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Figure 5: 3-D representation of synthetic data points. 
4. The method must be suitable for automation. No hu- 
man interaction should be necessary to correct param- 
eters. 
9. Reasonable demand on computer resources, i.e. time, 
memory, and storage. 
Matching aerial images typically renders a large number of 
data points, especially at the finer resolutions. Therefore, 
we have excluded all methods of least square fitting by poly- 
nomials or splines because of computational considerations. 
These methods would lead to a huge system of equations (in 
the worst case is one equation per point). In addition, hav- 
ing sparse data increases the risk of deficiency in the normal 
equation. Fitting a surface by piecewise polynomials, fur- 
nished with proper triangulation algorithm, stands a better 
chance for more efficient and realistic surface interpolation. 
However, the user must identify the set of break lines prior 
to the interpolation. Otherwise, a peculiar surface represen- 
tation would be obtained. 
The methods of weighted average are better suited for han- 
dling sparse data. Besides, they do not introduce new global 
extrema in the surface. On the other hand, there is no es- 
  
     
   
    
  
   
  
     
        
    
   
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Figure 6: Surface interpolation by weighted average method. 
tablished automatic strategy for defining the data subset for 
a point. Another concern is the fact that no a priori infor- 
mation about break lines can be included. Therefore, the 
231 
value of a point is computed based on data across break 
lines, creating undesired artifacts. Figure 6 shows the result 
of applying the weighted average method on the test data. 
The interpolated surface cannot be considered realistic. 
None of these methods provides explicit information for sur- 
face analysis. This quite different for fitting a surface by a 
thin plate (or membrane). Adopting the analogy of a phys- 
ical model allows exploring the mechanics of such model. 
Mechanical concepts, such as stress and bending moments 
of a plate provide the means for detecting break lines. Both 
models of thin plate and membrane are capable of achiev- 
ing surface interpolation and break lines detection. Judging 
from figures 7 and 8, the membrane produces a more realistic 
surface that the thin plate model. 
      
      
    
    
     
   
     
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Figure 8: Surface interpolation by a membrane. 
Figure 7 represents the interpolated test data by a thin plate. 
The problem of over-shooting between data points is clearly 
noticeable. Figure 8 shows the interpolation by a membrane. 
Here, the problem is interpolating between high frequency 
features. This is avoided by using the weak continuity con- 
straints. Interpolation by a weak membrane is shown in fig- 
ure 9. The discontinuities are now detected during the sur- 
face interpolation. Figure 10 shows the detected break lines 
superimposed on the surface.