ISPRS Commission III, Vol.34, Part 3A ,,Photogrammetric Computer Vision“, Graz, 2002
in other frames. For missing features, the noise model of
[Irani and Anandan, 2000] is inadequate, as will be seen
later. [Morris and Kanade, 1998] proposes a bilinear min-
imization method as an improvement on top of a standard
factorization. The bilinear minimization incorporates di-
rectional uncertainty models in the solution. However, the
method does not implement robust statistical techniques.
It is noted, that [Jacobs, 2001] have proposed a heuristic
method for dealing with missing data.
We have chosen to implement the proposed method in con-
junction with the factorization algorithm by Christy and
Horaud [Christy and Horaud, 1996]. This factorization
assumes perspective cameras as opposed to linearized ap-
proximations hereof. This yields very satisfactory results,
as illustrated in Section 5. In order to solve a practical
problem we propose a new method for Euclidean recon-
struction in Section 4, as opposed to the one in [Poelman
and Kanade, 1997].
2 FACTORIZATION OVERVIEW
This is a short overview of factorization algorithm. For a
more detailed introduction the reader is referred to [Christy
and Horaud, 1994, Christy and Horaud, 1996]. The factor-
ization methods cited all utilize some linearization of the
pinhole camera with known intrinsic parameters:
STij af len
SYij = bI tU | i | ( 1 )
s cT i
where the 3D feature, P;, is projected in frame i as (245, yi; ),
t; is the appropriate translation vector and a} ,b] and c;
are the three rows vectors of the rotation matrix. The used/
approximated observation model can thus be written as:
T ..
| 7 | = MP; @)
Yij
where M; is the 2 x 3 ’linearized motion’ matrix associated
with frame i.
When n features have been tracked in k frames, i € [1... k]
and j € [1...n], the observations from (2) can be com-
bined to:
S = MP 3)
where M is a 2k x 3 matrix composed of the M;, P is a
3 x n matrix composed of the P; such that:
Bit cot Tin
Xk1l ev
S = k kn
Vii ——'** Jin
Yet citm
The solution to this linearized problem is then found as the
M and P that minimize:
N-S-MP (4)
A - 16
Linear Factorization
of Structure and Motion
If Not
Modify Data to Approx. Stop
Perspective Camera
Figure 1: Overview of the Christy-Horaud algorithm.
where N is the residual between model, MP, and the data,
S. The residuals, N, are usually minimized in the Frobe-
nius norm. This is equivalent to minimizing the squared
Euclidean norm of the reprojection error, i.e. the error be-
tween the measured 2D features and the corresponding re-
projected 3D feature. Thus (4) becomes:
a D . KT 112
Sii MPIR mS 1% MPjls ©
=
where S; and P; denote the j*^ column of S and P, re-
spectively. In this case the solution to M. and P can be
found via the singular value decomposition, SVD, of S.
It is noted, that for any invertible 3 x 3 matrix, A:
MP = MAA‘P = MP (6)
Hence the solution is only defined up to an affine transfor-
mation. In [Christy and Horaud, 1996], a Euclidean recon-
struction is achieved by estimation of an A, such that the
rotation matrices, [a; b; ci]^, are as orthonormal as possi-
ble. Further details are given in Section 4.
2.1 Christy-Horaud Factorization
The approach we improve on by introducing arbitrary noise
models is the approach of Christy and Horaud [Christy and
Horaud, 1996]. This approach iteratively achieves a solu-
tion to the original non-linearized version of the pinhole
camera. These iterations consist of modifying the observa-
tions z;;, y;;, and hence S, as if they where observed by
an imaginary linearized camera, which in turn requires an
estimate of the structure and motion, see Figure 1. The
update formulae is given by:
M-(I-EDG-42) 0
Yij Yij Voi; i7
where (Z;5, §ii;) is the updated data and (z,,,, y,,,) is the
object frame origin.
3 SEPARATION WITH WEIGHTS
In order to deal with erroneous data, (5) should be com-
puted with weights:
HZ IV; (S; — MP;)IIZ (8)
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