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Table 1. Ground Residuals (RMSE:m) after Exterior Orientation 
  
Degree of Polynomial 
17 Control Points 
40 Check Points 
  
  
  
  
  
  
for Image Parameters mX mY mZ mS mX mY mZ mS 
Linear 3.56 6.29 23.61 24.69 8.48 6.82 20.23 22.97 
Quadric 1.69 3.87 2.60 4.96 8.51 7.01 8.10 13.68 
  
  
  
  
  
  
  
  
  
Figure 2. Topocentric Cartesian System 
coordinate system of  photogrammetric computation 
while processing SPOT imagery. Rigorous mutual 
transformations between map projection and TCS  co- 
ordinates involve following conversions: 
Gauss-Kruger Coordinates and Normal Height 
(Xg,Yg,h) «— Geodetic Latitude, Longitude, 
and Height (B,L,H) «— Geocentric Spatial Rec- 
tangular Coordinates (Xc,Yc,Zc) «— TCS Coor- 
dinates (X,Y,Z). 
The formulations are described in detail by Xiong. 
It is much time-consuming to perform these calcula- 
tions, and especially some conversions require it- 
erative refinement. A fast algorithm to substitute 
for the above complicated computation is required 
to meet the realtime needs of on-line  photogramme- 
tric mapping. One method that quadric polynomials 
perform the conversion between Gauss-Kruger and the 
TCS planimetric coordinates (Xo,Yo) on the refer- 
ence spheroid, and then the elevation is consid- 
ered, is proposed in this article. From (Xg,Yg) to 
(Xo,Yo), the conversion is: 
Xo=Ao +A, Xg+A, Ye +A; Xg* +A, XgYg +A; Ye? , 
Yo=Bo +B, Xg+B; Yg+B3 Xg”+B, XgYg+Bs Yg* (2) 
and reversely 
Xg-AÀ*A] Xo*A2YotA! Xo? £A; XoYotA? Yo? , | 
YgzBJ) *B/ Xo*B/! YotB3Xo* 4B] XoYo*B Yo* (3) 
where coefficients A;, B,, A!, and B] (iz0, !, ..., 
5) are solved in advance from rigorously computed 
coordinates of a number of known standard ground 
points (from scene parameters or existing maps) 
evenly distributed over the imaged area. Consider- 
ing the influence of height (illustrated in Figure 
3 where the reference spheroid is approximately 
considered as a sphere whose radius R is equal to 
the average size of the spheroid), the TCS coordi- 
nates of a ground point are 
H = 
X=Xo(1+}), Y=Yo(1+}), ZsH-zp(1+5) (Xo*+Yo®) (4) 
281 
  
Figure 3. Plane Section through Z-Axis 
and a Ground Point 
where H=h+ha (ha is the height anomaly of the 
area), and the reverse transformation begins with 
d R R 
SR*7) * Xo=X Yo=Y>— (5) 
Haz KH 
then h=H-ha and formula (3) are used. 
Computation has demonstrated that in an area 
tral B=39°46’and L=118°17.5’) of 60 by 60 km the 
maximal residuals (compared with rigorous  trans- 
formation) are 0.25m of planimetric coordinates and 
0.2m of elevation, much smaller than observation 
errors, and only occasionally existing far from the 
TCS origin. 
(cen- 
Solution of Exterior Orientation Parameters 
Instead of conventional relative and absolute  ori- 
entations of stereophotogrammetry, the image orien- 
tation parameters of two SPOT images are directly 
solved by means of spatial resection. For each 
  
  
point, its collinearity equations of left or right 
image are 
a, (X-Xs)+b, (Y-Ys)+c, (Z-Zs) 
x=-f 
ag (X-Xs) +b; (Y-Ys) +c; (Z-Zs) (e) 
a, (X-Xs) +b, (Y-Ys) +c, (Z-Zs) 
0=-f 
a5 (X-Xs) b; (Y-Ys) tc; (Z-Zs) 
where 
f is the focal length; Xs, Ys, Zs are dynamic 
exposure station coordinates calculated by (1); 
a; » b; , and c; (121, 2, 3) are direction cosines 
(Wang 14) determined by dynamic attitude 9, 2, K 
calculated also by (1); and X, Y, Z are TCS 
coordinates of the corresponding ground point. 
By linearizing (6) combined with (1), orientation 
parameters of two images, whose initial values may 
be obtained from orbit data, are computed in