In: Wagner W., Székely, B. (eds.): ISPRS TC VII Symposium - 100 Years ISPRS, Vienna, Austria, July 5-7, 2010, IAPRS, Vol. XXXVIII, Part 7B 
With the given rotation matrices the normal vector for the coil 
front v cn and the normal vector for the triangular shaped area 
spanned by the laser v Ln can be calculated in the fixed-place 
coordinate system, shown in Eqns 9 and 10. 
(°\ ( sin (^) \ 
VLn= R a ■ Rp ■ E- f ■ o = sin(a) cos(/3) (9) 
\lJ \cos{a) cos{(3) J 
e < 
Due to Eqn. 13 the gradient of the laser line k yx in the photograph 
is dependent on the linear trend of the coil windings represented 
by 9 and the laser rotation around the symmetry axis represented 
by /3. Both angles affect the gradient k yx in the same way and 
so an estimation using the Eqns 15 and 16 will fail because they 
can not be determined seperately by a single measurement. The 
solution to this problem is to make a reference measurement for 
the skewness of the laser line related to /3 by using a plane with a 
defined angle 9 instead of the coil front. A vertical plane as ref 
erence (0 = 0°) allows to extract /3 from Eqn. 13, (see Eqn. 17). 
(10) 
/3 = — arctan (k yx {9 = 0°) sin(a)) (17) 
0 = arctan (k yx tan(a) + (18) 
Then the direction of the intersection vector u between the coil 
front and the laser can be determined by the cartesian product of 
the according normal vectors v cn and v Ln, shown in Eqn. 11. 
Now the intersection of the coil front and the triangular shaped 
area spanned by the laser, results in the visible laser line on the 
coil with the coordinates (xl, Vl, zl). The intersection vector 
u is a scaled version (by a factor u) of the laser line with the 
same orientation, see Eqn. 12. Further the intersection vector u is 
independent of 7 so the orientation of the laser line is independent 
of an up or down tilting of the laser. 
U = V L n X V C n 
(H) 
( XL \ 
yL = V U 
(12) 
\ZL 
To eliminate the influence of the laser alignment and the linear 
trend of the coil front the remaining angles ¡3 and 9 must be calcu 
lated. Therefore the orientation of the laser line will be separably 
examined in the y,ax-plane and in the y,z-plane. Due to the setup 
the orientation of the laser line k yx in the y,x-plane is present in 
the observable image scene in Fig. 9 and can be measured by a 
trend estimation of the extracted laser line, the associated relation 
of the laser coordinates yz, and xl is shown in Eqn. 13. 
, _ Vl _ cos(o:) cos(/3) sin(6>) - sin(/3) cos(fl) 
yX XL sin(a:) cos(/3) cos($) 
The second laser line orientation k yz in the y ,2-plane is predeter 
mined by the laser light section technique given by Eqn. 2 and the 
associated relation of the laser coordinates yL and zl is shown in 
Eqn. 14. 
c yz — — = tan(</?) 
zl 
cos(a) cos(/3) sin($) — sin(/3) cos(9) 
sin(a) cos(/3) sin($) 
(14) 
With Eqns 13 and 14 for the orientation of the laser line it is 
possible to determine the remaining angles /3 and 9, shown in 
Eqns 15 and 16. 
Due the fact that the triangulation angle ip, and so a (= <p — 7r/2.) 
is not constant with respect to /3, a slight error occurs. Despite the 
fact that <p is influenced by /3, the approximation approach works 
excellent to determine the laser alignment and the linear trend of 
the coil front for values of /3 less than ±15°. For example a laser 
rotation by /3 = —7.5° results in an error of 9 of less than 0.01°. 
4 LASER LINE EXTRACTION ALGORITHM 
Before the coil profil can be determined, the laser line must first 
be extracted out of the image. Figure 10 shows a steel coil and a 
laser line projected on it. Due to a rotation of the camera of 90° 
(to obtain the profil length at a higher resolution corresponding to 
the camera chip with 1392 x 1040 pixel) the laser line is horizon 
tal in the acquired images. The laser line is clearly visible due to 
the use of optical filter mentioned in Sec. 1. 
200 400 600 800 1000 1200 
Width in pixel 
Figure 10: Post-processed image (for better contrast of the laser 
line) of a steel coil with a laser line. 
First of all the algorithm detects if a laser line is present in the im 
age by simply accumulating intensities along rows and searching 
for a clearly visible maxima. Also a first elimination of pack 
aged coils and a rough estimation of the laser position is done. 
For coils with packaging material a laser line extraction is mean 
ingless and therefore an early elimination is preferable. After 
the rough estimation of the laser position, the orignal image is 
cropped to a vertical region around the laser. Then a median fil 
ter with a rectangular mask (with 15x10 pixel) is applied at the 
cropped image to reduce noise and to smooth the laser line, the 
result is shown in Fig. 11. 
/3 = - arctan (v (15) 
0 = -arctan(j-7jyj (16) 
The detection of the maxima positions is an easy way to extract 
the whole laser line out of the cropped image. But due to the fact 
that the horizontal position of the coils in the images can vary in 
dependence on the coil size, the considered image area extends