Image Processing Reference

In-Depth Information

Figure 6.9:
While standard NRSFM approaches assume that the shapes lie on a linear subspace, the

true manifold can be nonlinear. This manifold can be better approximated by locally smooth manifold

learning
Rabaud and Belongie
[
2008
]. Courtesy of V. Rabaud. © 2008 IEEE.

explicitly computed, which removes them from the variables to optimize. This is similar in spirit to

the formulation of Section
4.2.2.2
for local deformation models, where the coefficients were directly

obtained from the mesh vertices. By assuming Gaussian noise over the measurements and over the

shape, the distribution over the measurements is also Gaussian. In this framework, NRSFM can be

formulated as maximizing the joint likelihood of the image measurements whose negative logarithm

can be written as

N
f

w
j

q
T
E
j
VV
T

+
σ
m
I
E
j
T

+
σ
2
I
w
j

q

1

2

E
j

E
j

L
=

−

¯

−

¯

j
=
1

N
f

ln

+
σ
2
I
+
N
c
N
f
ln
(
2
π) ,

E
j
VV
T

+
σ
m
I
E
j
T

1

2

+

(6.18)

j
=
1

where
w
j
is the vector containing the two rows of
W
associated to frame
j
,
E
j
replicates
d
j
R
j

across the diagonal, with
d
j
the scalar accounting for depth in frame
j
,
V
is the matrix whose
k
th

column contains the vectorized basis shape
S
k
, and

q
contains the vectorized mean shape.
σ
m
and

σ
are the Gaussian noise variance of the shape and of the measurements, respectively. This negative

log likelihood is minimized via an EM procedure, whose initialization is obtained using a rigid

structure from motion technique. A comparison of the results of the different algorithms proposed

in
Torresani
et al.
[
2008
] and of other techniques on the shark data is shown in Fig.
6.7
. Fig.
6.8

depicts the robustness to the number of basis shapes of the same algorithms. As before, the error

is defined as the ratio between the 3D distance to ground-truth and the span of the true shape.

Note that the error obtained by the EM procedure of
Torresani
et al.
[
2008
] is relatively stable with

respect to the number of basis shapes.

All the above-mentioned algorithms still rely on a linear subspace model to represent the

deformations of the object of interest. In practice, this only applies to relatively simple deformations,

¯

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