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4. EXPERIMENTAL RESULTS IN 3-D SPACE 
Encouraged by the promising first results, which are fast convergence, easy application to free form objects and relatively 
large zone of convergence, we now consider objects in 3-D space. We use a synthetic object to easily control all parameters 
like object size, point density and symmetry. A corner represented by a set of points forming its three sides is our choice. We 
now test how a copy of the corner object placed and oriented differently in space can be matched with its original. The 
number of iterations is fixed at ten, if not indicated otherwise. Figure 3 shows the model together with the test to be positioned 
at all 27 locations of the grid. Each position has a number which is referenced in the result plots. The error that corresponds 
to the mean square distance between test and model points will be indicated in square units. 
  
Fig. 3: Distribution of the starting positions 
Translation. The influence of translation and rotation on the matching shall be observed seperately. In a first experiment we 
keep the orientation unchanged and investigate how the convergence depends on the starting position. Figure 4 shows the 
final error of the matching for every starting point. A threshold, which is set at an error of one square unit, separates good and 
bad recognition cases (dashed line in figure 4). The black squares in figure 4 indicate the positions for which the matching 
failed. We see that the closest point matching algorithm can match the duplicate and the original for most of the starting 
positions. If we further investigate the positions for which the matching failed, we notice that the algorithm introduced a bad 
rotation at the first iteration step. This fact relies on the instable situation when many points in the test are coupled with only a 
few points in the model. 
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
error — z 
—————PM y 
M 2 3 uu: ; EMET 
el 4. BE 13[14]15] [22/23/24 
5E 
> "alo 16117 [18] [25/26/27 
3 
2 SUCCESS 
T E failure 
0 i position uU 
4 7 10 13 16 19 22 25 
Fig. 4: Results for translated corners 
Convergence. An iterative algorithm is only useful if it converges quickly. This criterion will be checked by observing the 
development of the error during the iteration. Figure 5 shows the measured matching error at successive iteration steps. 
Again the test objects are only translated. The errors are thresholded by appropriate limits at the iteration steps two, five and 
ten. At each step the result of the decision is the same and showed in the right half of figure 5. We observe that already after 
few iterations the successful cases with a minimum error can be extracted. This helps one to decide at an early stage of the 
iteration if a matching will be successful or not. Computing time can be saved because one does not really need an exact 
matching to find the interesting cases at the beginning of the object recognition procedure. 
IAPRS, Vol. 30, Part 5W1, ISPRS Intercommission Workshop "From Pixels to Sequences", Zurich, March 22-24 1995