ISPRS Commission III, Vol.34, Part 3A ,Photogrammetric Computer Vision“, Graz, 2002 
  
  
  
  
  
680r 
676r 
T 
X 
c. 
=. 
672: 
6654 878 882 886 
u [pixel] 
0-1; —— Chi Square Distribution 
^ [| Histogram of Mahal. Dist. 
4 | (1000 Experiments) | 
0.05 
  
  
  
  
  
  
  
  
  
  
% 10 
  
40 
Figure 3: Top: Back-projection of a 3D point using es- 
timated projection matrices. The projection is estimated 
based on randomly perturbed observations in the image 
and in the drawing. 5000 samples are taken to obtain the 
empirical 90% confidence region (solid), which is very 
close to the predicted one (dashed) from (5). Bottom: 
The Mahalanobis distance of the estimated projection ma- 
trix and the true projection matrix is x?, distributed. The 
histogram represents the distribution obtained by 1000 ex- 
periments on perturbed data. The curve indicates the true 
X1, distribution. The mean Mahalanobis distance is 10.87, 
compared to 11, which is statistically not different. 
the projection matrix P and the associated covariance ma- 
trix X are estimated, we can obtain the uncertainty of 
any 3D point projected in the image. To visualize this, we 
select 3D points on the ground plane and points above of 
each, in heights of 100, 200 and 300 pixels. The uncertain- 
ties of their projections are shown in Fig. 4. Note that the 
ellipses are plotted magnified by a factor of 10 for better 
visibility. 
Self-diagnosis can be bound to the estimated variance fac- 
tor 85 (cf. (4)). Here we obtain 62 = 1.5, thus 5 = 1.25. 
Assuming the constraints actually hold, we can conclude 
that the standard deviation of the measurements are worse 
by a factor 05 — 1.25, which practically confirms the as- 
sumed standard deviation. An indicator for characterizing 
the accuracy of the estimated projection matrix is the un- 
certainty of the camera's center of projection Z (cf. A.1). 
Computing the standard deviations for the camera center of 
the image shown in Fig. 4, we obtain standard deviations 
ofox = 11mm, ey z 37mm, and 97 = 10mm, where 
the indices X, Y, and Z indicate the coordinate axes of the 
   
Figure 4: Uncertainty of projected 3D points. The el- 
lipses show the covariances of 3D points on the ground, 
and points above of each in heights of 100, 200, and 300 
pixels. The ellipses are plotted magnified by a factor of 
10 for better visibility. In this case, the ground plane is 
not located on the actual ground, but at the top edge of the 
concrete foundations. 
drawing. In this case, the camera's principal axis is roughly 
parallel to the Y -axis of the drawing. Due to the fact that 
the intrinsic parameters, in particular the principle distance 
c, is estimated, this appears to be a reasonable result and is 
fully acceptable in many applications. For augmented real- 
ity applications, it is very useful to superimpose the image 
of an installation and the associated drawing. Fig. 5 shows 
such an overlay resulting from application of our method to 
another scene. All information contained in the drawing is 
directly connected to the respective components in the real 
world. At the same time, blending drawings and images 
can be used as a visual accuracy check of the projection 
matrix. 
6 CONCLUSION 
We presented a new method for single image orientation 
using scene constraints in conjunction with a drawing of 
the scene. We use vertical lines, located in the drawing or 
the map, the direction of horizontal lines, which may also 
be contours of cylinders, and points. Besides a direct so- 
lution, we provided a statistically optimal estimation pro- 
cedure for the projection matrix P. This allows us to ver- 
ify the results statistically, in particular to test whether the 
assumed noise model is correctly chosen. We applied the 
method to synthetic data to validate the correctness of error 
prediction. Statistical tests on experiments using real data 
showed that good results can be obtained using a compara- 
tively small number of constraints. The availability of this 
type of statistical tools for self-diagnosis and for charater- 
izing the performance increases the usefulness of image 
analysis considerably. 
A APPENDIX 
A.1 Uncertainty of the Camera Center 
Let the projection matrix be partitioned in the classical way 
P — (H| - HZ) - (H|h). 
   
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