Image Processing Reference
In-Depth Information
6.1 ORTHONORMALITY CONSTRAINTS
The first natural constraints that have been used to disambiguate NRSFM are orthonormality
constraints
Bregler
et al.
[
2000
]. As in rigid structure from motion
Tomasi and Kanade
[
1992
], they
were introduced to encode the fact that the rotation matrices are orthonormal. Therefore, the goal
is to find the invertible corrective transformation
G
that satisfies this property.
More specifically, from the formulation of Eq.
5.6
, one can write orthonormality constraints
for each of the individual blocks of
C
. This yields equations of the form
C
2
j
−
1
:
2
j
G
k
G
k
C
2
j
−
1
:
2
j
(c
k
)
2
R
j
R
j
T
(c
k
)
2
I
2
×
2
,
1
=
=
≤
j
≤
N
f
,
1
≤
k
≤
N
s
,
(6.3)
C
2
j
−
1
:
2
j
is a 2
C
corresponding to
where
×
3
N
s
matrix containing the two consecutive rows of
3 matrix containing three consecutive columns of
G
, and
c
k
is the weight of
the
k
th
basis shape in frame
j
. These constraints are quadratic in
G
, and typically need to be solved
by nonlinear optimization methods.
In the closed-form solution of
Xiao
et al.
[
2004b
], the authors proposed an approach to making
this step easier. To this end, they introduced new variables
H
k
=
frame
j
,
G
k
is a 3
N
s
×
G
k
G
k
. Given these quadratic
variables, the constraints are re-written as
C
2
j
−
1
H
k
C
2
j
−
1
−
C
2
j
H
k
C
2
j
=
0
,
1
≤
j
≤
N
f
,
1
≤
k
≤
N
s
,
(6.4)
C
2
j
−
1
H
k
C
2
j
=
0
,
1
≤
j
≤
N
f
,
1
≤
k
≤
N
s
.
(6.5)
The first constraint encodes both diagonal terms of Eq.
6.3
simultaneously, thus removing the
dependency on the unknown coefficients
c
k
. The second constraint represents the off-diagonal
terms. Only one such constraint needs to be added since
H
k
can be made implicitly symmetric,
which makes both off-diagonal terms identical.
Unfortunately, it was shown in
Xiao
et al.
[
2004b
] that this linearized version of the orthonor-
mality constraints is not sufficient to fully disambiguate the reconstruction problem. This led the
authors to argue that orthonormality constraints were insufficient on their own. However, as was
suggested in
Brand
[
2005
] and later proved in
Akhter
et al.
[
2009
], under noise-free observations,
orthonormality constraints are sufficient to overcome the corrective transformation ambiguity of
NRSFM. The reason for the remaining ambiguities found in
Xiao
et al.
[
2004b
] was that no con-
straint was added to force the rank of
H
k
to be 3. In
Akhter
et al.
[
2009
], it was shown that this
additional rank constraint was sufficient to determine the structure. More specifically,
G
k
can still
only be determined up to a Euclidean transformation, but this ambiguity has no influence on the
reconstructed structure. This is depicted in Fig.
6.1
, which illustrates the fact that a family of shapes
satisfies the orthonormality constraints, but that any point in this region gives the same structure up
to a 3D rotation.
While orthonormality constraints were shown to be sufficient to resolve ambiguities, solv-
ing the true constraints still involves a nonlinear optimization problem, which can lead to un-
desirable local minima. Several approaches to tackling this problem were proposed. For instance,
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