three triplets of planes
have a unique intersection. For example,
rank (IIo Il, IL») =
rank Ilo Ils Ha =
rank IIo II; II, zm
Noting that ho x no — lo, these rank constraints result in
the following sets of equations:
nl En;
min | =0
nl Gn»
hi Eh»
hi Fh;
hT Gh,
Mo X
T (a hft - ahi TY ly =0
E (aon! T) lo = 0
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