3.3 Attitude restoration 
We thus obtained the ground position of the lookpoint P as a 
function of the Euler angles and altitude: 
(3.3.1) Px , y*P Cr! (6,0, X)  H)zP (MC? 0, X) r, H2. 
Here vector r is only a function of the camera geometry, 
depending on the location of the pixel in the focal plane and on 
the focal length. The left part of equation (3.3.1) is obtained 
by measuring the components of the displacement vector, by 
comparing the time-sequential images for each detector array, as 
described before. Each detector array yields two equations, one 
for every vector component. Since the dependency of the vector 
components on altitude H is clearly linear, H can be omitted (set 
to 1), this corresponds to a simple scale change. In the 
particular case of nine arrays one obtaines the system of 13 
equations, as follows: 
X1=X(M(6,0,x) F1) ; 
Y1sY(M(o ,0, x) r3); 
(3.3.2) 
XgzX(M($ su: x)rg); 
Yg*Y (M(ó ‚w, X) rg) e 
We can now resolve (3.3.2) with respect to¢ sw and x. Since this 
system is overdetermined (the number of equations is greater than 
the number of unknowns) we apply the standard least-square 
fitting procedure. The solution is greatly simplified in the case 
of small angles, when the rotation matrix can be replaced by its 
linear approximation, as follows: 
1 Xoo wl 
(3.3.3) M(9,w,x)= |-x Hu lux 6 
0 X + - 4X i 
Formula (3.3.3) is obtained by multiplying matrixes M(s), M(w) 
and M(x) from (3.2.1) and replacing sin(a) and cos(a) by o and 1 
respectively. Now equations (3.2.2) can be re-written as follows: 
X'=Xp+XYp p! 
(3.3.4) y'=(=x+0w) Xp (1400) Yp oZ 
-w7 
ZEHN HN +Z 
p ^p 
and. substituting (3.3.4) into (3.2.3) and retaining only the 
first order terms with respect to ? , 9 and X we finally obtain: