Full text: Proceedings, XXth congress (Part 5)

International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B5. Istanbul 2004 
  
Plumb Line 
Sampling 
Nl Profile Line 
Circular Laser 
bert Beam Footprint 
     
  
Position of Range 
Measurement is 
Somewhere in 
Overlap Region 
Apparent Position of 
Range Measurement 
Sample Locations 
Figure 2. Beamwidth positional uncertainty. 
3. THE AMTF MODELLING APPROACH 
3.1 Equal angular increment sampling 
A three-dimensional scan of a scene can be compiled by 
mechanically deflecting the rangefinder laser beam in equal 
increments of arc in horizontal and vertical planes. A scanned 
scene can thus be parameterised in terms of range, p, as a 
uniformly sampled function of two independent variables: 
horizontal direction, ©, and elevation angle, ot, 
p. (86.0) S Y o. (mA. nA (8 - mA, o. nA) ; (1) 
m=-æ n=-0 
where p, is the sampled representation of the continuous scene, 
Pc. and & in this context represents the Dirac delta function. 
Note that a single sampling interval, A, has been assumed for 
both co-ordinate dimensions. 
3.2 Ensemble average functions 
The sampled representation of a scene given by Equation 1 is 
dependent upon the scene phase and, thus, is not shift invariant. 
To cope with this for digital imaging systems, Park et al. (1984) 
define the concepts of average system point spread function 
(PSF) and the average system optical transfer function. The 
average system PSF is an ensemble average function of 
randomly located point sources under the assumption that the 
independent variables are uniformly distributed on the sampling 
interval (Park et al, 1984). This permits application of 
modulation transfer function (MTF) analysis—restricted by 
definition to linear shift-invariant systems—to sampled imaging 
systems (Boreman, 2001). In this paper, the average MTF 
concept is applied to model both the sampling process and the 
laser beamwidth in order to derive a measure that accurately 
quantifies laser scanner angular resolution. 
3.3 Scanner sampling AMTF 
In the context of laser scanning, the average PSF concept is 
used to model the ensemble of possible random angular phase 
shifts of a scanned scene. Taking the average over one square 
(i.e. A x A) of the sampling lattice, in which the probability 
distribution is assumed to be uniform. the resulting sampling 
average PSF, APSF, is given by: 
  
APSE (8,0.)= T. 2) 
0 otherwise 
The corresponding MTF is given by the modulus of the average 
PSF’s 2D Fourier transform: 
sin(xAu) sin(xAv) 
AMTE v)= TAL ~~ RAV 
: (3) 
  
  
where u and v are the respective horizontal and vertical 
(angular) spatial frequency domain variables. Note that 
although a square sampling lattice has been assumed, the 
formulation is easily modified to accommodate unequal 
horizontal and vertical sampling periods or other sampling 
geometries (e.g., hexagonal). Though the present analysis is 
restricted to the directions of the co-ordinate axes, (0 and a; u 
and v), attention is drawn to the fact that the resolution 
measures derived herein are not applicable in other directions 
due to the angular dependence of AMTF; (Hadar et al., 1997). 
3.4 Scanner beamwidth AMTF 
For the beamwidth resolution model, the probability governing 
the angular position of a range measurement is assumed to be 
uniform over the projected laser footprint. Note that this does 
not refer to the irradiance distribution within the cross-section, 
which is typically Gaussian. Also, to keep the model generic, a 
beam diameter definition has not been specified, though the 
e? definition is most common. Integration over a uniform 
circular region with diameter 8 yields the beamwidth average 
PSF, APSF, 
  
4 2 2 e 
; 0 ro «— 
To" 4 
APSE, (8,0.)= ; (4) 
0 otherwise 
The corresponding circular beamwidth AMTF is given by: 
2J,| n6 u^ v^ 
neu v? 
AMTF, (u.v)= (5) 
where J, is the first order Bessel function of the first kind. 
Though a circular beam cross-section has been assumed, 
    
   
   
   
     
    
  
  
   
  
  
  
  
  
  
   
   
  
  
   
  
  
  
   
   
   
   
   
   
  
  
  
  
  
  
  
   
   
  
   
   
    
   
    
    
  
   
   
    
  
  
  
   
   
   
  
    
   
     
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