Image Processing Reference
In-Depth Information
Unfortunately, while in the weak perspective case
W
was known, here it depends on the
unknown perspective depth scalars
d
i
. As a consequence, the solution cannot be directly estimated
by a simple singular value decomposition. To overcome this difficulty, several solutions have been
proposed. In
Xiao and Kanade
[
2005
], an iterative procedure was introduced to alternatively compute
the structure and motion from fixed depths, and vice-versa. Initially, the depths
d
i
were set to 1.
In
Llado
et al.
[
2010
], some parts of the surface were assumed to move rigidly. Therefore, an initial
solution was computed using the results obtained with a rigid structure from motion techniques
on these parts and refined using a nonlinear optimization method. Recently, in
Hartley and Vidal
[
2008
], it was shown that the solution to perspective NRSFM could be obtained in closed-form
by exploiting the tensor estimation and factorization method of
Hartley and Schaffalitzky
[
2004
].
While this gives an exact solution in the noise-free case, the approach is sensitive to noise. As
observed in
Hartley and Vidal
[
2008
], this is mainly due to the fact that the tensor estimation and
factorization method they relied on
Hartley and Schaffalitzky
[
2004
] lacks robustness to noise, as
many purely algebraic methods do.
5.4 AMBIGUITIES OF NRSFM
Even though, in many NRSFM methods, the shape is already regularized by a linear subspace model,
ambiguities remain. This makes sense, since the shape basis also is an unknown of the problem. Fur-
thermore, while for template-based reconstruction going from weak to full perspective theoretically
yields a better-posed problem, perspective NRSFM still suffers from the same ambiguities as the
weak persective formulation.
First, the decomposition of
W
into
C
and
B
can only be computed up to an invertible
transformation. Indeed, for any invertible 3
N
s
×
3
N
s
matrix
G
, we can write
=
CGG
−
1
B
W
=
CB
.
(5.13)
This was also observed for the rigid structure-from-motion problem in the factorization method
of
Tomasi and Kanade
[
1992
]. This matrix
G
is known as the corrective transformation. Since, in
theory, any
G
would do, a way must be found to choose the best one. Typically, this is done by finding
a
G
that ensures that the rotation matrices are orthonormal. Details on the different manners to
exploit this will be given in Chapter
6
.In
Xiao and Kanade
[
2004
],
Xiao
et al.
[
2004b
], it was argued
that, even when enforcing orthonormality constraints, ambiguities remained in the reconstruction.
However, it was later shown in
Akhter
et al.
[
2009
] that all solutions in this ambiguous space yield
equal structures up to a 3D rotation.
In addition to the corrective transformation, other ambiguities inherent to NRSFM were
discussed in
Aanaes and Kahl
[
2002
]. One of them is the relative translation and scale between
the camera center and the object. As in the template-based case of Chapter
3
, it is impossible to
differentiate between a fixed camera seeing an expanding object and a camera moving closer to a
constant-size object. This, in general, is overcome either by fixing the object scale, or by imposing
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