The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences. Voi. XXXVII. Part Bl. Beijing 2008 
points, which need not be conjugated. In order to compensate 
for the non-correspondence between the line end points, we will 
introduce the necessary constraints to describe the fact that the 
line segment from the LiDAR strip 2 coincides with the 
conjugate segment from the overlapping LiDAR strip 1 after 
applying the 3D rigid-body transformation (Figure 4). The 
mathematical representation of this constraint for these points is 
shown in Equation 2. 
(iW.-rJ .OWi-r.) 
(Z r + z, — Z A ) (Z r + z 2 —Z A ) 
Where: 
*1 
T, 
— ^(n,<D,K) 
1 
1 
and, 
*2 
y 2 
— ^(n,d>,K) 
1 
1 
_*1_ 
L Z .J 
. Z 2_ 
XJ 
~x r ~ 
X" 
XT 
1 
Y T 
Z T 
1 
JN Jrc 
i 
1 
. X . ^ 
N 
1 
+ Z 1 
y b -y a 
Z B —Za _ 
(2a) 
~X T ~ 
X" 
XT 
~x B -x; 
(2b) 
y r 
Z T 
+ -^(n,<D,K) 
_ Z 2 _ 
Ya 
, Z A_ 
+ Z 2 
1 
^ N 
I I 
. OQ oq 
^ M 
1 
Where: 
(.X T ,Y T ,Z T ) T is the translation vector between the strips. 
R(0. d> K) re q uired rotation matrix for the co-alignment of 
the strips, and Aj and /L, are the scale factors. 
4.3 Noise Level Verification 
So far, we have introduced a methodology for detecting 
systematic errors in the data acquisition system. It is important 
to emphasise that for a LiDAR system with only random errors, 
the estimated transformation parameters should be zeros for the 
translation and rotation parameters. In other words, the 
expected values will not change with varying noise levels in the 
LiDAR point cloud. In this section, we are interested in a 
similarity measure for the evaluation of the noise level in the 
data. 
In this work, the noise level in the LiDAR data will be 
evaluated by quantifying the goodness of fit between conjugate 
primitives after removing existing discrepancies between 
overlapping strips. This can be accomplished by computing the 
average normal distance between conjugate linear features after 
applying the estimated transformation parameters. 
By subtracting Equation 2a from Equation 2b, and eliminating 
the scale factors (by dividing the first two rows by the third one 
in order) result in Equation 3 which relates the rotation 
elements of the transformation to the coordinates of the points 
defining the line segments. 
(x,-x A ) r <i (x 1 -x,)+r ì2 (y 2 -y ì )+r ì2 (z 2 -z ì ) 
(Z, -Z,) r»(x 2 -x,)+r 22 (y 2 -y,)+r 22 (z 2 -z,) 
(Y„-Y a ) R 2 ,(X 2 -X^R n (Y 2 -Y,) + R 2i (Z 2 -Z t ) 
(Z, ~Z, ) R„(X 2 -X,)+R, 2 (Y 2 - Y,)+ R n (Z 2 — Z,) 
The estimation of the two rotation angles (the azimuth, and the 
pitch angle along the line) is possible by writing the equations 
3a and 3b for a pair of conjugate line segments. On the other 
hand, the roll angle across the line cannot be estimated. 
Therefore, a minimum of two non-parallel lines is needed to 
recover the three elements of the rotation matrix (Q,0,K). To 
allow for the estimation of the translation parameters, the terms 
in equations 2a and 2b are re-arranged and the scale factors 
eliminated (by dividing the first two rows by the third one) to 
come up with Equation 4. Overall, to recover all six parameters 
of the transformation function, a minimum of two non-coplanar 
line segments is required. For a complete description of this 
approach refer to Habib et al., 2004. 
{X T+ x { -X A ) {X T+ x 2 -X A ) 
(Z T + z, —Z A ) (Z T + z 2 —Z A ) 
(4) 
5. EXPERIMENTAL RESULTS 
To evaluate the validity of the proposed QC methodology, 
experiments were performed using a real dataset. The dataset 
used in the experiments covers and urban area consisting of 
three strips as shown in Figure 5. The specifications of this 
dataset are shown in Table 1. 
Figure 5. Dataset used in the experiments 
Optech 2050 
-1000 m 
-0.75 m 
3 strips 
50 cm 
15 cm 
Sensor Model 
Flying Height 
Ground Point Spacing 
1 survey day 
Horizontal accuracy 
Vertical accuracy 
Table 1. Dataset specifications 
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