The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences. Vol. XXXVII. Part Bl. Beijing 2008 
space and object space. Straight lines, circles, ellipses and free 
form lines are examples of such representation. In this research, 
straight lines as well as free form lines converted center 
passing through a point on the image line must intersect the 
object line. In their approach, the standard point-based 
photogrammetric collinearity equations were replaced by line- 
circle based ones. Instead of the regularly used two collinearity 
equations, a single equation is established to ensure the 
coplanarity of a unit vector defining the object space line, the 
vector from the perspective center to a point on the object line, 
and the vector from the perspective center to a point on the 
image line. Furthermore, coordinate transformations are 
implemented on the basis of linear features. In this case, feature 
descriptors are related instead of point coordinates (Shaker, A 
(2004)). 
2.2 3D Affine LBTM 
Figure 1: Unit line vector representation in image and object 
space (case of linear array sensor) (Adopted from Shaker, A. 
2004) 
Vectors v 12 and ^ are unit vectors for conjugate lines in image 
and object space respectively (Figure 1). Any two points along 
the line segment in image and object space can define the two 
unit vectors. Suppose that point p x = (jc,,y x ) and p 2 = (x 2 ,y 2 ) 
are two points on the line in image space, then V 12 can be 
presented in matrix form as: 
Vi2=[*, a y °] r W 
where 
x 2 -x l and _ y2 ~>i 
> /(x 2 -x,) 2 +(y 2 -y 1 ) 2 ' sl(x 2 -x l f+(y 2 -y l ) 2 
On the other hand, suppose that points p { =(X l ,Y l ,Z l ) and 
P 2 = (X 2 ,Y 2 ,Z 2 ) are located on the conjugate line in the object 
space. Then, the unit vector v a is: 
v n =Ux A Af < 2 > 
where: t _ (*,-*,) 
x ~yj(x 2 -xy + (Y 2 -Y[) 2 + (Z 2 -Z,) 
A = (Äzhl 
V(A 2 -^) 2 +(y 2 -^) 2 +(Z 2 -Z,) 
A = (Z 2 -Z } ) 
Z ^X 1 -X l ) 2 +(Y 3 -Y l ) 2 HZ 2 -Z l ) 
It is worth to mention that points p x , p 2 and P [ ,P 2 on 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 high-resolution 
satellite imagery. 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 transformation into its conjugate vector in 
image space by applying rotation, scale, and transformation 
parameters as shown in equation (4.3): 
v = MAV + T (3) 
where V and V 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 T is a 
transformation matrix. 
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 x 
m,, 
"hi 
"hi 
0 
o' 
M 
\T X ] 
a > 
= 
«21 
"hi 
"hi 
0 
0 
Ay 
+ 
Ty 
0 
31 
"hi 
"hi. 
0 
0 
A 
4 
T z 
where A [ , A 2 , A i are scale factors, ftl u , m n ,..., W 33 
are the rotation matrix elements, (a x ,a) are the line unit 
vector components in the image space coordinate system, 
(A x ,A y ,A z ) are the line unit vector components in the object 
space coordinate system, and T x , T Y , T z are the components 
of the transformation 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