Peter Lohmann 
  
In the linear prediction values u; are estimated from the measurements z and the described covariance information 
contained in C. The interpolated surface in point P; is given by: 
Sc CZ 3) 
5 =JC(RR) C(RR) -- CR 
V, CBP) - CAP) 
Cz CAP) Ve (BE) U; predicted value 
= ; "e ; c covariances between point to 
be interpolated and 
[CFA ) CAR) Va measurements 
E c covariance matrix 
71 Z vector of centered 
z measurements 
2 
zZ = 
  
The vector c contains the covariances between the point P; and the other measurements. The matrix C contains the 
covariances between the measurements, the main diagonal consist of the variance V.. = 1.0 of the centered 
measurements. All measurements are regarded to be of equal accuracy. Since the vertex-value of the signal-covariance- 
function C(0) is restricted to 0.99, interpolation and filtering is performed. 
The area of investigation is devided into meshes of equal size. While processing the points of one mesh (processing unit 
= 1 mesh) the bordering meshes are also considered (8 surrounding meshes). The coefficients ay, a;, a; are computed by 
the measurements of the height-points P; within the 9 meshes using a least-squares adjustment. This means the moving 
plane is adjusted to all points within the area of consideration (j = 1,...,n; n ^ number of points within the area of 
consideration). 
Two tolerance factors are defined in the dialogue between DTMCOR and the user. First the user enters a tolerance- 
factor /,, with respect to the moving plane. The height values whose deviations are above this tolerance are considered 
as outliers and are excluded. The trend-splitting is enforced iteratively, i.e. repeated several times, until no more height 
value is rejected. The height-outliers in the processing unit are deleted. The values as such recognized in the bordering 
meshes are available for computation of the next mesh again. 
The second tolerance-factor /,,, checks the difference between centered measurement value z; and value calculated by 
prediction u;. If the residuals z; are bigger than the predefined factor, these points are also eliminated. 
i =z; —U; (4) 
4. DUAL RANK FILTERING 
One method based on morphological filtering for the elimination of points above ground in a laser scanner data set has 
been implemented within a standard image processing software, the HALCON package, which offers a library of 
functions (operators) and a programming environment called Hdevelop (MVTec 1998). 
Points above ground can be characterized by a large change of the height at the transition from the ground to the object 
and vice-versa (local discontinuity). This is quite obvious at buildings, where the outline of such height-alterations 
reflects the shape of the building. But also within vegetation such large height-alterations often appear. 
The impementation is established by a series of modules in a processing chain within the HALCON environment. In 
order to detect the changes in height, a standard deviation filter is applied in a first step to the image. A small mask was 
used for the filter in order to obtain sharp edges. A size of 3x3 pixel was supposed to be sufficient. The filter generates a 
new image based on the value of the standard deviation within the mask (see Figure 2). Next a threshold is applied to 
the standard-deviation image in order to find the areas of large changes in height. The necessary threshold-value, 
however, can not be calculated precisely beforehand, because of the à priori unknown structure of the terrain. Therefore, 
in the beginning it is necessary, to determine this value on the basis of a test-window, which contains only true ground- 
  
542 International Archives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part B3. Amsterdam 2000. 
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