Heikkinen, Jussi 
  
3 IMAGE BLOCK ESTIMATION 
The model used is based on image bundle blocks and collinearity condition. Another alternative would have been to use 
independent stereo models as primary computation units. It is true that block adjustment based on stereo models and 
coplanarity condition do not include unknown 3D object points in estimation, which was stated earlier. But geo- 
metrically thinking those unknown points are still there, and in a case of bundle of rays you can always eliminate the 
unknown 3D points out of a LSQ type estimation like (Mikhail, 1976): 
( 10) 
A = |A, A, ] 
(11) 
N= AA N Ng 
N, Nj 
Nip Nod, b, 
Nua N» ]Lx; b; 
The normal design matrix A is partitioned in two sub matrices where the columns of the matrix represent the 
coefficients of the photo orientation or 3D point co-ordinate values as depicted in equation ( 10 ). The normal matrix N 
can then be reduced to the size of the sub matrix Ny; of the original normal matrix as shown in equation ( 11 ) (Mikhail, 
1976). 
(12) 
(N NN N,)x, - b, NAN] b, 
By using the linear model we do not need to re-eliminate the 3D point unknown parameters, but since we have chosen 
to use a nonlinear type model we are forced to resolve corrections also to approximations of point unknown parameters. 
The re-elimination is depicted in equation ( 13 ). 
( 13) 
X5 -N3(b-Nx) 
The idea of eliminating the unknown point parameters from the estimation might be rather beneficial. Since we are 
going to have numerous images included in a single circular image block, we will also most likely have numerous 
unknown 3D points. One additional photo increases the number of unknown parameters by one, but one additional 3D 
point increases the number of unknowns by three. As the normal matrix N can be updated by observation equations 
iteratively, there is no need to construct the design matrix A at all. Also, the elimination of unknown parameters can be 
done iteratively. So the elimination can be done point by point. Though we have to take care that all observation 
attached to that point are updated consecutively to the sub matrix of Nj, and N,,. After this the reduced normal matrix 
can be updated by using equation ( 12 ) and we can continue by processing the observations of the next point. The 
number of steps to construct the reduced normal does not differ much from the number of steps used to construct the 
original N. Calculating the re-elimination increases the number of steps but the computing time in this task is 
tremendously short compared to time spent for computing the LSQ solution for a large normal matrix N. 
3.1 Aspect of geometry 
In chapter 2 we introduced the camera configuration of a circular image block. We mentioned that point intersection 
would be poor unless we do not use image observations from a second co-centric image block. The difference between 
these two co-ordinate systems is only an angle between their x-axes. So the angle can be estimated from observations of 
common points. 
When thinking about single block estimation we can find that the same geometrical problem appears. The angle of 
image rays for the unknown 3D tie point will be rather small (Figure 5). So the position accuracy for such a point is 
questionable. Even though those tie points are not going to be used for modeling purpose, the unreliability of those 
observations also affects camera orientations. 
  
362 International Archives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part B5. Amsterdam 2000.