e CO- 
f the 
ation 
ue to 
on of 
t any 
995 
  
153 
A 
Qu 0 
Co(apriori) 
4 ni (t 
Jn ook 
The partial redundancies are called r;. The values for d and K are determined with the help of Q. 
i = 
3) Proposal of Huber: 
Vi or vj| «c 
d R vi) T El 2e e 
The value of c can be chosen by the user. 
4) Proposal of Hampel: 
vj for [vi] <a 
a-sign(v;) for a <|vi| <b 
nens a (v -c-sign(v;))/(b - c) for b «l|vilsc e 
0 for vil >C 
  
The values of a,b,c can be chosen by the user. 
Iterative reweighting procedures with objective function orientation 
The methods with objective function orientation have to be distinguished from the above mentioned trial and error methods 
because they describe an alternative algorithmic procedure to the method of linear programming. In particular there is 
1) Approximation of L; -norm-method: 
pi(t+1)=1/|v;(t)+c| with c » 0 m 
2) Lp-norm-estimates with 1<p<2: 
pitt +D =1/(jy ff +c) withec>0;1< p<2 (8) 
3.2. Least Median of Squares Procedure 
One further robust estimation method is the Least Median of Squares (LMS) procedure (ref. Rousseeuw, Leroy 1987), which 
minimizes the median of the square sum of the residuals. Die objective function is given by: 
median; (v2) — min (9) 
With this strategy Rousseeuw was able to equalize the influence of the geometry of the observations. Unfavourable geometries can 
led to the case that single observations become restrictions and influence the parameter estimation result like leverage points (ref. 
chapter 4). 
Besides the high computation time the LMS-method can generate wrong results. An example out of the literature (table 1) (ref. 
Fellbaum 1994) demonstrates this fact. Through the introduction of a single error in the y-co-ordinate of point 2 the LMS estimator 
breaks down. The error is not detectable while looking at the residuals. The parameter estimation was done with the Program 
PROGRESS (Program for Robust reGRESSion) (Rousseeuw, Leroy 1987). 
The example shows clearly that the LMS-method is barely suitable for error detection. In this case even the non robust L;-norm- 
method was able to detect the error. Therefore a different strategy for equalization of the geometry of the observations is used in 
the next chapter. 
IAPRS, Vol. 30, Part 5W1, ISPRS Intercommission Workshop “From Pixels to Sequences”, Zurich, March 22-24 1995