(2.8) 
-0 
(2.9) 
(2.10) 
entation 
1 matrix. 
ice, the 
ation for 
(2.11) 
(2.12) 
! 
x 
J Q1 
ior (IO) 
1eters, 
.7)) the 
as pass 
nmetric 
larly be 
ere are 
circles 
exterior 
If the 
overed, 
equired 
res and 
do not 
ever, a 
RO. A 
pass circle in 2-image overlap contributes 4 equations, 
and in 3-image adds 9 equations. 
Extended Relative Orientation (ERO). The coplanarity 
condition of the base vector B, and two image vectors p PR 
is given by: 
Bp, xPp) =0 (2.14) 
Alternatively, it is given by 
x 
[x y 1], 1,M,KMp1, |y| =0 (2.15) 
Hr 
in which 
0 -B, B, 
K=|B 0 -B, (2.16) 
-B, B, © 
10 -x 
Ing = 10 1 -y (2.17) 
00 fi. 
E-M,KM x is called the essential matrix and is used for 
calibrated cameras when  x,yyf are known, while 
F-IM EM q i5 called the fundamental matrix and is 
used for uncalibrated cameras. Since the rank of K is 
2, |F|=0 (Barakat, 1994). Further, the 9 elements ofF 
are recoverable to a scalar multiple, hence the 
maximum number of independent parameters in F is 7. 
Consequently, ERO of a stereopair can only recover 2 
IO elements in addition to the classical 5 EO elements. 
Partial Absolute Orientation For complete absolute 
orientation (AO) of a relatively oriented stereomodel, 
control linear features are needed. Each control 
straight line contributes four independent equations to 
the recovery of the 7 parameters of AO. Therefore, a 
minimum of 2 non-coplanar such lines is required. 
Frequently, no "control" lines may be available, and 
instead geometric constraints which yield partial 
absolute information exist. These may then be used to 
recover additional rotational elements depending upon 
the available constraints (horizontal or vertical lines, 
etc.). 
Block Adjustment. This is the general method which 
when based on unified least squares and carries all the 
parameters and constraints as a priori weighted 
information, can be used to perform any of the 
operations discussed separately above. 
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B3. Vienna 1996 
     
   
   
    
    
    
    
   
   
   
   
    
    
  
    
     
   
   
    
   
   
   
   
     
   
    
    
   
    
   
   
    
   
   
  
    
   
    
   
    
2.5 Experiments and Results 
A large number of experiments, with both simulated 
and real data, were conducted to test the developed 
mathematical models and study the effectiveness of the 
exploitation of linear features in photogrammetric 
applications. The results are: 
Case A - Simulated Data A pair of photographs with 
strong convergent geometry were simulated such that 
the set of perfect image coordinates were perturbed 
with errors having 0,70, -0.01 mm. Two experiments 
were conducted using the two photo block triangulation. 
Experiment #S1 is a regular two photo block 
adjustment, which recovers 12 exterior orientation 
parameters, using 10 control lines and 10 pass lines. 
Experiment #S2 attempts to recover both interior and 
exterior, 18, orientation parameters of the two photos 
using 10 control and 10 pass lines. Tables 1 and 2 list 
the RMS for dX,dY,dZ computed at 5 points on each 
pass line for experiments S1 and S2 respectively. For 
each point; dX,dY,dZ are the differences between 
X,Y,Z computed using the a priori known line 
descriptors (q,B,,B,,8;) and X,Y,Z computed using 
their estimated values after the block adjustment. 
Case B - Real Data (Bangor Imagery): The data set 
consists of two nearly vertical aerial photographs flown 
over an urban area in Bangor, Maine, at a scale of 
about 1:8660. Regular two photo block triangulation 
(i.e. solving for 12 parameters) was performed using 6 
control lines and 9 pass lines. Table 3 lists the 
differences in the camera parameters between the 
original and the recovered parameters while table 4 lists 
the RMS for dX,dY,dZ computed at 5 points on each 
pass line. The results show the applicability of using 
lines in the two photo block to recover both the camera 
and pass feature parameters. 
3. INVARIANCE-BASED OBJECT 
RECONSTRUCTION 
3.1 Invariance Versus Photogrammetry 
Image invariance theory is based on a premise which is 
fundamentally different from photogrammetric theory. 
Image invariance deals with invariant quantities under 
perspective projection (transformation). The cross-ratio 
is the classic invariant of the projective line. For four 
points on a line, under projective transformation, the 
ratio of ratios of distances is invariant. In most 
photogrammetric activities, very careful modeling of the 
sensor elements as well as imagery acquisition 
parameters is central to the techniques used. By 
contrast, image invariance is almost totally built on the