Heikkinen, Jussi 
  
2.2 Estimation problem, Approach II 
The image block estimation based on bundle of rays model can also be written by using different set of parameters. As 
we are not using any exterior control points and we are constructing our own co-ordinate system we can make some 
assumptions. As we have already defined origin to be the revolution center and x-axis lying at the direction of the first 
photo projection center, we can also state that projection centers of the camera will lie on a plane parallel to a co- 
ordinate plane. We can choose the y-axis to point upwards so the all projection centers will lie on the xz-plane. This is 
due to avoid situations where the denominator of the collinearity condition will go to zero. Now we can fix the y-co- 
ordinate of all projection centers to be a constant and express the x- and z-coordinates in a polar coordinate system: 
(7) 
X; =r-cosa, 
Y, =constant 
Z; =r-sina, 
l 
Now we do not need to put any constraints to force parameter values of the projection centers to lie on a plane nor to be 
in a distance of r from the revolution center, cause all this information is included in equation. What we have not yet 
included in the model is the constant angle between optical axis of the camera and the position vector of the projection 
center. In a special case where we have zero tilt (@) and spin (K) angles, our rotation matrix differs from the first camera 
rotation matrix Ra, only by the difference Re, «. Note that the angle values in $ increase in the opposite direction 
than a values. In practice this kind of configuration is quite easy to arrange. But more generally, the rotation of camera 
by o angle can be expressed in 2D rotation on plane: 
(8) 
Ro Az HR 
0, .K; ; (d Ko 
where, 
(9) 
cosa, 0 —sina, 
R, =| O0 1 0 
sing; 0... Cosa; 
Since the rotation angles à, à; and K; depend on the first photo orientation matrix by the angle oy, there is no point to 
estimate those dependent parameters. Instead, we should linearize the nonlinear observation equation respect to first 
photo rotation angles, radius of the circle r and separate o; angles. Now, the total number of photo unknowns will be 
4+n. The effect of one image observation on normal matrix N is depicted in Figure 4. 
ee r i XYZ 
  
  
  
  
Figure 4 Effect of one image observation on normal matrix 
  
International Archives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part BS. Amsterdam 2000. 361