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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