Image Processing Reference

In-Depth Information

6.2.1 FROMBASIS SHAPESTOBASISTRAJECTORIES

As mentioned in Chapter
5
, while most NRSFM approaches represent the shape with a linear

subspace models, some recent works have proposed different formulations. Among them, the method

of
Akhter
et al.
[
2008
] introduced an alternative approach to enforcing temporal consistency by

formulating NRSFM in trajectory space. In other words, instead of reconstructing the whole shape

at each time instant, the trajectory over the whole sequence of each 3D point is estimated.

To this end, the usual shape basis is replaced by a trajectory basis, as shown in Fig.
6.4
.

More specifically, the
x
−

T

,
y
−

, and
z
−

trajectories of a 3D point
q
i
=[
x
i
,y
i
,z
i
]

in
N
f

frames

=[
x
i
,
···
,x
N
f

=[
y
i
,
···
,y
N
f

=[
z
i
,
···
,z
N
f

T
,
t
i

T
, and
t
i

are defined as
t
i

T
, respectively.

Assuming that these trajectories can be described as a linear combination of
N
t
basis trajectories
θ
k
,

this yields

]

]

]

i

i

i

N
t

N
t

N
t

a
i,k
θ
k
,
t
i

a
i,k
θ
k
,
t
i

a
i,k
θ
k
,

t
i

=

=

=

(6.12)

k
=
1

k
=
1

k
=
1

where
a
i,k
,
a
i,k
, and
a
i,k
are the
x
−

coefficients for point
i
and basis trajectory
k
, and
θ
k

is an
N
f
-dimensional vector. Given this formulation, NRSFM can be re-written as the factorization

problem

,
y
−

, and
z
−

⎡

⎤

a
1
,
1

a
N
c
,
1

···

⎡

⎤

⎣

.

.

.

⎦

β
1

⎣

⎦

a
1
,N
t

a
N
c
,N
t

β
1

···

⎡

⎤

a
1
,
1

a
N
c
,
1

β
1

···

R
1

.

.

.

.

⎣

⎦

W

=

...

,

(6.13)

R
N
f

a
1
,N
t

a
N
c
,N
t

β
N
f

···

β
N
f

a
1
,
1

a
N
c
,
1

···

β
N
f

.

.

.

a
1
,N
t

a
N
c
,N
t

···

θ
1
,

,θ
N
t
]

contains the
j
th
element of all
θ
k
, and
W
is the same measurement

matrix as before.
W
is then factorized into
and
, and the resulting corrective transform is

estimated by ensuring that the rotation matrices are orthonormal. In practice, the
Akhter
et al.

[
2008
] method assumes that the basis trajectories
θ
k
are known and can be generated from the

Discrete Cosine Transform. While this might seem restrictive, it was shown to generalize to many

different trajectories. The results are compared to those of
Torresani
et al.
[
2008
] and
Xiao
et al.

[
2004b
]inFig.
6.5
. Note that the reconstructions of
Akhter
et al.
[
2008
] correspond more closely

to what is depicted by the images.

where
β
j

=[

···

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