art B3. Istanbul 2004 
vector have to be 
24,4, + A} + A; (4) 
uted as expressed in 
ite parameters (73, 75, 
(5) 
's can be computed as 
nputing intermediate 
International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B3. Istanbul 2004 
d. 25; d. 
“sin pcos X + —U cos vg + —sin x 
; s 5 (6) 
Q = arcsin re = 
sin” x + (sin p cos x + U cos e) 
Readers interested in the derivation of the above formulas can 
refer to (Morgan, 2004). So far, the relationships between the 
non-linear and linear forms of the parallel projection model are 
established. It is important to remember that the linear model is 
preferred in cases where GCP are available, while the non-linear 
model is preferred if scanner navigation data are available. 
Section 4 deals with the latter case. But first, a pre-requisite 
transformation prior to handling the scenes according to the 
parallel projection model is discussed in the next subsection. 
3.4 Perspective To Parallel (PTP) Transformation 
Original scenes captured by linear array scanners comply with 
the rigorous perspective geometry along the scan lines. 
Therefore, before dealing with the parallel projection model, 
Perspective To Parallel (PTP) transformation of the scene 
coordinates are required. Such a transformation alters the scene 
coordinates along the scan lines in such a way to make them 
comply with parallel projection (Ono et al, 1999). The 
mathematical model for this transformation requires the 
knowledge of the scanner roll angle, v, and is expressed as: 
| 
Mey i 
1-2 tan (v) 
ré: 
where: 
C is the scanner principal distance; and 
y,y are the coordinates along the scan line according to 
parallel and perspective geometry, respectively. 
Therefore, any coordinate along the scanning direction can be 
transformed from its perspective geometry, y, to another 
coordinate according to parallel projection, y’. One has to note 
that the mathematical model in Equation 7 assumes a flat 
terrain. Morgan (2004) used GCP for estimating the scanner roll 
angle together with the parameters associated with the parallel 
projection. This model is expressed as: 
oe 
I 
ELXT AYA Z+ A, 
AX cr Y udo 44. (8) 
  
j, nv) tt. diy dn 
c 
In the above model, measured coordinates from the scenes (x, y) 
are used directly to estimate the 2-D Affine parameters together 
with the scanner roll angle. The next section deals with an 
alternative approach for obtaining the scene parallel projection 
parameters — that is when scanner navigation data are available. 
BETWEEN THE 
PARALLEL 
4 THE RELATIONSHIP 
NAVIGATION DATA AND 
PROJECTION PARAMETERS 
Referring to the discussion in Section 3.1, space scanners can be 
assumed to travel with constant attitude (c,, @,, &;) and constant 
velocity V,(AX, AY, A7) during the scene capture, starting from 
(Xo, Yo, Zo) as the exposure station of the first image. The 
parallel projection vector can be computed as: 
A — ^ ^ C 
[z M N]= Ks Fos sd (9) 
where 7,11 to 7,45 are the elements of the scanner rotation matrix. 
l'he scene rotation matrix A in Equations 1, may differ from the 
scanner rotation matrix and it can be computed as: 
Rss. yz] (10) 
where x, y and z are unit vectors along the scene coordinate 
axes with respect to the object coordinate system. They can be 
computed as follows: 
T 
V eir T» Fu] 
LAN (11) 
V, xy] 
X=yxz 
The scale value of the scene can be computed as: 
where: 
(Xo, Yo, Zo.) are the components of the exposure station at 
the middle of the scene; 
Zo is the average elevation value; and 
€ is the scanner principal distance. 
Finally the two scene shifts can be computed as: 
s(r,U mU )X,, dr s(r4U = M. T. s(r, U E M. 
s(n; V mo M. + Lo F ln Y, + 8. F-r 7 
(13) 
Ax 
Ay 
where U and V can be computed from Equations 3. Derivation 
of this transformation is not listed in this paper due to its 
complexity. Interested reader can refer to (Morgan, 2004) for 
detailed derivation. So far, the derivation of scene parallel 
projection parameters, Equation 1, using the scanner navigation 
data is established. The next section deals with experimental 
results to verify the derived mathematical models and 
transformations using synthetic data. 
5. EXPERIMENTS 
Synthetic data are generated and tested to achieve the following 
objectives: 
e Verifying the transformation from navigation data to 
scene parallel projection parameters (as discussed in 
Section 4); 
e Verifying the transformation between the non-linear 
and linear forms of the parallel projection parameters 
(as discussed in Section 3.3); 
e Analyzing the effect of PTP transformation, based on 
the true/estimated roll angles, on the achieved 
accuracy; and 
e Comparing direct and indirect estimation of the 
parallel projection parameters.