r 
Fig. 3 Representation of a straight edge segment 
Each edge segment with start point S and end point E is 
described in a local (u, v) coordinate system which lies near 
the principle axis of the edgels. In this local coordinate sy- 
stem the covariance matrix of the straight line parameters m 
and b of v 2 m: u- bis diagonal and can be derived from the 
extraction process by error propagation. The v-coordinates 
of the end points of an edge element will be correlated due 
to the common factor m and will have following variances 
[FORSTNER W. 91]: 
2 2 2 42 
= 0p t+ uyo,, 
vA 
2552552 2.7 
Ope = 04T UBER (1) 
2 = 7 É 2 
Fave: =: 06 TUAUEC, 
The u-coordinates of the two end points can be treated as 
uncorrelated because the accuracy in u-direction is mainly 
influenced by roundig errors. Therefore the uncertainty of the 
the u-coorinates could be represented by having the variance 
~ 1/12[pizel], or any other reasonable variance takeing the 
edge extraction process into account. 
Thus the covariance matrix and especially the weight matrix 
of an edge segment in the local coordinate system (u,v) will 
have the following structure: 
gl : 0 0 0 
0 2 0 c 
yw) = VA ” VAVE (2) 
0 0 0s, 0 
Ü qua XU gs 
wit 0 0 0 
2 
(u,v) 0 w, 0 Wu 
NW à 0 =! 0 3) 
0 Won U we 
Transforming the edge segment back into the image coordi- 
nate system (r, c) yields the covariance and weighting matrix 
yino) = yw) s RT (4) 
wre) =n. wo» 3 RT (5) 
with the rotation matrix 
-f7fnR, Q0 
Rr ( 0: R, ) (6) 
The rotation matrix R is only depending on the individual 
direction $ of the edge segment in the image coordinate sy- 
stem. 
This kind of representation of the uncertainty of an image 
edge segment does not depend on whether the start or end 
point are longitudinally linked to the model edge. The edge 
segments are treated as units holding the line information 
with possibly undefined start and (or) end point position. 
594 
This enables handling of different cases of matching an image 
edge to a model edge shown in Fig. 2. 
Homologous lines having different length now can be repre- 
sented by treating their end point pairs like normal points, 
but with a joint covariance or weight matrix resp. . The spa- 
cial resection is based on these properly weighted end point 
pairs. One simply had to set the weights (cf. equation ??) se- 
perately for the start and end point correspondencies. For 
example case d in Fig. 2 could be represented by setting w,, 
and w,, to zero. 
This favourable property will be used in the robust estima- 
tion (cf. section 3.3). At the beginnig all weights are set 
corresponding to case a) (hypothese : complet match). The 
weights will be iterativetly modified in the robust estimation 
realizing the partial matches. 
3.3 Determining the Final Position via 
Robust Estimation 
The clustering procedure is done seperately for each control 
point model in the aerial image leading to a list of possibly 
corresponding straight line segments for the whole image, 
i.e. for all control points. In addition to the partial matches 
mentioned in section 3.2 (wrong start and end point corre- 
spondencies), this list contains two kinds of gross errors. The 
first one is due to the design of the conservative test in that 
matching procedure which results in wrong edge correspon- 
dencies though the control point model was located correctly. 
The second and more severe error is a partially or completely 
incorrect location due to a weak model or weak image infor- 
mation. Both error sources lead to wrong correspondencies 
between image and model edge. 
To clean these inconsistencies a robust estimation [HAMPEL 
86 / HUBER P. J. 81] is applied for the final common fit 
of the control point models and the evaluation of the orien- 
tation parameters. Wrong correspondencies are interpreted 
as outliers or as observations with large deviations from the 
mean value. 
The used robust estimation, which is a ML-type estimator, 
is able to cope with outliers up to 3096. In comparison whith 
least square (LS) technique which minimizes 
f 2 X ej?wj (7) 
with the weights w; — zr 
one has to minimize a less increasing function 7 
0 - 2 T (e;/w;) (8) 
of the normalized residuals leading to the normal equations 
Y rn (e; wi) Ai; W; = 0 (9) 
with r'(z) being the derivative of the minimum function 
7(x). 
There are two ways to modify a standart LS technique to 
solve these systems (eq. 8 and 9). One could use adap- 
ted weights or adapted residuals, both leading to an itea- 
tive LS procedure. Here the method of modified weights is 
used because this simplyfies the handling of the partial mat- 
ches mentioned in section 3.2. Using a weight function 
f(t) = m(t)/t and the following iteration scheme for calcu- 
lating the weights for the k-th iteration