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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