The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences. Vol. XXXVII. Part B5. Beijing 2008 
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Figures 2 and 3 illustrate the geometry of the left panorama, 
which is acquired with the camera being mounted on the left of 
the stereo bar, as shown in Figure 1. There are four coordinate 
systems considered in the geometric model, the object 
coordinate system (X-Y-Z), the local camera coordinate system 
(x-y-z), the cylindrical camera coordinate system (r-^-q), and 
the image coordinate system (i-j). The local camera coordinate 
system (x-y-z) is an auxiliary coordinate system. Its origin, o, is 
the intersection point of the rotation axis and the stereo bar, and 
its z-axis coincides with the rotation axis. The cylindrical 
camera coordinate system (r-^-q) is a coordinate system adapted 
to describe this special camera geometry. In this system, r is the 
radius of the cylindrical panorama, ^ is the angle between the 
positive y-axis and the rotation radius, measured clockwise, and 
q is the same as z. When the rotation radius R, the swing angle 
Q, and the camera’s focal length c are fixed, r is then fixed 
( r = Vr 2 +c 2 + 2RccosQ ), and each point corresponds to unique 
(^, q) coordinates. 
The relationship between a point’s object coordinates (X, Y, Z) 
and its image coordinates (i, j) is defined by a sequence of 
coordinate transformations. The first transformation is a rigid 
body rotation and translation from the object coordinate system 
(X-Y-Z) to the local camera coordinate system (x-y-z): 
- 
(r 
Ml 
r 27 
r i2 
r 22 r 23 
Y-Y 0 
\ Z J 
/31 
r 32 r 33 , 
0 
N 
1 
N 
A h = horizontal angular resolution of a pixel 
Thus the collinearity equations for the left panorama are: 
f 
( \ 
\ 
f x Ì 
Q - arcsin 
RsinQ 
+ 4 
arctan — - 
[yj 
[v* 2 + y ¿ J 
N_ 
2 
cz 
tJx 2 + y 2 -R 2 sin 2 Q -RcosQ 
+ n 0 
/A. 
(4a) 
(4b) 
The observation equations that describe the transformation from 
the object point to the image point are given by incorporating 
Equation 1 and Equation 4. 
In Equation 4, N and A v are fixed values associated with the 
camera. A h is dependent on the exposure time and rotation 
speed, while % 0 , q 0 , c and R are acquired through calibration. 
For the right panorama, the collinearity equations are: 
f 
( ■ \ 
\ 
f x ^\ 
Q'- arcsin 
R' sin Q' 
arctan — - 
UJ 
(5a) 
/=- 
N' 
c z 
yjx 2 +y 2 -R' 2 sin 2 CS - R' cosQ' 
+ %' 
/A v ' ( 5b > 
where (X 0 , Y 0 , Z 0 ) are the object coordinates of the origin, o, of 
the local camera coordinate system; r n , r ]2 , ..., r 33 are 
components of the rotation matrix 
where q 0 ', £, 0 ', A h ', A v ', c', N', R', Q' are the corresponding 
parameters of the right panorama. 
In the second transformation, the point (x, y, z) is projected 
onto the cylindrical panorama. 
£ = arctan(—) - 
y 
n = 
Q-arcsin/ÜL) 
V 
x + y 
cz 
■yjx 2 + y 2 -R 2 sin 2 Q - RcosQ 
(2a) 
(2b) 
The parameters in collinearity Equations 4 and 5 have the 
following relationships, although some of them are difficult to 
realize: 
1) if the same camera is used to take both left and right 
panoramas, then c = c', A v = A v ', and N = N'; 
2) if the two stereo camera positions are equidistant from 
the rotation axis, then R = R'; 
3) if the two swing angles are the same, then Î2 = Q'; and 
4) if the two stereo camera positions have the same height, 
then q 0 = q 0 '. 
Equation 2b represents the pin-hole perspective projection of 
the linear array cameras in the off-axis panoramic scanning 
system. 
The last transformation transforms (£, q) into the image 
coordinates (i, j): 
i= l±A ( 3a ) 
A 
. = N_ _ n + Ro (3b) 
J 2 A v 
where E, 0 = the bias of coordinate E, 
q 0 = the bias of coordinate q 
N = resolution of the linear array CCD 
A v = pixel size of the linear array CCD 
If all the above conditions are met, a point in the object space 
will be projected onto the same image row in the two stereo 
panoramas. Also, the image rows in the right panorama will be 
the epipolar lines of corresponding rows in the left panorama. 
2.4 Stereo Geometry 
Figures 4 and 5 illustrate the stereo geometry of multi 
perspective panoramas acquired by this prototype system. The 
collinearity equations are used to model the optical rays from 
the left and right cameras to point P. From Figure 5(c) we can 
see that for any point at a constant distance from the mast (on a 
circle) in the object space, the stereo intersection angle is the 
same. That explains why the off-axis configuration of the 
cameras enables the system to have uniform 3-D measurement 
accuracy in all 360° directions. But the accuracy varies at 
difference distances in each radial direction.