The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences. Vol. XXXVII. Part B5. Beijing 2008
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Figure 4: Data of the Riegl distance experiment series.
Each bullet represents the data of one target at a certain
range and k (product of reflectivity and cosine of
incidence angle). The vertical axis is the mean intensity
per target. Targets are grouped by distance, and for each
distance the least squares fit 3 rd order polynomial is
shown.
0 10 20 30 40 50
Figure 5: Polynomial coefficients (fat black lines) of the
functions of Figure 4, i.e. cubic polynomials in the
monome form. These polygons are approximated by
cubic polynomials each (thin gray lines), split for
ranges below and above 15m.
Experiment
series
g4-
Model
ct 0 abs.,
do rel.
Distance
Cubic
0.00297, 1%
0.00255
Distance
Log.
0.00659, 2%
0.00614
Angle
Cubic
0.00241
5% better
Angle
Log.
0.00656
7% worse
Table 6: Precision values for the different data driven
models (2 nd col.) of the Riegl data. “ct 0 abs.” is the a-
posteriori precision (Eq. 5), and “ct 0 rel.” is the portion
of ct 0 on the maximum mean target intensity. The 4 th col.
shows the r.m.s. of the residuals, and the results can be
compared between the distance experiment, used for
parameter estimation, and the angle experiment (5 th
col.).
The coefficients of the cubic polynomials for each distance step
can also be regarded as discrete observations of curves: one
curve for the constant term, one for the linear, etc. The vertices
of these curves can again be used for fitting a curve model. If a
cubic polynomial is chosen, then the resulting patch is bi-cubic.
For g 4 not only cubics, but the following functions were tested.
• Logrithmic function, with constant offset and scale factor.
This is suitable, if the conversion from optical to electronic
power is a logarithmic amplification.
• Linear scaling, suitable if received optical power and
intensity value are directly proportional, termed “Scale”
below.
• Linear function: like “Scale”, but adding a constant offset, e.g.
background noise.
• Cubic polynomial, providing flexibility but still over
determination in the estimation procedure.
For g 5 always cubic polynomials were used, as the data does
not follow the Lidar equation (Figure 3). However, some
similarity of the Optech data to the curve shape of Figure 1 can
be noted.
Next, the results of the different models will be presented. Only
those models describing the data reasonably well are presented.
Also the Optech data had to be split into two patches for ranges
above and below 18m. As it can be seen in Figure 5, the data
cannot be approximated well for the last distance step (50m).
These measurements were excluded from further analysis.
The model is determined from the range series as explained.
The angle series can then be used to verify the suitability of the
model. It should be kept in mind, that the mean intensity values
of the Riegl angle experiment data drop below the lowest mean
intensities of the distance experiment (0.1156 vs. 0.1207). Thus,
there is a certain amount of extrapolation with respect to k. For
the Optech data this is not the case. The Riegl results are
summarized in Table 6, the Optech results in Table 7.
Experiment
g4
Co abs, (T 0 rel
Distance
Cubic
0.00218, 1%
0.00180
Distance
Linear
0.00613,3%
0.00508
Distance
Scale
0.00681,3%
0.00564
Angle
Cubic
0.00814
452% worse
Angle
Linear
0.00716
41% worse
Angle
Scale
0.00765
36% worse
Table 7: Precision values for different data driven models
for the Optech data. Columns as in Table 6.
6. DISCUSSION
With the data acquired in the previous experiment (Pfeifer et al.,
2007) with the Riegl scanner the precision a was 0.0108. The
precision obtained now for the separation approach is
comparable and approximately worse by 10%. The parameters
also changed, which is attributed to the new data distribution in
the experiments. Nonetheless, the behaviour of the device,
namely decrease and increase of intensity depending on the
range with a minimum at ~15m was the same. The precision
obtained with the new data driven approaches are much higher
and the separation method will not be discussed further.
The two scanners show different, i.e. opposite, behaviour of
intensity vs. range. Both deviate strongly from the pure Lidar
equation, showing that it does not hold for the distances
investigated. To reach a better agreement the model would have
to be extended, depending on the system design. The effect of
overlapping footprints, simulated in Sec. 3, appears to be
similar to the function shown for the Optech scanner in Figure 3.
For the Riegl scanner the situation is more complicated, and
currently we do not have a good explanation. However, this
behaviour was also found in Pfeifer et al. (2007) and the
stability of the device could be confirmed.