1ON 
values 
erence co-ordinate 
by fixing seven 
of this local co- 
rresponding to the 
get realistic r.m.s. 
visional tie curves, 
ibed above, allows 
ige orientation. The 
d out later on (see 
tted out from the 
os for the image 
p of calculation is 
s: the end points of 
ints. End points of 
cause significantly 
iminated by robust 
tment can be seen 
  
r.m.s. error in 
image 
  
5 mm 
  
  
  
of a provisional tie 
an approximation 
in object space are 
ays with the cone 
f another image. 
tes the nodes in 
of curve points. In 
nodes was chosen 
ber of curve points 
1 order to improve 
roximation process 
istment with the 
d to be constant. 
is quite inaccurate. 
al tie curves could 
>. 
le to refine the tie 
je orientation. The 
tion" task are: the 
le positions of the 
ect co-ordinates of 
of the orientational 
e constant. The 
rve reconstruction 
y applied: 
f nodes 
     
  
   
  
   
   
    
   
  
   
  
  
  
   
   
  
  
  
   
  
  
   
  
   
  
  
  
  
   
   
     
    
   
    
   
   
    
  
  
  
   
   
  
   
   
  
     
  
   
   
  
   
  
   
     
2) insertion of additional nodes in the intervals 
containing points with the greatest mean residuals 
So, the flexibility of the adjusting curve is improved step 
by step by inserting additional nodes. If the insertion of 
additional nodes does not result in significantly reduced 
residuals, the optimum reconstructing curve has 
obviously been found. The refinement of the provisional 
tie curve at the right back door is shown in figure 6. 
oy Nea 287 
A i M0 
p" ^ AS: 
24 NIB 
p a 3S 
Ud = - 
3 ; 
> + 
Te S NS 
vi ERES EO 
7 5 QUEE Ye 
> > v = wc 
, > - N 
2 won NS d 
A , 7 AN 
2 * Fx s 
IE. i Cal 
v > 1 
P 3 Y 7 y di : 
^. / d 
N Pd 
LT x. * 
s E / Le ; 
A AN An / NR / 
bs f f \ # 
d NS A ' / 
2 + se fe ys | / 
> M i f ud 
& = ve 
X 7 \ i 
X 
N 7 
^. PA v. 
e d i ue 
^. I : + 
~~, pi i "E 
; i T 
Nee v7 ses NY un 
~ E i 
x Cor 
Figure 5: Approximated provisional tie curves 
curve points from 
different images 
(every tenth image 
point plotted) 
  
  
    
  
  
  
  
   
     
   
^ e| g o V 
———— adjusted curve 
with node point 
curve adjusted to the 
approximated object points, 
improved initial 
node arrangement 
refined curve adjusted to the 
approximated bundles of image rays 
Evi S 
2 
2€ TV IN 
Ns 
Figure 6: Refinement of a provisional tie curve 
Up to now, 29 provisional tie curves are available. In 
order to get initial values for the remaining 11 curves, 
the provisional values for the image orientational 
parameters have to be improved by a further step of 
adjustment. In addition to the orientational parameters, 
the object co-ordinates of the nodes and the curve 
points are assumed to be unknown during this step. 
The positions of the curve points along the curve are 
considered to be constant in order to save 
computational time. The result, obtained after five 
iterations, can be seen in table 2. 
  
No. tie No. nodes No. curve r.m.s. error 
curves points in image 
  
  
  
  
  
  
29 104 5380 55 pm 
  
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B3. Vienna 1996 
Table 2 
Using the improved image orientation, the remaining 11 
provisional tie curves can be obtained by the 
approximation algorithm as described above. 
4.2 Image Orientation using short tie curves 
After having determined all provisional tie curves the real 
image orientation can commence. At the same time, the 
real shape of the tie curves is reconstructed in object 
space. Unfortunately, long complex tie curves, as 
obtained from the line extraction procedure described 
above, are not optimally suited for image orientation: a 
high number of nodes might be necessary to build up a 
curve which is flexible enough for the spatial 
reconstruction of the tie curve. However, a very flexible 
curve would fit to the bundles of rays even if the relative 
orientation is weak. Consequently, the shape of the tie 
curve has to be rather simple, for instance U- or S- 
formed, in order to force an accurate image orientation. 
So, short tie curves have to be extracted from the 
provisional tie curves in object space. 
curvature 
4 
| 
  
"node positions 
«> 
region of max. 
curvature 
7 ' ' 
region of min. 
curvature 
^ Se + 
Le IN 
  
  
Figure 7: Extraction of tie curves from a provisional curve 
Two different configurations of tie curves have been 
examined in the course of this test project: 
a) low bent tie curves extracted from regions of minimum 
curvature of the provisional curve, and, 
b) highly bent tie curves extracted from regions of 
maximum curvature. 
The tie curves of configuration a) are initially built up by 3 
nodes (thus describing a parabola), those of 
configuration b) by 4 nodes (see figure 7). 
199