tool
reseau
—
reseau
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wd
zl x
= \
^ ^ \
P4 / v y
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nt “rente e x
£N oA ! YA s
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apparent camera positions
Fig.6: Arrangement of the beam splitters
Fig.6 shows the way the beam splitters were arranged in
front of the lenses to obtain a constant distance to the
tool so that the object remains in the range of focus while
the cameras are shifted parallel to the reseau grids. It is
necessary to shift the cameras because the cutting edges
of the tools lie in different distances to the rotation axis
and at different heights. By turning the tool the edges
can always be put in a plane parallel to the reseau grids
so that no movement in depth is necessary. Whether the
cutting edges or parts of the reseau grids are projected
onto the CCD-arrays is determined by the illumination of
the desired spot.
With the use of three beam splitters per camera it seems as
if there are six cameras. Because all components are fixed
together the exterior orientation of one image is enough to
determine the orientation of the whole optical system. As
the images have a very narrow angular aperture there is no
influence of lens distortion detectable. The parameters of
the inner orientation are highly correlated with the exterior
orientation, so it is sufficient to have only rough values.
The mathematical model for reseau crosses in the first
image is described by the well known collinearity equations:
Xa7 f (91,01, 51, Xo1, Yo, Zo1) (3.1)
X;; : image coordinates of the independent first image
£1 — Zoi : parameters of the exterior orientation
The geometric relation for the other five dependent images
are determined by the orientation parameters of the first
image and the parameters of the relative orientation bet-
ween the first and each of the other five images j.
X;i=f(p1,w1, k1, Xo1, Yo1, Loi, do, dw;, dk;, b. b b.;)
(3.2)
X;; : image coordinates of the dependent images
£4j
d; — b; : parameters of the relative orientation
As there are several images with a constant relation to
each other it is not enough to compute 5 parameters for
the relative orientation. 6 Parameters are necessary to ob-
tain a homogeneous scale for the whole imaging system.
Simulations have shown that the standard deviations of the
exterior orientation parameters for layout 1 and 2 are al-
most identical, that means that the unknown object points
do not contribute to stabilize the computation of these pa-
rameters. Table 1 summarizes the standard deviations, as-
suming an accuracy of 1 um for image coordinates and an
angle of 80 gon between the bundles of rays in horizontal
direction.
pi wi K1 Xo1 Yoi Zo
[mgon] | [mgon] | [mgon] | [pm] | [um] | [um]
8.5 3.9 3.3 1.9 0.8 0.5
Tab. 1: theoretical standard deviations of the exterior
orientation parameters (layout 1 and 2)
The X-axis lies in horizontal direction , the Y-axis in verti-
cal direction and the Z-axis perpendicular to X and Y. As
expected the determination of ¢; and Xo, is more difficult
compared to the other parameters. But it is of no use to
widen the opening angle to a farther degree because of
the limited range of focus.
The differences between layout 1 and 2 lie in the corre-
lation of the unknowns. Because of the extrapolation in
layout 1 the correlation between ¢; and X; amounts to
0.77 and the correlation between w, and Y; to 0.42. These
values are much higher than in layout 2, where the corre-
lation between ¢; and X; and between v; and Y; amounts
only to 0.09. The different correlations cause the diffe-
rent standard deviations for unknown object coordinates
as shown in Table 2.
x[um] y[um] z[um]
layout 1 3.3 23 6.5
layout 2 2.0 2.0 6.3
Tab. 2: theoretical standard deviations for unknown object
points
In order to control the theoretical values empirical inve-
stigation were made and have given the results shown in
Tab. 3. They were derived by repeated measurements of
reseau crosses as unknown object points. Both tests were
made under similar conditions.
x[um] yim] z[um]
layout 1 13 4 9
layout 2 3 3 9
Tab. 3: empirical standard deviations for unknown object
points
