ISPRS Commission II, Vol.34, Part 3A „Photogrammetric Computer Vision“, Graz, 2002
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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.