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where: 
C is the effective focal length 
S is a scale factor 
x and y are the photo coordinates of a point 
X, Y and Z are the ground coordinates of the same point 
XO, YO and ZO are the perspective center coordinates 
M(x), M(o) and M(o) are orthogonal matrices as a function 
of rotation angles (kappa, phi and omega) 
between ground and photo coordinate systems. 
A SPOT panchromatic image is built by combining 6000 
linear strips recorded every 1.5 millisecond as the satellite 
moves in its orbit. An important feature of the SPOT dynamic 
image is that the spatial bundles of the reconstructed image 
rays are restricted to a plane (the CCD size for each pixel of 
panchromatic image is 13 by 13 um), and the sensor's field of 
view is very narrow (Michele 1981). This results in a high 
correlation between the projection center displacement and 
the sensor's tilt in the same direction. Another important 
feature of the dynamic SPOT image is that although the 
orientation elements are continually changing, they are 
changing in a highly predictable way, because the satellite is 
moving along a well defined orbit path and is always 
pointing toward the center of the earth (Gugan 1987). 
The photogrammetric space resection formula (Equation 1) is 
expanded in order to include the effect of time dependent 
parameters and for the purpose of deriving design matrices 
of the linearised solution, the dynamic space resection 
formula for SPOT 1A image can be written for any image 
point (p) as follows (Farrag 1991): 
| 0 | | X(P)- (XO* ai*x(p)* br*(x(p)}") 
p) 7 s* Mo.o* |Y(P) - (YO a*x(p)* bz*(x(p)))I —Q) 
c | | Z(P) - (ZO* as*x(p)* bs*(x(p))) 
Where: 
c isthe effective focal length (the nominal focal length of 
SPOT HRV is 1082 mm). 
s isa scale factor. 
XO; YO; and ZO are the coordinates of the perspective 
center corresponding to the center of the SPOT scene. 
X(p) is image coordinate of point (p) along track (line 
number). 
y(p) is image coordinate of point (p) across track (it is a 
function of pixel number); 
Y(p)=(pixel number - (6000+1)/2) * 0.013mm. 
X(P); Y(P); and Z(P) are the ground coordinates of the same 
image point (p). 
MqLo isa compound orientation matrix as follow: 
Maro =M(Q)*M(L)*M(O); where: 
M(O) is an orthogonal matrix as a function of reference 
rotation angles Ko, Po and wo, between the 
working and the image coordinate systems, 
corresponding to the scene center. 
M(L) is an orthogonal matrix as a function of linear rates 
of change (ki, v; and 1) in the reference rotation 
angles and the image line number x(p). 
M(Q) is an orthogonal matrix as a function of quadratic 
rates of change (K2, v» and o» ) in the reference 
rotation angles and the image line number x(p). 
al; a2; and a3 are linear rates of change in (XO), (YO) and 
(ZO) coordinates respectively. 
247 
bl; b2; and b3 are quadratic rates of change in (XO); (YO); 
and (ZO) coordinates respectively. 
Accordingly the dynamic space resection formula of the 
SPOT image include the following unknown exterior 
orientation parameters: 
i- The coordinates of the reference position of the 
perspective center XO; YO; and ZO (corresponding to the 
center of the SPOT scene), (three unknowns]. 
ii- The attitude reference values Ko; qo; and o, (ie. the 
elements of matrix M(O) {three unknowns}. 
iii: The linear rates of change (ki; qi; and 01) of the attitude 
elements, (ie. the elements of matrix — M(L) ), {three 
unknowns}. 
iv- The quadratic rates of change (ko; qo; and €?) of the 
attitude elements, (1.e. the elements of matrix M(Q) ), (three 
unknowns}. 
v- Although the values of the linear and quadratic rates of 
change in the coordinates (XO; YO; and ZO) of the reference 
position can be determined as a function of the well defined 
earth rotation and satellite tracking speed, further 
corrections are given to these parameters during the solution. 
This will take care of random changes in the satellite tracking 
speed {this leads to six additional unknowns}. 
Then the total number of unknown exterior orientation 
parameters of a SPOT image is 18; accordingly the minimum 
number of the required control points is nine control 
points. 
2. RESAMPLING ALGORITHM 
Resampling algorithm of SPOT level 1A images for 
orthoimage production is carried out in two steps: 
Step 1: Locating pixels of the output image (orthoimage) on 
the original image plane; 
Step 2: Assign a gray value for these pixels. 
Special computer programs were written by the author in 
order to carry out the resampling algorithms of this study. 
The two steps of resampling SPOT images for orthoimage 
production will be briefly discussed. 
2.1 Step 1 
This step can be carried out applying anchorpoints technique 
or pixel by pixel technique. 
2.1.1 Anchorpoints Technique 
Anchorpoints technique involve the following tasks (Farrag 
1991): 
- Determination of the image coordinates (in the original 
image plane) corresponding to the DEM grid points 
(anchorpoints). 
- Determination of the image coordinates (in the original 
image plane) corresponding to the pixels of the output image 
within the DEM grid. 
Equation (2) cannot be used directly to determine the image 
coordinates of the DEM grid points because we do not know 
exactly which scan-line orientation parameters to be used. 
This can be overcome by applying an iterative approach as 
follows: 
i- Determine approximate position of the DEM grid points on 
the original image plane, i.e. the pixel No. and the scan-line 
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B4. Vienna 1996