Ilkka Niini 
  
4 BLOCK ADJUSTMENT STEPS 
The projective block adjustment method is a stepwise method. First, find an optimal combination of image pairs by Using 
a suitable optimization algorithm (Niini, 1998). Second, solve the singular correlation and interior orientation parameter, 
and possible non-linear parameters (Niini, 1996). Third, solve the rotations of the images. Due to the equation 1, the 
rotations can now be computed from the adjusted singular correlations and interior orientation parameters without furthe 
adjustments. Fourth, solve the positions of the images along with the model co-ordinates. Fifth, change to the physic 
model (equation 6), and make a new adjustment for all physical parameters simultaneously. The projective stage consi 
of the steps 1-4, and the physical stage is the fifth step alone. The model is re-computed after the physical stage, too. 
The new constraints (equations 1 and 5) are added in the second step of the projective part of the method. The experimen 
made in this article show that the new constraints make the results from the projective stage better. 
4.1 Block optimization in the projective stage 
There can be n(n — 1)/2 image pairs in an arbitrary block. In the optimization stage, the goal is to find the best imag 
pairs among all possible ones so that the inner geometry of the block, in terms of the determinability of the chosen relative 
projective parameters, would be optimal. It has been shown earlier that, in general, the correct number of independent 
block parameters is obtained when the block of n images is arranged in the form of n — 2 image triplets (trifocal planes, 
or triangles) and 2n — 3 image pairs (Niini, 1998). The optimization is made as a triangular network optimization. Th 
optimization algorithm can be tuned so that the number of observations also affects to the search result, thus, the mor 
observations, the better chances there are to be taken into the adjustment. 
The block is solved as a system of 7(2n — 3) projective parameters, and m < 5n — 8 interior orientation parameters, which 
are constrained with 3(n — 2) trifocal constraints, with 2(2n — 3) interior orientation (Kruppa) constraints, and with n -? 
new trifocal interior orientation constraints. The number of free parameters in the block is then 7(2n — 3) — 3(n — 2) = 
11n — 15. 
One reason for the optimization is also the need to avoid certain degenerate cases of three images inside the block (collinex 
projection centres, parallel rotation axes, certain symmetric orientations) (Niini, 1998). The coplanarity of the projection 
centres of three images and an object point is also a dangerous case, unless the point is obsereved from a fourth imag, 
projection centre of which is not in the same plane. Otherwise, if an object point and the projection centres of three or 
more images lie in the same plane, the 2n; — 3 singular correlation equations between the observations of this point ar 
no more independent. In fact, due to the coplanarity, only n;, — 1 singular correlation equations are independent for the 
observations of this point. 
4.2 Pointwise optimization in the physical stage 
In the physical stage, the adjusted observation pairs can be chosen for each object point individually, so that the corre 
number of independent equations per object point is obtained. Again, there can be at most 2; — 3 independent equations 
between the observations of this point when the point has been observed from n;, images. Any additional equation for thi 
point p would be redundant and make the corresponding residual coefficient matrix B singular in the general adjustmen 
system Az + Bv = f. The same optimization algorithm that was used in the general block optimization before tlt 
projective stage can be used here, but separately for each point. The datum was fixed by using the seven inner constraint 
applied to the base components. 
4.3 Bundle adjustment 
An enhanced collinearity model (Melen, 1994) was used in the bundle adjustment, in order to get the treatment of interit 
orientation and non-linear lens distortion parameters similar to the one used in the projective block adjustment. The bundk 
method allows any combination of image observations to be used by requiring only that the observations from differeil 
images are consistent with a single 3-D object point. The datum was fixed with seven inner constraints, applied to the 
image positions only. 
5 TESTS WITH REAL DATA 
The two versions of the new block adjustment method were compared with a free-network bundle adjustment. Varios 
properties could be compared: the size of the system (number of unknowns), processing time, mean errors of the parame 
ters, root mean square error (RMSE) of the 3-D model, etc. Here, only the system sizes and the RMSE values of the 3D 
models were compared. 
  
646 International Archives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part B3. Amsterdam 2000. 
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