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2. MATHEMATICAL MODELS 
2.1 Space Resection 
Photogrammetric collinearity conditions used for 
SPOT panchromatic imagery with linear CCD 
arrays are: 
  
x;+a0,+al,s;+a2,57 mu 
Aux = Xos) t 812; (Y. = Yoz) +H a13,(Z; = Zoj) 
In Eq.(2) d is the separation between two neigh- 
boring sensor stations k and k + 1; station j lies 
between k and k + 1, and j is away from k in s 
units. For space resection, Eqs.(1,2) serve as our 
functional model in least squares adjustments. 
Based on weighted ground control points, we es- 
timate mainly the position and attitude param- 
eters .. Xo. > Yon: 207 3 ks Ok Kg... and the self- 
  
a31, (Xi = Xo) d azz; (Y; "T yi) + ass, (Zi = Zoi) 
(1a) 
az, (X: — Xo,) +422, (1H: — Yo; ) + tas, (Zi — Ze, ) 
  
yi+a0y+al,y;+a2,y? = —C 
  
where c : camera constant; 
X;,Y; : image coordinates of point : and z; = 0; 
Sj : Strip coordinate in units of length or time; 
Xo; Yo; Zo; : time-dependent position param- 
eters at sensor station 7 when point ¢ is im- 
aged; 
411,;...433, : elements of an orthogonal matrix; 
they are functions of time-dependent atti- 
tude parameters w;, $;, &j; 
Xi, Yi, Zi : object-space coordinates of point 7; 
a0,...a2, : additional self-calibrating parame- 
ters. 
In order to be practical for implementation, the 
time-dependent parameters of exterior orienta- 
tion Xo; » Yo; » 70; » Wj, Dj, K; are described by piece- 
wise continuous linear models: 
X,, 2 (1— s/d)X,, + (s/d)X 
Ok+1 
(1 — 8/d)Yo, + (S/d) Yo, 
(1 = s/d) Zo, + (s/d)Zo,,, 
(1 = s/d)wr + (s/d)wr+1 (2) 
(1 — s/d)®x + (s/d)®x+1 
( 
1 — s/d)kr + (s/d)Ki41 
Y 
Z 
wj 
bj 
Kj 
646 
(Yı — Yo,) + 433, (Zi — Zo,) (1b) 
calibrating parameters. 
2.2 Integrated Approach to Image Matching and 
3D Positioning 
The method of least squares image matching for 
stereo images can be written as 
Vy — g'(zj,y;) — ro — 1g" (27, y; Pg: (3a) 
in which g', g" : gray-value functions evaluated 
at two corresponding image points 77, y; and 
a"! yl; 
254429) 
To9,T1 : radiometric additive and multiplicative 
parameters; 
vo Pol : gray-value residual and its associated 
4 i 
weight. 
Logically and in a straitforward manner, the 
collinearity conditions Eqs.(1 with 2) can be used 
for pairs of image coordinates in Eq.(3a); and in 
vector notation, we get 
vu = g'(Zi p^ a) - ro - rig" (Zi, p^, a)i py: 
(3b) 
where p/, p" : vectors for position and attitude 
parameters along single- and double-prime 
orbital paths, respectively; 
a : vector for self-calibrating parameters.