International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B3. Istanbul 2004 
  
7 
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Figure 2. Parallel projection parameters 
Equations | represent a non-linear form of the parallel 
projection model. They can be reduced to a linear form, 
Equations 2: 
Xze4XE4Y44.7-t4, Q) 
ys X AQ Zt A, 
where A; to Ag are the linear parallel projection parameters, and 
will be referred to as “2-D Affine parameters” (Ono et al., 
1999). It is important to mention that Equations 1 are useful if 
scanner navigation data are available (the derivation of scene 
parallel projection parameters from the navigation data is 
presented in Section 4). On the other hand, if GCP are available, 
the linear model in Equations 2 becomes easier to use. The 
relationship between the linear and non-linear forms is 
summarized in the following subsection. 
3.3 Transformation between Parallel Projection Forms 
3.3.4 Transformation from the Non-Linear to the Linear Form 
Knowing the scene parallel projection parameters, intermediate 
parameters (U, V) can be computed as expressed in Equations 3. 
Afterwards, the 2-D Affine parameters are computed. 
U- 1h trader 8 
rh brad N. (3) 
Fb MAIN 
bade t tn M tr NX 
4, = se, = nU) À, = s(n; T nV) 
4, = slry = r,U) dm sin = rV) 
sir. E SU) À, = is V) 
A, = Ax 
| 
> 
"cm 
™ 
va 
I] 
A, 
where r;, to 73; are the elements of the rotation matrix R. The 
derivation of this transformation can be found in (Morgan, 
2004). Next subsection deals with the reverse of such 
transformation. 
3.3.2. Transformation from the Linear to the Non-Linear Form 
Knowing the 2-D Affine parameters, scene parallel projection 
parameters can be derived in three steps: 
Step 1: Components of the projection vector have to be 
computed as expressed in Equations 4. 
  
A; 
L'= = 3 
A rud. ur. oN al Ad ed A) aE 
ArT Alli YH A lM AS 
dde {jar Ven Ada rit dant c } 
MA A 
m 
vus dos Mta 
A, 
N° 
C=t 
DA 
Me 
[N 
N = [A 
Step 2: The scale factor can be computed as expressed in 
Equations 5, by first computing intermediate parameters (7, 75, 
T. A.B. C. U and D). 
  
T aded. ud 
T, =A; + A; + 4; 
BL, = À 4. + 4. 4, + 4.4, 
A=F Tl, 
B= 27, +1, -TT, 
Cal; 
sfr BEN HACL 
24 n 
pub fry’) 
f uU 
snb, (5) 
]12-U* 
Step 3: Finally, the scene orientation angles can be computed as 
expressed in Equations 6, by first computing intermediate 
parameters (D, E, and F). 
D=U FRI 
  
  
  
  
A: 
Z = gu 4. 2V — 
S S 
A dz 
E : >—] 
Sf S 
E+ VE -4DF 
@ = arcsin = 
: A; 
#4 sing+-> 
K = arctan Sos A 
U sin @ + — 
Sn 
cos @ 
    
  
   
     
    
  
    
     
   
  
  
  
  
   
   
  
  
    
     
   
  
  
  
   
  
   
   
    
     
   
   
   
  
    
   
  
  
   
  
  
  
   
  
  
   
     
    
    
  
  
   
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