ISPRS Commission II, Vol.34, Part 3A „Photogrammetric Computer Vision“, Graz, 2002 
  
REFERENCES 
ArcTron, 2002. http://www.arctron.de/ (accessed 13 March 
2002) 
Axelsson, P., 2000. DEM generation from laser scanner data 
using adaptive TIN models. International Archives of 
Photogrammetry and Remote Sensing, Vol. XXXIII, Part B4, 
Amsterdam, Netherlands. 
Borgefors, G., 1986. Distance Transformations in Digital 
Images, Computer Vision, Graphics and Image Processing, 
CVGIP 34 (3), pp. 344-371. 
Burmann, H., 2000. Adjustment of laser scanner data for 
correction of orientation errors. International Archives of 
Photogrammetry and Remote Sensing, Vol. XXXIII, 
Amsterdam, Netherlands. 
Elmqvist, M., Jungert, E., Lantz, F., Persson, A., Sódermann, 
U., 2001. Terrain Modelling and analysis using laser scanner 
data. International Archives of Photogrammetry and Remote 
Sensing, Volume XXXIV-3/WA, Annapolis, Maryland, USA. 
Filin, S., 2001. Recovery of systematic biases in laser altimeters 
using natural surfaces, International ^ Archives of 
Photogrammetry and Remote Sensing, Volume XXXIV-3/W4, 
Annapolis, Maryland, USA. 
Journel, A. G., Huijbregts, Ch. J., 1978. Mining Geostatistics. 
Acad. Press, New York. 
Kraus, K., 1998. Interpolation nach kleinsten Quadraten versus 
Krige-Schátzer. Osterreichische Zeitschrift für Vermessung & 
Geoinformation, 1. 
Kraus, K., 2000. Photogrammetrie Band 3. Topographische 
Informationssysteme. 1* ed., Dümmler Verlag, Kóln. 
Kraus, K., Pfeifer, N., 1998. Determination of terrain models in 
wooded areas with aerial laser scanner data. ISPRS Journal of 
Photogrammetry and Remote Sensing 53, pp. 193-203. 
Pfeifer, N., Stadler, P., Briese, Ch., 2001. Derivation of digital 
terrain models in the SCOP-- environment. Proceedings of 
OEEPE Workshop on Airborne  Laserscanning and 
Interferrometric SAR for Detailed Digital Terrain Models, 
Stockholm, Sweden. 
Riegl, 2002. http://www.riegl.com/ (accessed 1 July 2002) 
Schenk, T., 2001. Modeling and Recovering Systematic Errors 
in Airborne Laser Scanners. Proceedings of OEEPE Workshop 
on Airborne Laserscanning and Interferometric SAR for 
Detailed Digital Terrain Models, Stockholm, Sweden. 
Vosselmann, G., Maas, H., 2001. Adjustment and Filtering of 
Raw Laser Altimetry Data. Proceedings of OEEPE Workshop 
on Airborne Laserscanning and Interferometric SAR for 
Detailed Digital Terrain Models, Stockholm, Sweden. 
ACKNOWLEDGEMENTS 
This research has been supported by the Austrian Science 
Foundation (FWF) under Project No. P14083-MAT. 
APPENDIX A. ROBUST LINEAR PREDICTION 
In the following the basic functions for linear prediction with 
individual weights are presented. This means that a surface is 
interpolated from a cloud of points, but each point has an 
individual accuracy. It can be used for robust interpolation by 
modulating the weights (accuracies) depending on the filter 
value (negative residual) of the observations in an iterative 
manner. The filter values of one iteration are used to determine 
the weights for the next surface interpolation. More details can 
be found in (e.g. Kraus and Pfeifer, 1998 and Kraus, 2000). 
A.1 Linear prediction with individual weights 
Given are n points P; with the heights z, which have been 
reduced by subtracting a trend surface (e.g. a plane). After this 
reduction the expectancy of the observed heights is zero. The 
height z at a position P is determined by Eq. (1): 
z = clCz (1) 
with: 
e = (C(PP),C(PP,),...C(PP,))” (2) 
zoe numer (3) 
V. on C(DP) .. CPP) 
Cus E Ce. @ 
ZZPn 
The function C(P;P,) describes the covariance between two 
points on the surface in the following way (Gaussian model): 
LB y 
CP) = C0)e * (5) 
with: 
C(0) = covariance for a distance of zero 
PP, = horizontal Euclidian distance between the two 
surface points P; and P, 
C — factor (estimated from the given points) for 
controlling the steepness of the covariance function 
Vom, in Matrix C of Eq. (4) is the variance of the given points, 
which is the sum of C(0) and the variance of the measurement 
07. The points are considered to have the same 00° (a priori 
accuray), but different weights p;. The accuracy o? of a point P; 
is obtained from: 
  
2 
e s T (6) 
Pi 
The variance of each point P; can be computed by: 
V. zxc00ro? (7) 
zp; 
A.2 Robust weight function 
The weight p; depends on the filter value f, which is the 
oriented distance from the prior computed surface (in the first 
step all points are equally weighted) to the measured point. The 
weight function (a bell curve), which can have different 
parameters (a, b) for the left and right branch, is given by: 
1 
zn 8 
Po rea - 8) e 
f, — filter value 
— shift value, determined automatically from the filter 
values 
a = 1/h 
b = 4h-s 
with the half width value / and the slant s at the half weight. 
Additionally thresholds are used to set the weight of a point 
with a higher or lower filter value to zero so that it is excluded 
completely from the modelling process.