hifts 
ence, 
pe. 
n the 
nding 
.S ba- 
ojec- 
orre- 
' Sen- 
ition 
ordi- 
f the 
f the 
(7) 
es to 
a ray 
o two 
into 
on of 
ystem 
) gi- 
(8) 
n ma- 
ctors 
ector 
nd is 
'amera 
'orre- 
nsor, 
e, is 
ind Oy 
(9) 
jeome- 
on of 
y the 
f one 
n the 
) and 
ne by 
(10) 
jd va- 
nates 
0 Eq. 
The above formalism gives 3-D coordinates of one 
point of the workpiece surface for each profile po- 
sition in the computer image, i.e. one point for 
each line or column of computer image. 
EXPERIMENTAL SETUP WITH HIGH SYMMETRY 
Transformations for evaluation of 3-D data become 
much easier for the special case of measuring ben- 
ding angles in a setup with high symmetry. This 
setup is characterized by the following assump- 
tions, which can be easily fulfilled by the experi- 
mental setup: 
1) The center of the scanner field on the surfa- 
ce of the optical bench defines the origin of 
the world coordinate system. 
2) The profile is taken along the y,-axis, i.e. 
the light plane is identical to the y,-z,-pla- 
ne and Eq. (5) simplifies to x,=0. 
3) The origin of the camera coordinate system 
lies in the x,-z,-plane, i.e. t, - O. The me- 
chanical adapters of scanner and camera to 
the optical bench are constructed to fulfil 
this condition. 
4) The optical axis of the camera is directed on 
the origin of the world coordinate system. 
Eq. (6)-(7) do not change, but transformation for- 
mula Eq. (8) from camera coordinates to world coor- 
dinates simplifies. Assumption 3) and 4) allow the 
description of camera orientation by only one angle 
0 and the translation vector T(t,,t,,t,). The angle 
$ denotes the angle between z,-axis and (-z,)-axis, 
which is identical to the triangulation angle O in 
this setup. Hence Eq. (9) can be rewritten as 
x -cosû O0 sind)|Px Cx 
yl -Al 0..1.-9.1lp,] +10 (A€R) (11) 
Zi sind O0 cosŸ/|z, t]. 
[94 
Considering assumption 2), the parameter À is de- 
termined by Eq. (12) 
zt. 
me 12 
P,cosû - z,sinù (12) 
Finally the 3-D coordinates of the point of the 
workpiece, which correspond to the position (f,, fy); 
in the computer image, resp. (P,,PyrP;) according to 
Eq. (6), are given by Eq. (13) 
x 0 
t, Dy 
yl = p,cos0 - z, sint (13) 
t, (p, sint -*z,cos) 
Z "t; 
w P,cosû - z,sinŸ ; 
The result of Eg. (13) was gained with the very 
simple model of perspective projection and many 
other influences (for example lens distortions) 
have been totally neglected. Suitable experiments 
will show the limits of this simple model, which 
can lead to a basic understanding of photogramme- 
tric considerations, even if the model proves to be 
insufficient. It is very likely, that this model 
has to be improved to fit the requirements for mea- 
suring bending angles. 
EXPERIMENTAL RESULTS 
First measurements with the symmetrical experimen- 
tal setup show promising results. Fig. 8 shows 
plotted profile data for a symmetrically bent sheet 
metal (material: St 1403, size: 200 mm x 50 mm x 
2,0 mm) with a bending angle of 89? 53' (measured 
with a standard mechanical goniometer). Camera co- 
ordinates were t, - 559 mm and t, - 805 mm, i.e. 
the triangulation angle was O - 34.8?. For imaging 
a usual 1:1.8/50 mm TV objective was used. From 
working distance and focus length of the objective 
the focal distance was calculated to z, - -52.7 mm. 
Plotted in Fig. 8 are profile positions in each 
  
  
s LN 
3 re 
a = 
f S 
x N 
N 
0 64 128 192 256 320 384 448 512 
Line Number 
  
  
  
Profile Position 
  
  
  
  
300 
  
  
  
  
Fig. 8: Result of the off-line setup 
line vs. line number, resp. xy-coordinates vs. yy- 
coordinates of the computer image. The positions 
are evaluated with subpixel accuracy using the 
Gaussian interpolation algorithm according to Eq. 
(1)-(3). Measuring of the bending angle requires 
calculation of the intersection angle of the two 
legs of the profile in world coordinates. The fol- 
lowing procedure will give this angle: 
First step is regression analysis for each leg of 
the profile in the computer image. For linear re- 
gression analysis in each leg of the profile a num- 
ber of 250 data can be used. The standard deviation 
of evaluated profile positions from the regression 
line is about 0.16 pixels. This is a rather poor 
accuracy for Gaussian interpolation with a systema- 
tical error of about 0.01 pixels. But fortunately 
the profile data are centered very good around the 
regression line and the deviations show statistical 
behaviour. Hence linear regression is a very effec- 
tive tool for eliminating statistical deviations 
from column to column and the slope of each regres- 
sion line can be calculated with a standard devia- 
tion of only 1.5-10* rad (resp. 1/2'). 
For the next step any two points (AA), and 
(Bx/By)¢ of each regression line are transformed 
into the world coordinate system according to the 
formalism of Eq. (6)-(13). The straight line 
through these two points represents the regression 
line in world coordinates. Essential for evaluating 
the bending angle is the slope of this straight li- 
ne, which is given by Eq. (14) 
Zp-Za 2 BIA. 
  
= cos? ; (14) 
Yp7Ya (ALB, - A,B,) e (B,-A,) sin 
Last step is the calculation of the bending angle 
from the two slopes for the regression lines for 
each leg of the workpiece. Application of Eg. (14) 
for the data of Fig. 8 yields to slopes for legs of 
the workpiece of 44.80° and -45.08°, from which the 
bending angle is easily inferred to 89.882 or 
89°53', which is exactly the same value as measured 
by the mechanical goniometer. 
The simple model of the detection system yields to 
a measuring accuracy of the bending angle, which is 
comparable with conventional mechanical goniome- 
ters. In despite of this promising result there 
probably might be more calibration efforts necessa- 
ry, if the system must deal with any given experi- 
mental setup. 
ERRORS AND ITS SOURCES 
The CCIR standard video interface between camera 
and frame storage is suggested to be the main sour- 
ce of error. Different authors have investigated 
the electronic transfer chain from CCD sensor to 
frame storage in detail (see for example (5], 
[11]). The analogous video signal in combina- 
tion with the lack of standards for the number of 
pixels in a horizontal line cause statistical de- 
viations in the synchronization of the frame grab- 
ber's A/D-converter, the so-called line jitter. In 
combination with the different number of 756 sensor 
elements and 512 frame storage pixels there is no