Image Processing Reference

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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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