The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences. Vol. XXXVII. Part Bl. Beijing 2008 
a x ~ (^¡^ll^X A^lfl^Ay ^3 w i3-^z) 
a y = (A i m 2l A x + ^m^Ay + A 3 m 23 A z ) + T r (5) 
0 = (A l m 3l A x + A^tn^Ay + A 3 m 33 A z ) + T z 
After substituting the scale factors A. multiplied by the 
rotation matrix coefficients YYl iv , and the translation 
components T x _ z , by the new coefficients b t , the 
transformation equation can be rewritten as: 
a x = b x A x + b 2 A Y + b 3 A z + b 4 
ciy = b 5 A x + b 6 Ay + b 7 A z + b% (6) 
where b x to b 3 and b 5 to present the rotation and scale 
factors, and b 4 , b % are translation coefficients. Equation (6) 
represents the mathematical form of the 3D affine LBTM. 
2.3 Rectification by 3D Affine LBTM 
The research in this thesis will apply this 3D affine LBTM for 
high-resolution satellite imagery rectification, and then 
evaluate the accuracy of the result. The basic image 
rectification procedure is studied in Chapter 1, and in this 
section, we will analysis rectification arithmetic using the 
program language by matlab step by step, and all these 
arithmetic made by Shaker, A., 2004. 
Firstly, determine several ground control points for coefficients 
calculation and accuracy analysis and numbers of ground 
control lines for coefficients calculation. Acquire image space 
2D coordinates (*., y.) on the image and object space 3D 
coordinates ( < X i ,Y i ,Z i ) on the ground. It needs to get two end 
point coordinates for every GCLs and calculate the length of 
every line on the image and ground. But these points on the 
lines need not to be conjugate points between image spaces and 
object spaces. 
Secondly, the parameters will be inversely calculated in the 
software, the main principle of which is expressed in Equation 
4.6 and the whole prudence for arithmetic 3D affine LBTM. 
Then the eight-parameters will all be computed from the 
transformation model and the characteristic linear features with 
the method of least square in the program. 
Finally, all the pixels on the image will be defined by new 
coordinates by 3D affine LBTM. Achieve the image 
rectification. Accuracy analysis will process by root mean 
square of ground coordinates for GCPs. 
3. TEST ARTIFICAL DATA 
LBTM is a new kind of non-rigorous mathematics 
transformation model, the stability and adaptability for 
different kinds of terrain, different linear feature attitude and 
distribution need to be considered first, so the date for practice 
must be relative perfect, the error must in tolerance range and 
can be evaluated. And the accuracy influence will be claimed 
mainly come from forecasted errors and transformation model. 
Due to the reasons above, artificial data is needed first for this 
project. 
Data sets covering a 10x10 square kilometers area were 
simulated to test the developed mathematical model discussed 
in this paper. Three different elevations of terrain conditions 
were established for the artificial data. The first condition was 
simulated for flat terrain by an elevation difference of less than 
50m. The second condition was simulated for hilly terrain by 
an elevation difference of about 500m. The third condition was 
simulated for mountainous terrain by an elevation difference 
over 1000m. Thirty well-distributed ground points were 
established for every condition in the object coordinates system. 
The three groups of the ground points for three conditions of 
terrain were considered to have the same planimetric locations 
and the only difference for them was in their elevations. The 
original coordinates of the four comers are clockwise: (10km, 
10km), (10km, 20km), (20km, 20km), (20km, 10km). Figure 2 
shows the distribution of the ground points in plane, which is 
made in autoCAD interface, while Figure 3, a b and c shows 
the three different terrain shapes of the artificial data. 
(10km, 20km) (20km, 20km) 
(10km, 10km) (20km, 10km) 
Figure 2: Distribution of the artificial ground control points in a 
plane 
(a)Flat terrain 
(b) Hilly terrain 
(c) Mountainous terrain 
Figure 3: Three different terrain shapes of the artificial data 
1173