International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B3. Istanbul 2004 
2.2 Space resection with adjustment 
If we have more than three control points the space resection 
should be solved with an adjustment. We adjust only the 
projection center coordinates. The rotation angles can be 
calculated separately in one step at the end. Let’s list the steps 
of the adjustment procedure. 
STEP 1. Let the number of control points be 7 . We will 
group them by three in every possible combination and for each 
group we solve the space resection directly (formulas 2-6). In 
general case, at each group we will get more than one solutions 
for the projection center. 
STEP 2. From each group we choose common solutions, it 
means we choose those solutions where the sum of square 
differences is minimal. 
STEP 3. By the error propagation law we calculate the 
M , covariance matrices for each solution by the following 
formula: 
7 
M. =F MF 9) 
Yi x14 
where 
M.:; covariance matrix of control points considered at the 
. 
geodetic measurements. 
The F" dispersion matrix can not be derived directly with 
partial derivation from the equations 3-6, so we construct this 
matrix from the differences gained from the original solution 
and from solutions where each image coordinate is incremented 
with a small value one by one. Finally we got the FT matrix 
with the dimensions of 3x6, this matrix well approximates the 
matrix containing the partial derivatives. 
STEP 4. We determine the weight matrices for each solution 
by the following equation: 
P=0";=c(M,}' (10) 
! 
i 
where Cis a scalar factor, at the space resection we take its 
value as 1/1000. 
STEP 5. We calculate the adjusted values of unknowns by the 
Jacobian Mean Theorem as follows: 
Xs By'«N nr) (11) 
X = OQ X S (PL) 
where 
X, 
L = Y (12) 
Z 
and it contains the solutions from each group. 
2.3 Gross error detection 
During the adjustment we can detect the control points with 
gross errors. A gross error can exist in the ground coordinates 
or in the image coordinates. By the following procedure we can 
detect them affectively and no matter where the gross error is. 
Let's group four control points in every combination and solve 
the space resection with adjustment (11). Since we made the 
space resection in every possible combination, our duty now to 
determine which group has a gross error and finally we can 
conclude exactly which point or points caused the gross error. 
For example if we have 5 control points, we can group them by 
four as follows: 
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Let's suppose that the control point No. 1 has a gross error, it 
means that the first four solutions will be wrong and only the 
{2,3,4,5} group gives a good solution. So, by this logic we 
conelude that only the point No. 1 can be the cause for a gross 
error. À similar logic can be proved when the number of points 
are more than 5 or the number of points with gross-error is more 
than one. The only limitation for the detection is that finally at 
least four error-free control points should remain (otherwise no 
reason to do the space resection with adjustment). 
Here is the procedure which helps to decide whether a space 
resection made by four control points has a gross error or not: 
STEP 1. After getting the adjusted projection center we can 
calculate the residuals by the following: 
V. -X- L;, where V, E y: (13) 
STEP 2. After this we can calculate the 77, weight unit error 
(14) and the RMS for each unknown (15) with the help of the 
D s covariance matrix: 
    
7 
3 V; hr, 
My 7 —- 
0 (14) 
3n—3 
where M means the number of control points. 
M, = sg 
x = M qa 
Sy 23 
my = My VI. (15) 
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