ISPRS Commission III, Vol.34, Part 3A „Photogrammetric Computer Vision“, Graz, 2002 
  
  
  
  
  
  
S Noise This BFGS Flop 
kxn | Ratio | Method Ratio 
20x40 | 0.02 | 1.200407 2.32¢+08 | 19.33 
>? 0.10 | 1.58e+07  581e+08 | 36.73 
>? 0.50 | 5.50e+07 4.226+08 | 7.67 
40x40 | 0.02 | 7.20e+07 1.99e+09 | 27.58 
-". 010 | 1.15e+08 . 3.64et09 | 31.73 
>? 0.50 | 3.59e+08 — - 
80x40 | 0.02 | 5.17e+08 1.78e+10 | 34.41 
-". 0.10 | 8.00e+08  7.08e-10 | 88.52 
-" 0.50 | 2.300409  8.74e*10 | 37.93 
  
  
  
Table 1: Computational time comparison of the proposed 
  
algorithm with MatLab's BFGS (fminu()), — denotes that 
the optimization did not converge due to ill-conditioning. 
   
   
   
Christy-Horaud with 
Weights 
   
  
  
If Not 
le Stop 
  
   
Calculate New Weights 
Figure 3: Overview of the proposed approach for arbitrary 
error functions. 
Rangarajan, 1996]. This is achieved in the presented setup 
via Iterative Reweighted Least Squares (IRLS). Where IRLS 
works by iteratively solving the "weighted" least squares 
problem and then adjusting the weights, such that it cor- 
responds to the preferred error function, see Figure 3. A 
typical robust error function is the truncated quadratic: 
1 || Vij 
Us; = k2 
M vr NVA 
where N;; is the residual on datum 47, wj; is the corre- 
sponding weight and k is a user defined constant relating 
to the image noise. If an a priori Gaussian noise struc- 
ture, X;;, is known for the 2D features, the size of the 
residuals Nj; is evaluated in the induced Mahalanobis dis- 
tance, otherwise the 2-norm is used. In the case of a priori 
known Gaussian noise, X;, it is combined with the trun- 
cated quadratic by VV, — wj Ed otherwise VV; = 
WjW; . 
  
Ey SR 
=. To (10) 
  
4 EUCLIDEAN RECONSTRUCTION 
The objective of Euclidean reconstruction is to estimate the 
A in (6), such that the a;,b; and ¢; of (1) are as orthonor- 
mal as possible. In the paraperspective case [Poelman and 
Kanade, 1997], which is the linearization used in Christy 
and Horaud [Christy and Horaud, 1996], the M;'s compos- 
ing M are given by: 
A - 18 
  
Figure 4: A sample frame from the 
Eremitage sequence. 
  
Figure 5: A sample frame from the Court 
sequence. The test set was generated by 
hand tracking 20 features in this sequence 
of 8 images. 
where (zi, yoi) is the projection of the object frame origin 
in frame 4. 
Since the paraperspective approximation is obtained by lin- 
earizing +c?-P; the orthonormal constraints are restricted 
to a; and b;.With Q = AAT these constraints can be 
formulated as [Christy and Horaud, 1996, Poelman and 
Kanade, 1997]: 
Vi al Qa; =bIQb; = 
ITO, de Q4 
1-4 22; " 1-4 32; e 
Vi alQb; 2-02 
Tool; Ql)  voiyo(J7 QJi) _ 0 
2(1 +22.) 2(1 +2) 
Vi 0 
  
  
Vi IiQJ;— 
With noise, this cannot be achieved for all i and a least 
squares solution is sought. In order to avoid the trivial null- 
solution the constraint al Qa, = blQb, = 1 is added 
[Christy and Horaud, 1996, Poelman and Kanade, 1997] 
and the problem is linear in the elements of Q. 
This approach has the disadvantage, that if Q has nega- 
tive eigenvalues, it is not possible to reconstruct A. This 
problem indicates that an unmodeled distortion has over- 
whelmed the third singular value of S [Poelman and Kanade, 
1997]. This is a fundamental problem when the factoriza- 
tion method is used on erroneous data. 
  
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