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International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part Bl. Istanbul 2004 
  
correlation arising from the Kalman filtering of the combined 
IMU/GPS solution is stressed, to avoid biased mean values and 
too optimistic accuracy estimates from the calibration. 
Restricting the functional model of calibration to the 
misalignment angles, he argues that correlations between 
different updates in the KF solution can be modelled by an 
exponentially decaying function: 
p(t, t+ At) = e (4) 
where T is the bias variation correlation time of the gyroscopes 
and At the time interval between two images. 
Following his suggestion, we further extended our two-step 
model to account for a more complete stochastic model, on the 
AT side as well as on the IMU/GPS side. 
The functional model is again provided by equations (3), 
written in a slightly different way: 
tof Tra Lh. 
Far Sr Wwwsps * Ry a 
(5) 
hop pnm l/pi^ p. 
R ar" Re RyRR! eRCAaUGPS 
to stress that r,7" is the perspective centre position as computed 
by the AT and the product of the four rotation matrices on the 
right side yields the image attitude from IMU measurements. 
For each image of the calibration block, we have 12 therefore 
observed quantities: (X,Y,Z)ar, QGY,Z)mwaps. (0, 6, X)Ar and 
(0, , K)imu/aps- 
We write three equations for the projection centre position and 
three for the attitude angles. Unlike (Bäumker and Heimes, 
2001 and Skaloud and Schaer, 2003), out of the nine elements 
of R (both AT and IMU/GPS) we just take rj», r;; and rj; (the 
elements corresponding to c, 6, « in case of small rotations) to 
have truly independent information, since each rotation matrix 
depends originally on just three computed attitude values. 
From a general L.S. model with condition equations and 
observation equations: 
Dy =Ax+d (6) 
where y and x are the expected values of the observations, we 
linearize (5) with respect to the observations as well as with 
respect to the unknowns. The matrix D in (6) is therefore the 
Jacobian with respect to the observations and the matrix A is 
the Jacobian with respect to the unknowns (the design matrix). 
As far as the stochastic model is concerned, the covariance 
matrix of the observations is in principle a block diagonal 
matrix made of two blocks only, one representing the 
covariance matrix of the bundle adjustment, the other the 
covariance matrix of the Kalman filtering for the IMU/GPS 
solution. To keep things reasonably simple (thought no 
optimization of the computation has yet been performed) we 
only considered the 6x6 diagonal blocks referring to the EO 
parameters of each image from the bundle block. As far as the 
IMU/GPS observations are concerned, we followed the model 
suggested by Skaloud, considering time correlations for 
IMU/GPS positions and attitudes to be modelled by (4). Since 
we didn't have actual estimates for the variances from the KF 
nor specific information on the quality of the GPS data acquired 
during the calibration flight, we simply used the accuracies 
stated by the manufacturer for the IMU (8 mgon for pitch and 
roll, 10 mgon for yaw) and 6 cm in E,N and 10 cm in elevation 
for the GPS positions. 
521 
3. EXPERIMENTAL RESULTS 
3.1 Calibration parameters 
As mentioned previously, no attention has been paid yet to 
speed up computing time for the computation of the LS 
solution; the computational burden, though, increases 
significantly with the number of images in the calibration block. 
This has still prevented, at the time of writing, to compute the 
estimates of the new method using all images (see Table 3). 
With a smaller number of calibration images, using just four 
strips (2 perpendicular, flown twice in reverse, 5 GCP — figure 
6) this has been possible (see Table 4). 
  
Calibration a^ [cm] R. [deg] | 
method | Ex | Ey | Ez Q $ K | 
"two-steps"|-18.7| -5.0 | 27.5 | 180.6665 | -0.1365 | 179.9778 
"one-step" |-20.3| -6.9 | 23.2 180.6675 | -0.1360 |179.9778 | 
  
  
  
Wil 9 
  
  
  
  
  
  
  
  
Table 3 - Estimates for the calibration parameters from the 
whole block, with both methods 
  
Calibration a’ [em] R.° [deg] 
method | ex | ey | Ez © $ I + 
"two-steps"|-11.7| -1.2 | 39.5 | 180.6664 | -0.1377 | 179.9791 | 
"one-step" |-19.5| -7.0 | 29.6 | 180.6678 | -0.1359 179.9790 | 
“two-steps |-11.2| -2.5 | 33.4 | 180.6716] -0.1379 | 179.9779 | 
w.corr.” | 
  
  
  
  
  
  
  
  
  
  
  
  
Table 4 - Estimates for the calibration parameters from 4 strips, 
with both methods. 
As it is apparent, estimates for the misalignment angles do 
agree much more compare to offsets. Differences are better than 
2 mgon for the same dataset (and much more for k, whose 
accuracy is the highest). Differences in offsets are not 
significant statistically, tough. 
The one step method proved much more difficult to handle to 
reach convergence with respect to the OEEPE test where it was 
first applied. Indeed, unless the values of the two-step methods 
are used as approximations, at the first iteration the offset 
parameters can jump to meters. In the following iterations, 
though, angles converge more quickly. Correlations between 
offset and misalignment were also found higher than in the 
OEEPE test data, though they also strongly depend on the 
accuracy of the pseudo-observation on IMU/GPS data. Despite 
this problem, the one-step solution is the most coherent overall 
across the two calibration datasets. 
3.2 Discrepancies at the check points 
Different sets of calibration parameters have been obtained for 
each of the three methods, by varying the calibration block 
configuration either in terms of number and type of strips as 
well as of ground control provided. Each calibration dataset has 
been applied to the IMU/GPS data of the whole block to get the 
correct EO parameters. Then the coordinates of the 164 check 
points have been computed by forward intersection, so the 
results are directly comparable with those of the manual Aerial 
Triangulation and with the reference values from the terrestrial 
GPS network. 
Table 5 shows the RMS on the check points for the calibration 
computed over the whole block.