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The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences. Vol. XXXVII. Part B4. Beijing 2008 
Equation 5 accounts for two translations, three rotations and 
non uniform scaling and skew distortions within image space. It 
could be regarded as a first order rational function with the 
denominator equal to one. 
Strictly speaking, the imaging geometry of a satellite push 
broom scanner can be described as perspective in the line of the 
linear array and parallel in the along-track direction. However, 
with a very narrow field of view, the assumption that the 
projection is parallel stands true in practical cases (Fraser &. 
Yamakawa, 2004). 
1.3 DLT Model 
Direct linear transformation, which is a first order rational 
function, maps 3D object space into 2D image space via 11 
parameters: 
LX +LY + L Z + L 
L X + L Y + L Z + 1 
LX+LY+LZ+L 
(6) 
y = 
LX + L Y +L Z +1 
Where L = DLT parameters 
DLT parameters can be computed directly using EOP and IOP 
parameters or indirectly using GCPs. This model is considered 
as an approximate model for linear array scanner since the EOP 
parameters are not the same for all image lines (Morgan, 2004). 
2. VIRTUAL GCP GENERATION 
As explained before, fitness accuracy of the 3D affine model as 
compared with the RFM model is evaluated by generating 
virtual GCPs. These points are calculated via the RPCs 
accessible in the IRS-P5 metadata. This is explained in the 
section that follows. 
2.1 RFM Intersection 
The rational function model which is presented as equation (1) 
is always referred to as forward RPC model which provides a 
mapping from geographic coordinate of object (#>, to 
image coordinate (5, L ). Denormalizing equation (1) yields: 
x = r((p,A,h),y = p(tp,A,h) 
x = 
F’(X,Y,Z) 
F’(X,Y,Z) 
■x -x 
(7) 
F'; (X,Y,Z) 
y =—, TY,-y. 
f! (x,y,z) 
Where 
A-A (p-<P h - h n 
X = ~,Y = - -,Z = 2 
/1 (p h 
Because of nonlinearity of the equation, applying the Taylor 
series expansion gives (Grodecki et al., 2004). 
x =r(<p 0 ,A 0 h 0 ) + 
dr 
dw 
(8) 
y = P(<P 0 ’ À o h o) + 
dw 
dw 
The derivation can be calculated through some partial 
derivations as follow (Grodecki et al., 2004): 
dr 
dr du dv 
dw T du dv T dw T 
dp dp du dv 
(9) 
dw T du dv T dw T 
Where: 
u =[1 ,X ,Y ,Z,XY ,XZ,YZ,X 2 ,Y\Z\XYZ, 
X\XY \XZ\X 2 Y ,Y \YZ 2 ,X 2 Z,Y 2 Z,Z 3 ] 
v =[x Y z],w =[A,(p,h\ 
To evaluate the object space coordinate of a point we should 
measure the image coordinate in at least a stereo image. In this 
solution we use two stereo forward and afterward images. The 
equation would read as follow (Grodecki et al., 2004). 
x f = / (tp, A, h) + s\ 
y F = P F (<p,A,h) + e y 
x 1 = r A ((p, A, h) + £ A 
y A = p\(pA,h) + £ A 
(10) 
The final observation equations follow with (Grodecki et al., 
2004). 
A dw = L 
(11) 
Where: 
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