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p second 
order polinomials using some control points. With those 
polinomials, the approximate value of Line number x can 
be computed for a ground grid point ( i.e. anchor point ) 
and the estimated orientation elements could be given out. 
Then, xg ( here k is the number of iteration ) /can be 
calculated using the first collinearity equation for 
getting the coordinate in line number. That value should 
be zero according to the SPOT imaging geometry. In prac- 
tical calculations, one should set up a threshold, which 
is generally of 0.1 pixel. If xg is less than that, it 
converges to the solution, and 
X= x txg 2 
where x’could be considered as the line coordinate for the 
corresponding image anchor point. If x is not less than 
the threshold, we have 
Xgai 7Xk*X&-1 (3 
then compute x/ using formula (2), and estimate the orien- 
tation elements again. 
When the iteration converges to the solution, the 
coordinate y can be calculated by means of the second col- 
linearity equation. As the ground coordinates are those 
of tangent plane system ( when using collinearity equa- 
tions), the image coordinates x',y' must be transformed to 
the raw image coordinates (line and pixel sequence number) 
for grey level interpolations. 
Because the iterative calculations are time 
consumed for the rectification at each point, it is needed 
to determine a regular grid ( i.e. anchor point grid ) in 
ground coordinate system. The interval of that grid could 
be defined as of 200--500m along both x and y direction. 
The calculations mentioned above should be performed only 
for the grid points. 
As the three dimensional coordinate for ground 
system are needed for the resections, one has to prepare 
the digital elevation model (DEM) in advance. The DEM grid 
could be defined as same as anchor point grid so that two 
grids are overlayed with same interval distance. 
4. The employment of direct approach 
Generally, the direct approach is never used for 
digital image rectification, because its result is not a 
regular grid and must be further interpolated so that the 
algorithm is more complicated and more time-consuming. 
But for SPOT image, it will have more advantages, 
since it needs no iterations for computing the orientation 
elements. If the line number is known, one can calculate 
directly the orientation elements with higher accuracy. In 
this case, the direct approach is available. 
Firstly, one must determine a grid in the image 
window to be rectified. The interval of this grid could be 
same as that of DEM, but the range should be smaller than 
that of DEM. Then one éan calculate the ground coordinates 
X; Y for each image grid point according to the following 
formula, 
a,y-a3f 
X=(7-7s)——- —+Xs 
c,y-caf (D 
b, y-bzf 
Y-(Z-25)-————— —+Ys 
c, y-c3f 
where a; b; c; (i=l, 2,3) are the functions of attitude pa- 
rameters and Xs,Ys,Zs are the position parameters for the 
line concerned. The value Z can be obtained from DEM and 
f is the equivalent focal length. Here, the key is the Z 
value which should be interpolated from DEM. Fitting up 
the second order polinomials is necessary for calculating 
ground coordinates X'Y'using image grid point coordinates, 
and then one can perform the interpolation for 217 
should be transformed to the Z value in tangent plane 
system, Here, Z', as a approximate value, must be checked, 
Thus, .X',Y', Z' should be transformed to tangent plane 
coordinates and then be used for resection to get corres- 
ponding image coordinates which'will be compared with the 
image anchor point coordinates, If the deviation of image 
point is less than 0.1 pixel, Z value can be accepted. 
Otherwise, one has to take in use the values 2’'45, Z'i10 
33 
(unit in meter) to perform the resection and comparison 
in order to chose Z' value which should meet the need of 
precision. 
After the calculation using formula (4), X ,Y must 
be transformed to Gause-Kruger coordinates in order to 
form a coordinate set for a grid which is irregular. Fur- 
thermore, the interpolation is needed for a regular grid 
using that coordinate set as the control by means of first 
order polinomials only. 
5. Rectification for getting orthophoto 
After having the pair of coordinate sets of anchor 
point grid, one can complete the rectification for whole 
image window, Since there are four points in each case 
within the grid, the first order polinomials are sufficient 
for the interpolations of image coordinates one point by 
one point so that one can perform the grey level interpo- 
lations from raw image window. [Each pixel of rectified 
image has the same size as that of raw image so that the 
resolutions are identical, 
Because the interval of anchor point grid is of 
200-500m, each grid case could be considered as a tilt 
plane on ground surface, so that there is only the dis- 
placement and rotation of image distortions and the first 
order polinomials are sufficient to express this kind of 
distortions. 
6. Results of experiments and analysis 
of precision 
6.1 The solutions of orientation elements and 
their precision 
The SPOT image window selected as test area is of 
good quality, so that one can easily choose the control 
points and check points, Since the preliminary values of 
exterior orientation elements and the equivalent focal 
length have been reasonably determined, and the approach of 
alternate iteration for the solution of linear orientation 
elements and angular elements has been taken in use, the 
convergence to the solutions is very quick and generally 
three or four times of iterations are enough to converge 
the solutions with higher precision. The Mean Square 
Error (MSE) of 14 control points is of only 0.56 pixel. The 
MSE of 37 check points is of 0.9 pixel. Thus, it makes up 
the good basis for the geometric rectification to reach to 
the higher precision. 
6.2 Coordinate calculations for anchor points and 
the accuracy 
The coordinate set of anchor points is one of the 
keys in rectifications. The precision of those coor- 
dinates is very important to predict if the process will 
be successful or failure. When taking the indirect 
approach, 50 check points have been selected to examine 
the accuracy of anchor point calculations, The mean abso- 
lute errors (M.A. E) for those points are dx- 0.61(pixel) 
dy= 0.56(pixel) and 94% of those points can reach to the 
precision of subpixel. 
One the -other hand, the traces in iterations show 
us that one or two times of iteration can converge to the 
solutions during the indirect process, so the calculating 
rate is very high and it reduces effectively the time of 
rectification, 
Since there is no iteration during the procedure 
of direct approach, the calculation is relatively quicker. 
As to the precision of anchor point coordinates, the 
checks in 4087 anchor points by resections show us that 
the error of point is very small. The MAE for those points 
is of only 0.000006 pixel. The maximum is 0.0000236 pixel. 
The minimum is 0.0000038 pixel. Therefore, the precision 
of anchor points is very high, and this shows us that 
the calculations of X Y and the interpolation of Z can 
perfectly assure the accuracy of coordinate Z for anchor 
point. 
Two kinds of approaches have created two files of 
anchor point coordinates, In order to estimate the preci- 
sion of further interpolation within the cases of anchor 
point grid, we have computed the MSE of point after fit-