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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