dont 
Jj; Ai) 
j SHEA 
imized, 
utliers, 
esiduals 
weights 
weights 
ails get 
res (or 
akdown 
12 ie. 
'ssed in 
cates a 
| to or 
weight 
normal 
"7 t 
eA or 
' parts, 
  
The number À could be increased until 20/3. preserv- 
ing the reliability of the observations, globally and locally, 
according to geodesists community suggestions (Benciolini 
et al. '82), if the amount of suspected outliers isn't too 
large. The breakdown point decreases, obviously, but not 
too much, so that the procedure continues to be effective. 
These methods are grouped together and generalized by 
means of the definition of the S-estimators. 
3. Examples 
The presented methods are already tested and discussed 
in the scientific literature by statisticians. Unfortunately 
whilst downweighting methods have been broadly studied 
by photogrammetrists too, since the last fifteen years (Ku- 
bik 80, Fôrstner '86), the redescending estimators seems to 
be not popular, but for the simple Hampel estimator. 
On the other hand, redescending estimators with a very 
high breakdown point have been recently introduced in 
the survey and mapping disciplines (see Carosio's research 
team: Wicki '92 a et b). Therefore some examples of pho- 
togrammetry and cartography are welcome, with the aim 
to spread out information. 
The most interesting examples in photogrammetry and 
cartography involve S-transformation, fitting and match- 
ing. The first two classes of examples are common between 
photogrammetry and cartography, whilst the last class of 
examples is central for photogrammetry and it constitutes 
the experimental conclusion of this paper. 
Image matching can be done in image space, as well as in 
object space, by using complanarity condition or collinear- 
ity equations, respectively, adding a grey level model and, 
eventually, an object model. As well known, the compla- 
narity conditions is one of the most critical examples, con- 
cerning well-conditioning, reliability and robustness. 
For these reasons, the relative orientation of a couple of 
images is adjusted, by using redescending estimators with a 
high breakdown point, where the amount of outliers ranges 
until m/[3, being M the number of observations. The data 
collect three series of observations, according to Ackermann 
suggestions (Ackermann ’79), in the canonical points, with 
different combinations of outliers. 
The outlier location shows 2, 4 and 6 outliers in a series 
of observations in the canonical points (see Figure 1), pre- 
serving the global and local reliability. Least squares and 
downweighting methods fail the adjustment, because their 
breakdown point is zero or too low. On the contrary, re- 
descending estimators with a very high breakdown point 
catch all outliers, in all combinations of them. 
147 
1st & 2nd series 3rd series 
  
  
® e e 
2 outliers ® ® 
® ® 
  
  
  
  
® e 
e e 
@ ® 
6 outliers ® e : : 
e e 
Fig. 1 
The strategy of application of the presented procedure (the 
  
  
  
  
  
  
  
  
  
  
same of Barbarella, Mussio '85) is an adjustment of the best 
observations, after a preliminar least squares adjustment. 
Successively the suspected outliers, which don't show blun- 
ders, leverages or small outliers, are forward accepted by 
using the Hawkins test: 
Ho : P(HÍ?(oJ2) « H, € HU? (14a/2) 21-0 
being V = | — n the degrees of freedom, where | € m the 
number of observations actually processed at the present 
step of adjustment (remember, Mm is the number of obser- 
vations), and 1 the number of unknowns parameters. 
The expected value H, is computed, as follows: 
being 0; the residuals, T; the recursive residuals, 62 the 
variances of the residuals, óg the squares sigma zero and V 
the degrees of freedom. 
The critical values, for a parametric test on two sides, are 
derived from the Hawkins probability distribution (Hawkins 
'80), defined as follows: 
H, = maz((x,),)/x, 
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B1. Vienna 1996