nd 
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ons 
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ter 
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[6], 
chms 
ob- 
beam 
sian 
(1) 
(2) 
which reduces for the usual case of adjacent pixels 
(xX,7X9 = -1 and x,-x, = +1) to Eg. (3) for the sub- 
pixel center position C of the Gaussian distribu- 
tion curve 
L-1i 
eum 3I, 
(3) 
Gaussian interpolation proved to be a very effecti- 
ve and precise subpixel accuracy algorithm. It is 
more precise than simple centroid calculation, whe- 
re especially the choice of the evaluated pixel 
range is very critical. Gaussian interpolation 
needs not too much calculation efforts, but proved 
to be a resistant algorithm with little systematic 
errors of about 1% of a pixel period [8]. 
Linear regression analysis 
Next step for evaluating the bending angle is sepa- 
ration of the two legs of the profile and linear 
regression in each leg. Regression analysis will 
give improved values for the slopes of the two legs 
of the profile and, simultaneously, statistical 
prediction for the resolution of the measuring sy- 
stem. 
EVALUATION OF 3-D COORDINATES 
For evaluation of the bending angle the relations 
between 3-D coordinates of the workpiece surface 
and profile positions in computer image have to be 
known. Only an exact knowledge of this relations 
gives the chance to evaluate the bending angle of a 
workpiece from the profile in the computer image. 
The functional relations become calculable by per- 
forming all coordinate transformations through the 
detection system. But the equations contain a lot 
of parameters (for example internal and external 
camera parameters, lens distortion, digital image 
acquisition parameters), most of which can not be 
measured directly. These parameters have to be ca- 
librated, which is a familiar procedure in close 
range photogrammetry (see for example [9]). In 
the following the functional relation of 2-D com- 
puter image coordinates and 3-D workpiece coordina- 
tes are to be evaluated. 
3-D sensing by triangulation is based on calcula- 
ting the intersection point of two rays in space. 
One of these two rays is given by the direction of 
illumination. However, in our system not a ray but 
a light plane is defined by the scanner path. The 
intersection of this light plane with the object's 
surface yields to the 'profile' of the object. Hen- 
ce the coordinates of the light plane give a first 
geometric locus for the surface of the workpiece. 
Transforming back one profile position in the com- 
puter image through all steps of the detection sy- 
stem provides 3-D coordinates of a ray, on which 
one spot of the profile must lie. This ray gives 
the second geometric locus for one point of the 
workpiece surface. 
World coordinate system 
First step is definition of a 3-D world coordinate 
system, to which all other coordinate systems re- 
fer. In this system the bending angle has to be 
calculated finally. 
The schematical drawing of the experimental setup 
in Fig. 3 depicts also the cartesian world coordi- 
nate system (x,,y,,;Z2,) and the camera coordinate 
system (Xx,,y,,Z2,). The origin of world coordinates 
is defined by the point of impact of the laser beam 
in the center of the scanner coordinate system. X,- 
and y,-axes lie on top of the table and are paral- 
lel to the corresponding axes of the scanner coor- 
dinate system. Z,-dimension is the height above the 
table. To get a right-handed world coordinate sy- 
stem the y,-axis has opposite direction than y,- 
axis, hence y, = —y,. 
Camera coordinate system 
The z,-axis of the camera coordinate system is gi- 
ven by the optical axis of the imaging system. The 
origin lies in the principal point of the objective 
and the y,-axis is assumed parallel to the -(yy)- 
413 
axis. Due to the opposite direction of y, and y, 
the y,-axis is aligned with the corresponding frame 
storage axis yy. The translation vector of the ca- 
mera system is T(t,,t,,t,), the rotational orienta- 
tion may be denoted, for example, by the direction 
cosines between the axes of the coordinate systems. 
Scanner coordinate system 
  
  
SA Scanner-Control DE2000 
    
    
   
  
  
  
Scanner | 
XY2026S 
  
  
  
   
serial 
(RS232) 
  
     
  
  
  
  
  
c 
= 
S 
S 
s 
£ 
= parallel | 
s (Centronics) | 
d | 
F= | 
Loser Diode rer LS 1 
Power Supply * EDD: Euro Digital Driver 
* EDG: Euro Digital Generator Heckel 
Fig. 7: Components of illumination system 
Fig. 7 shows the components of the illumination 
system and illustrates definition of the scanner 
coordinate system. The scanner electronics address 
the working field of the scanner (maximum deviation 
angle $. can be adjusted up to £20? optical) by 16 
bit integers, i.e. scanner coordinates stretch from 
0 to 65535 LSB (Least Significant Bit) in each di- 
mension. The center of the scanner working field is 
addressed by the scanner coordinates (X,y), - 
(32767, 32767). 
Correspondence between world coordinates and scan- 
ner coordinates is simplified by the intrinsic geo- 
metric flat-field correction of the scanner elec- 
tronics. The scanner program generated by the host 
computer is sent to the 'Euro Digital Generator' 
(EDG) via a standard 'Centronics' interface. The 
EDG stores and executes the scanner program inde- 
pendently. During scan vector processing the EDG 
calculates 'micro-steps' every 150 us and sends 
them to the 'Euro Digital Drivers' (EDD), which 
control the movement of the x- and y-mirrors. The 
EDG calculates flat-field corrected micro-steps 
based on a stored grid correction table, which is 
specific for each working distance and scanner con- 
figuration. 
Hence the relation of scanner coordinates and world 
coordinates on the surface of the table (z,-0) is 
simply given by the linear transformation of Eq. 
(4) 
x x,-32767 
5296; — de tan (0, d 
yl «| y. 32767 
- . t o 
er RO, 
Z) w w 
where 
d, working distance (from the last scanner mir- 
ror to the optical table) 
maximum scan angle in x,,y, direction 
Bry 
World coordinates of the light plane 
The light plane is defined by three outstanding 
points in the world coordinate system: The first 
point is given by the world coordinates of the last 
scanner mirror (X;,y;,2Z;)r, which are determined by 
the experimental setup. Two other points are the 
starting point (X,,¥,2,=0) and the final point 
(X3,Y3:2370) of the scan vector (on top of the ta- 
ble), which are calculable with Eq. (4). Hence Eq. 
(5) gives the light plane in the usual three-point-