The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences. Vol. XXXVII. Part Bl. Beijing 2008
expanded along the plane, and the modified weight matrices are
calculated by the orientation of the plane and the expanded
error ellipse. In Equation 4, Xuvw is the modified variance-
covariance matrix of the selected point on the plane, and the
relationship between XYZ and UVW spaces is explained by the
rotation matrix R, which is derived from plane equation
parameters. As shown in Figure 5, the direction of W axis is
parallel to the normal vector of the plane, and Mjj and M v are
applied to the variance-covariance matrix to expand the error
ellipse along the plane. At last, the modified weight matrix P is
derived using the R and Zuvw matrices as shown in Equation
4. The system calibration parameters are obtained by a least
square adjustment procedure with the modified weight matrix.
This approach is sequentially described using the flow chart in
Figure 6. The main advantage of this procedure is having a
single algorithm for simultaneously handling conjugate points
and planar patches, if available. The limitation of this approach
is its dependency on the presence of planar patches in the
available data.
Figure 5. Conjugate planar patches and expanded error ellipse
p ' = &xyz) 1 = (R T I^uvw R ) 1
7 u + M u
a uv
°uw
^vu
<7y + My
<jyw
a WU
(Jwv
a W
(4)
parameters). The next section outlines the optimal configuration
of the control patches for a reliable estimation of the calibration
parameters, while avoiding possible correlations among these
parameters.
2.4 Optimum Configuration of Planar Patches
When using planar patches as alternative primitives, one should
give attention to the optimum configuration of planar patches
because that condition is the one that yields an accurate
estimate of the parameters while avoiding any possible
correlations among them. In general, we do not expect that
significant errors exist in the directly measured spatial offsets
between the GNSS/INS and laser scanner of a LiDAR system.
However, if we can de-couple the spatial and rotational offsets
relating these components, we can simultaneously estimate the
angular and spatial bore-sighting parameters. For the control
patches, the ideal configuration is shown in Figure 7.a, which
illustrates orthogonal patches in the XY, XZ, and YZ planes.
Unfortunately this situation is not realistic (i.e., it is not always
guaranteed that such a configuration is available in a LiDAR
data captured from an airborne system).
A more realistic planar patch configuration is shown in Figure
7.b. For this configuration, horizontal and sloping planar
patches are used for the calibration process. It is important to
have sloping planar patches with different aspects (e.g., some of
the patches can be parallel to the X-axis while others are
parallel to the 7-axis).
To test the performance of this configuration together with the
impact of the slope of such patches, we simulated a LiDAR
strip using a linear scanner system at 1,500 m flying height with
25 degree scan angle. The simulation process started with a
surface model and system trajectory. Using such information,
we produced synthetic LiDAR data including system raw
measurement such as GNSS, INS, and laser scanner
measurements, which were then used to estimate the bore
sighting parameters. After preparing simulated LiDAR data, the
discrepancies were analyzed to compare the original and re
constructed surfaces using system parameters and raw
measurements.
Figure 7. Optimal (a) and realistic (b) planar patches for the
LiDAR system calibration
Figure 6. The calibration procedure using planar patches and
modified weight matrices
To ensure a reliable estimation of the system parameters, the
utilized planar patches must represent sloping surfaces with
different aspects (i.e., to avoid possible correlation among
Figure 8 shows the accuracy of the reconstructed coordinates
using the recovered bore-sighting parameters from well-
distributed 5 control patches along the LiDAR swath with
varying slopes.
