International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B3. Istanbul 2004 
  
Assuming that the line 1 — x, x x, is sufficiently parallel to the 
reference 1 = X; X X., it approximately holds 
d 
2 
E(x: —x, | + |X —x.]) 
zE(:-x |) + EU Fe —x, D 
  
and in a symmetric situation, 1. e. 
  
) = E(x. — x, |), 
—e 
it holds 
  
ó denoting the expectation value of the one-sided shortening at 
each end point of a line or edge segment. 
Given N observed end points x1,...,xw of linear features with 
their reference points xy, .. ., Xn, the mean one-sided shortening 
ó and the variance 05, of the one-sided shortenings ôn =| En — 
Xn | may be estimated from 
= 1 N 
à = N SN Ön (1) 
and 
n=l 
4.1.3 Noise characteristics of point extraction. The noise 
sensitivity of point extraction algorithms may be characterized in 
terms of the quality of the point localization under varying image 
noise, quantified with the bias 5. — b(o2) and covariance ma- 
trix X..(02) of extracted points dependent on the image noise 
variance c2 S 
Given © independent point observations x1, . . . , x x of equal pre- 
cision, an estimate x for true point x is given by the mean 
N 
A ] 5 
X = N Xn (3) 
n=} 
and the bias b of the observations and their covariance matrix 
51,4, may then be estimated from (cf. (Luxen, 2003)) 
^ E = ~~ Er 1 = =F 
b-s—£ wd Xx i S Gs -£x4.--£)' (4) 
7i 
4.2 Test procedures 
With a calibrated digital camera Kodak DCS 460, images of a 
polyhedral object were taken from 46 different perspectives, cf. 
fig. 4. All images were corrected referring to distortion. 
  
Figure 4: Images of a polyhedral object (sample). Image size: 
2036 x 3060 [pel]. 
4.2.1 Characterizing the shortening of linear features The 
shortening of linear features provided by the feature extraction 
software FEX is investigated by comparing the end points of ex- 
tracted straight lines and edges with ground truth resulting from 
the multiple view approach. 
Reference data estimation by multiple view approach. In case 
of a precise polyhedral object, the end points of straight line and 
edge segments coincide with imaged object corners, and refer- 
ence values for the image coordinates of object corners can be 
considered as reference for the end points of straight line and 
edge segments. 
Therefore, to estimate reference data, the object corners were ex- 
tracted from each image using the corner extraction proposed by 
(Forstner and Giilch, 1987). Based on approximate values for 
the image orientations as well as for the object coordinates, the 
point correspondence problem was solved and spurious features 
were eliminated. A bundle adjustment was carried out for simul- 
taneously estimating the projection matrices P; of all images /; 
and the coordinates X; of the corresponding corners C'; in object 
space. As the comprehensive exposure setup realizes heteroge- 
neous viewing angles for almost every object point and due to the 
fact that in the estimation process the redundancy is very high, 
effects of small errors in the mensuration process were assumed 
to be negligible and the result of the object reconstruction to be 
complete. Therefore, the estimated coordinates X; and the esti- 
mated projection matrices P j Were considered as reference data 
in object space. Reference data in the image domain was obtained 
by projection 
Xi = Xi; = P,X;, (5) 
resulting in reference values x;; for the image coordinates x;; of 
each corner C; in each image /;. 
Analysis of extracted lines and edges. The feature extraction 
software FEX was applied to each image, with the control pa- 
rameters being optimized by visual inspection. For each image 
Ij, the end points x,;,, and x,;j,« of all extracted line segments 
lr; were matched to the reference points x;; by employing a dis- 
tance threshold e — 20 [pel], for each reference point x;; leading 
to a set 
£i == deest) | [a - Xi; |< 2 U 
U Tenis) | | Xkje — E |< 2 (6) 
of point-to-reference-point correspondences. Using eq. l, the 
one-sided-shortening ó was estimated junction-wise for each £j, 
leading to estimates ó;;, image-wise over Z; = | J; Z;;, leading 
; (ET ; ; : ; ; ; 
to estimates ô; ) and for each junction over all images, i. e. over 
adi ctimates S 
J;z LU, £i, leading to estimates 9;" '. 
4.2.2 Characterizing the noise sensitivity of corner extrac- 
tion. 
Reference data from multiple resolutions approach. To in- 
vestigate the robustness of the corner extraction (Fórstner and 
Gülch, 1987) with respect to noise, image pyramids were gen- 
erated for all images. 
x . 3 ; 
The third level image / « ? was taken from each pyramid, em- 
. ; : T 3 . . 
bodying an almost noiseless image I; = I ) with real image 
> . ~(3 : 1 a 
structure. Reference coordinates gn for the object corners in the 
third level images were derived from the reference coordinates 
Xj (eq. 5) by scaling, 
i 4h (7) 
   
Interr 
  
Anal 
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mean 
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N = 
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Using 
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4.3 
First | 
propa 
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4.3.1 
cerni 
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As to 
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error | 
fig. 5) 
than 
cause 
param 
stand: 
0. 7[p: 
As sn 
(cf. (« 
cause 
amou 
14 p— 
0.8 
0.6 
0.4 
0.2 
  
Figure 
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Empir 
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increa