Herbert Jahn 
  
Figure 3 shows the resolution in case of double point resolution (distance between the two spots with respect to the 
pixel size) as a function of the PSF parameter o: Here it was assumed that the used optics has a f-number = 4. Then for 
the diffraction limited optics the equivalent o is lum (see Jahn, Reulke, 1998) and for the real optics we have g - 
2.6um (calculated from Schlienger, 1996). 
In case of neglectable PSF the limit for the double point accuracy (3) is valid. With increasing PSF-width opsp 
especially for the staggered arrays the MRD also increases, while the resolution for single arrays keeps constant. We 
find the minimal MRD for staggered arrays in case of diffraction limited optics. Increasing Ops, further the MRD be. 
comes worse. For the used optics the resolution is in the same range as in the ideal case. The comparison of the MRD's 
of all different line types is shown in Figure 4. 
  
  
   
  
  
  
  
Figure 4. double point resolution for the different line types 
The normalized MRD of the three different CCD-lines for the width Ops; of the used optics is 
a] 2516 2514.2. (11) 
ô ó ó 
The resolution of the staggered line is about 9096 of the MRD of the linear 24k line! 
The accuracy investigation for the single point can be performed with the signal-calculation (9). Because there is no 
resolution criterion like the Rayleigh approach, we are looking for the error of the image point determination. For that 
purpose a simple interpolation algorithm is used. 
After calculating the signal, a Gaussian will be fitted to the signal. From this fit, the point position can be derived. The 
error in the point position is a measure of the resolution. The interpolation is only possible for a Ops, greater than a 
minimum value (e.g. the diffraction limit). For smaller values of Ops; there are not enough points for interpolation 
(e.g. in the case of vanishing PSF and single arrays only one pixel generates a signal, which does not allow the 
application of an interpolation algorithm). With the application of the interpolation algorithm the location error € or the 
MRD decrease dramatically. Only in the case of a small PSF the phase dependence generates a small error. The 
location error € is much smaller than a tenth of a pixel. The MRD for the three cases is 
MRD(12k) > MRD(2-12k,aggerea) > MRD(24K) or &» : > a (12) 
Fora PSF of the real optical system no interpolation error can be recognized. That means that all information can be 
retrieved from the signal. This means that the sampling theorem is fulfilled. 
The introduction of an interpolation approach is a essential for the resolution considerations. If the sampling theorem is 
not fulfilled, errors in parameter estimation occur. If the PSF-width increases, the parameters become more accurate. 
This approach is in contradiction to the classical Rayleigh criterion and will be explained in the following chapter more 
in detail. 
3 RESOLUTION AND THE SAMPLING THEOREM 
If a continuous signal has a band limited Fourier transform then the signal can be uniquely reconstructed without errors 
from equally spaced samples /(iA) (-e» « i « oo) if the spatial sampling or Nyquist frequency 
  
168 International Archives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part Bl. Amsterdam 2000. 
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