International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B3. Istanbul 2004 
27 (x» y2) are two points on the line in image space, then Yi 
can be presented in matrix form as: 
v,=la, a, Of (1) 
Xx, —X, and n y,-y 
J 
Q, m op Qm ocu 
NL = X, ) + (y, = Y y (x; Tas X, Y + (y, = Yi ). 
    
On the other hand, suppose that points P,= (X, Y,, Z;) and P,= 
(X5, Y», Z5) are located on the conjugate line in the object space. 
Then, the unit vector 7 is: 
12 
  
  
  
[7 T 
V ea A, A; ] (2) 
where: 
490 (X, X.) 
asy y EZ AZ) 
AE (py) 
S MN -Xyeq-k)aQ zy 
um 23 
XS Y HZ. -Z) 
"y 
prix y) 
pis. y) dry 
Vig 
Image a 
coord. o: 
system 
X 
v EF nd 
Z 2 Vio Ax 
Xx: Pi(X Y,. 2) 
Yı ds Ground 
0 E „X coord. 
system 
Figure 1 Unit" line vector representation in image and object 
space (case of linear array sensor). 
It is worth mentioning that points p;, p, and Py, P; in image and 
object spaces are not conjugate points, but the lines they lie on 
are conjugate lines. As was mentioned earlier, the relationship 
between image and object space can be represented by 3D 
affine transformation for HRSI. The same relationship between 
the two coordinate systems is used to represent the relationship 
between vectors in image and object space. Any vector in 
object space can be transformed into its conjugate vector in 
image space by applying rotation, scale, and translation 
parameters as shown in equation (3): 
P=MAV +T (3) 
where y and J/ are vectors of line segment in image and object 
space respectively, M is a rotation matrix relating the two 
coordinate systems, A is a scale matrix (a diagonal matrix 
providing different scales in different directions) , and 7 is a 
translation matrix. 
852 
The elements of M are functions of three sequential rotations 
about the X, Y and Z (object) coordinate axes and are the same 
as used in the derivation of the collinearity equations used in 
photogrammetry. Substituting the various presented matrices 
into equation 3 gives: 
a, mu my Mm |A 0 0/4, T. (4) 
a, |=im mao malo 4.0]|4;]|rl7, 
0 Hsc. Hm; | 0... 0 
Ss 
> 
EN 
N 
~ 
N 
where À, A, A; are scale factors, my, m,....m;; are the 
rotation matrix elements, (a, a,) are the line unit vector 
components in the image space coordinate system, (Ay, Ay, 47) 
are the line unit vector components in the object space 
coordinate system, and Ty, Ty, Tz are the components of the 
translation matrix between the image and the object coordinate 
systems in X,Y and Z directions. The previous form is valid 
only if the scale factor is equal to +1 since the transformed 
vectors, in this case, are unit vectors. This condition is 
necessary and should be sufficient to validate the equation. 
Then, the equation will lead to the following individual 
equations 
a, 2 (Am, A, t Am, A, t Ama, A, ) e T, (5) 
a, = (Amy Ay +A,mp A, + Ama 4; ) * T, (6) 
0 (Am, A, * Amy, A, c Am, 4,)4 T, (7) 
After substituting the scale factors Ai multiplied by the rotation 
matrix coefficients m,, and the translation components 7x-z, by 
the new coefficients b; the transformation equation can be 
rewritten as: 
a, =b, A, +b,4, +b,4, +b, (8) 
a, =b,A, +b,A; +b, A, +bp (9) 
where b, to b; and b; to b; present the rotation and scale factors, 
and b, b, are translation coefficients. Equations 8 and 9 
represent the mathematical form of the 3D affine LBTM. 
The model is similar to the ordinary 3D affine model used by 
Fraser et al. (2002), with the only difference being the use of 
line unit vector components instead of point coordinates in 
order to calculate the model coefficients (image parameters). A 
unique solution for the new model coefficients could be 
calculated by using four Ground Control Lines (GCLs), which 
are conjugate image/object lines. If the number of observations 
available (number of GCLs) is more than the minimum amount, 
then a least squares adjustment is used. 
2.2.1 The unit vector problem: As mentioned above, unit 
vectors do not provide a unique representation of a line and thus 
the LBTM expresses the relationship between a group of lines 
in image space and any other parallel group of lines in object 
space. Comparing the coefficient values calculated from the test 
data with the use of the ordinary 3D affine model and GCPs 
with the coefficient values calculated with the use of the 
developed 3D affine LBTM and control lines (unit vector 
components) shows: a) very small differences between those 
coefficients representing rotation and scale factors; b) large 
differences between the translation coefficients. This finding 
suggests that there is a problem in the LBTM concerning 
calculation of translation coefficients.