Heikkinen, Jussi 
  
2.1 Estimation problem, Approach I 
We can see the estimation problem as n+1 number of relative orientations, or even n number if we include relative 
orientation between first and last image. By choosing the model of independent stereo models we can avoid resolving 
the object point unknowns. That is good since we are primarily interested in image orientations in an image block. 
Alternatively, we can end up using the bundle block estimation and applying collinearity condition. Then we will have 
n number of exterior orientations to be solved. 
In both cases we have a datum what is insufficient. We can apply freenet type solution and use minimum norm solution 
or we can fix some parameters in order to get datum become sufficient. Fixing the x-axis of a defined co-ordinate 
system in the direction of the first image projection center solves this problem. 
As the camera is rotated around the origin, the rotation angles will change respect to the co-ordinate system but the 
angle between the image plane and position vector of the projection center will stay constant. By applying this 
knowledge and the fact mentioned earlier that all projection centers lie on the path of the same circle, we can set 
constraints to stabilize the estimation process. 
(1) 
pJ=r 
or 
(2) 
  
=r 
Equations ( 1 ) and ( 2 ) state that all projection centers are at the same distance from revolution center, but this does not 
say anything about them lying on the same plane. This can be expressed by setting a constraint between projection 
centers and a normal vector n of the plane. 
(3) 
Pen=0 
We can also consider N as a new parameter vector to be estimated. By giving a big weight in LSQ estimation for this 
observation equation we force the system to retain this condition. The constant angle between the position vector of the 
projection center and the optical axis of the camera can be forced by adding in estimation a constraint like: 
(4) 
o Pp 
R, [0 |e —— = constant 
a 
By using independent stereo models and coplanarity condition we will have in LSQ estimation n*5+3 unknown 
parameters (n is included in unknowns) plus n*2 constraint equations. With bundle ray model we will end up to 6*n43 
plus m*3 unknown parameters, where m denotes the number of tie points. With stereo models the condition equation 
will take the form (Mikhail, 1976): 
65.) 
A t v) t Bx zd 
Cx=g 
and with bundle of rays based on collinearity condition, the equivalent form is: 
(6) 
Ax=v+| 
Cx=g 
  
360 International Archives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part B5. Amsterdam 2000.