224 
Ke = Pr HL (50) [G7 ) Pr HE (32) +R | ©) 
and the Jacobian of the measurement equation 
  
  
oh, oh, oh 
E LEAD met 9 6 : 
b. ah, (xy, X k Yk k 
HS = = | (10) 
Xy = lan, Oh, oh, 
Xx XX | DC lio mé THE 
at af a 
As H, depends on the predicted state vector the sequence of Kalman gain matrices K has to be computed on-line. 
The initial estimate for the spatial vertex position is computed by applying the above mentioned triangulation method to 
the first two measured image points corresponding to the vertex. 
4. RESULTS 
The approach has been tested in an application with a stationary CCD camera observing 3-D workpieces on a conveyor 
belt. Both the motion parameters and the internal camera parameters are estimated off-line by a robust test field 
calibration method (Otterbach and Gerdes, 1994a). The spatial vertex coordinates as obtained from Kalman filtering 
allow for the computation of lengths and orientations of 3-D contour segments, which can be compared with the true 
dimensions of the observed object. 
  
Errors of segment lengths Errors of segment angles 
  
  
  
  
  
  
mi TN 
60 
m N e. 
40 | S 10 
mm 30 | À 
Dil : 
10 xU 
A ir EE. 
10 1 2 d. ———78-6--77T 8 9 10 
201 
k 
| Deraki — -— DetaK2- — — DeiaK3 —- Delta K4 | [—— Deka Wi2 — Dea W23  — — Deka W34 | 
  
  
  
  
  
Fig. 7. Results of 3-D structure estimation by means of Kalman filtering. 
The diagrams in Fig. 7 show the measurement errors as functions of a parameter k, which numbers the images in the 
processed sequence. The corresponding test object is depicted in Fig. 8. A rather short sequence of about 10 pictures 
yields an error of less than +1 mm and £2 deg for lengths and angles, 
Ke 20 mm Wi 2s 20 eT respectively. The mean distance between camera and workpieces was 
= mm , - ! DN . . . . * . 
K3 = 109 mm W3,4 = 90° | LA Ps about 1.40 m in this example. The vision system was equipped with a 
ka = 101 mm | | standard CCD camera (756 X 581 sensor elements) and a standard lens 
with a nominal focal length of 25 mm. Table 1 summarizes the computing 
times, which are achieved by the VMEbus system described in Section 2. 
Because of the nonlinearity of the Kalman Filter the covariance matrices 
of process and measurement noise as well as the covariance matrix of 
the initial state estimate are primarily regarded as tuning parameters, 
which control the convergence of the filter. Let S, denote the horizontal 
scaling factor of the imaging system, which relates video memory 
coordinates to coordinates in the CCD sensor plane. Empirically, the 
covariance matrices have been specified as follows: 
  
Fig. 8. A test object. 
IAPRS, Vol. 30, Part 5W1, ISPRS Intercommission Workshop “From Pixels to Sequences”, Zurich, March 22-24 1995