closest point §; = min 18 992951 is the conjugate point in 
Figure 1(c). 
2.3 Precision and Reliability 
Precision and reliability are the two basic factors for quality 
analysis of adjustment systems. The theoretical precision of 
unknown parameters and the correlation coefficient matrix are 
also an important basis for the procedure of least squares 
solution (Li and Yuan, 2002). The theoretical precision O; 
and the correlation coefficient can be estimated from a co- 
factor matrix of unknown parameters. 
0, = 000; » Ju E Qu 7 (4'PASP)" (14) 
To detect the gross error of the observation, a simple and 
efficient weight function is used in our robust estimation 
routine. 
0 |v,|» Ko, 
= (15) 
lL |v ds Ko, 
In equation (15), when the gross observed value is detected, its 
weight will be set to zero P — () , in other case, the weight of 
the observed value will be set to one P =] In our 
experiment, when the constant K is set to 6 or 7, the 
adjustment system has a good performance to estimate the 
unknown parameters. 
2.4 Compared with Existing Methods 
The procedure of LS3D proposed above is separated into two 
parts, the adjustment model and the conjugate points rule. The 
adjustment model can be derived form the formula of the 
Euclidean distance. So, it is easy to adapt to new conjugate 
points rules and good for some special applications. In this 
section, we show the differences between our method and 
existing methods. 
* compared with Gruen's method 
The most important factors for adjustment process are the 
coefficient items and constant item of the error equation. Gruen 
and Akca (2005) derived the error coefficients under certain 
assumptions and lacking of rigorous mathematical formula. 
They proposed that the vector 
[0D / 0t,, 0D / 0t, , OD / Ot,] is only related with the 
coefficients of a local triangle plane. Hence, if the matching 
point does locate the same triangle, the vector values are 
constant. But under the rigorous derivation of our formula, the 
vector [OD / 0t,, 0D / Ot, , OD / Ot, Y may be changed, 
even if the local triangle has not changed during the iteration 
procedure and the vector values are related with the three 
coordinate components of current conjugate points. Another 
difference is the constant item of error equation. In Gruen’s 
method, they define the conjugate points distance as constant 
item directly, but we use the square value because of the 
smaller amount of computation. Especially, we found Gruen's 
method maybe need quite good approximations and we are less 
after similar iteration times, under the same priori weights of 
the unknown parameters. 
* compared with ICP method 
ICP method is a linear squares solution for estimation of the 6- 
parameters between two point sets. So, ICP needs a relatively 
    
high number of iterations in some tests (Pottmann et al. 2004). 
Another difference is that conjugate points of LS3D can be 
obtained using interpolation on a 3D surface but ICP needs real 
points. So LS3D can achieve higher accuracy in many cases, 
especially in the co-registration routine of different resolution 
point clouds. 
* compared with Rosenholm's method 
Rosenholm's adjustment model can be considered as the 
special form of our approach. If we define the conjugate points 
as Figure 1(c), the equation (5) can be derived to 
Dz(z—z Ne , Which is the same as Rosenholm's model. In 
many cases, the registration accuracy of this model is limited, 
because it can only meet the discrepancy in z direction of two 
point sets. 
3. LS3D AND STRIP ADJUSTMENT 
Strip adjustment is a relevant problem for the post-processing 
of airborne laser scanner data. 3D surface matching is a typical 
data-driven method of strip adjustment. The transformation 
parameters of the adjacent strips can be estimated by a least 
squares routine. Each strip can be seen as a single surface, and 
the conjugate points can be interpolated by one of the finite 
element methods discussed in section 2.2. 
  
  
  
* m 
&. * 
* * 
x wl 
(a) (b) 
Figure 2. Surface matching for laser scanning strips: (a) 
overlapping area on conjugate surfaces, (b) Estimating 
transformation parameters with conjugate points. 
  
In this work, we are aimed to use primitives, which can be 
derived with minimal pre-processing of the original laser 
scanner point clouds. To satisfy the 3D surface matching 
procedure, we chose one strip of the original points, while the 
other strip is represented by a triangulated irregular network 
TIN). 
  
  
  
Figure 3. Interpolation of conjugate points between 
template surface and searching surface in LS3D routine. 
  
  
  
In Figure 3, ¢ can be interpolated by the vertex coordinates 
of local finite elements in Tri( A, D, C) : 
4. EXPERIMENT RESULTS 
Two practical examples are given to show the capabilities of 
our proposed method. All experiments were carried out using 
software based on C code that runs on a MS Windows 
operating system. In order to increase the accuracy of 
conjugate points between adjacent strips, it is necessary to 
label terrain points and off-terrain points by a fast filtering