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International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B5. Istanbul 2004 
For the computation of the rotation angles in each image we use 
vanishing points. The advantage of using vanishing points is 
that can work with only on object in the image and in the worst 
case scenario with only a portion of a building facade provided 
that we can find lines parallel to the outline of the building. We 
assume a local coordinate system in which the X axis of the 
image is the X axis in the building façade (width of the façade), 
the Y axis of the image is parallel to the Y axis in the building 
façade (height of the façade) and the Z axis is the distance from 
the sensor position. The two coordinate systems are shown in 
figure 7. 
Building Facade 
  
Figure 7 The two coordinate systems 
As shown in [Petsa E, Patias P., 1994] we are able to calculate 
the three rotation angles in a local coordinate system parallel to 
the object using only two directions on the image plane. In 
figure 8 we can see the determination of the vanishing points. 
The rotation angles and focal length can be computed with the 
following equations: 
    
  
Cz4-XpXp —YnYyy (1) 
Xy ; 
tan x = — (2) 
Xr 
c 
taf ooo (3) 
Fg Sin K — xy cos 
c 
fam z-————— (4) 
Xp SINK + yp COSK 
Vv 
Figure 9 Vanishing Points computation 
After the determination of the rotation angles we proceed in the 
computation of the translation. In order to achieve this task we 
use the previous information of the rotation angles. From the 
rotation angles and a known line segment in one image we can 
compute the new distance of the sensor from the object. 
Lr i : j i i 5) 
Rijxig + Ray via — Rayei _ Rjyxip + Roy Vip — Riyci 
  
Rjyxi, € Ry, - Risci R{3xiy + Rha yi — Rizci 
3 
M 
i1Va * opa - Rig) — Rips * Ro ya - Rayo 
  
(6) 
  
RIA. * R3 yj, = Ric Riaxj; pe Ry yj = Rise 
if we name the denominator of equation 5 A and the 
denominator of equation 6 B, and take the ratio of the two 
equations we have: 
S 
p qu. 
8 edo (7) 
#, 5 R A 
B 
Using equation 7 we can compute the range of the second 
image knowing the range of image 1, the coordinates for a 
known line segment in both images, and the rotation angles of 
the images. From these equations we are able to derive the DZ 
component in our local coordinate system. 
We proceed by creating a quasi rectified version of the two 
images using the rotation angles. In order for the image to be 
fully rectified we have to use the projective transformation. In 
our case we use the rotation angles to rotate the image points in 
plane parallel to the plane of the façade. As a result the two 
quasi rectified images have the same orientation. In figure 10 
we can see the procedure, in the left image we have the two 
sensor position and the rotation angles as recovered from the 
vanishing points, while in the right image we can the system 
after the quasi rectification procedure. We will use these two 
images to compute the translation of the sensor along the X and 
Y axis of our coordinate system. 
p T 
Figure 10 The quasi rectification procedure 
We can compute the scale of each point in the images 
using the rotation information and image range, DZ that we 
have already determined. 
1 > Rx Ray = R33C 
scale Z— Zn 
Where Z-Z, is the range of the image in our local coordinate 
system and refers to DZ. Computing the scale of the four points 
in the two images we can compute a mean scale for each image 
and furthermore to compute a relative scale between the two 
images. In order to compute the DY, and DX we will use the 
translation of the points from the quasi rectified images, and the 
scale factor between the two images and the scale of the known 
orientation image. 
  
(9) 
DX j = Rs, * SC; * dx; (10) 
Dy, Rs, eS dig d 
Where Rg; is the relative scale between the two images, Sc; is 
the scale of image i and dj, dy; is the translation of the points 
in the quasi rectified images.