Image Processing Reference

In-Depth Information

BA-Quad

BA-Lin

EM-LDS

Figure 6.12:
Comparison of the results obtained with global quadratic models
Fayad
et al.
[
2009
], global

linear models with bundle adjustment
Del Bue
et al.
[
2007
] and the EM-LDS algorithm of
Torresani
et al.

[
2008
]. Note that the quadratic models are better suited to model these large deformations. Courtesy of

A. Del Bue.

points is expressed as

⎡

⎣

⎤

⎦

x
1

···

x
N
c

y
1

···

y
N
c

···

z
1

z
N
c

x
1

···
x
N
c

=

y
1

···
y
N
c

Q

,

(6.19)

z
1

z
N
c

···

x
1
y
1

···

x
N
c
y
N
c

y
1
z
1

···

y
N
c
z
N
c

z
1
x
1

···

z
N
c
x
N
c

Q

where
,
, and
are 3

3 matrices containing the coefficients of the linear, quadratic, and mixed

terms, respectively. This formulation relies on the availability of a rest shape
ˆ
Q
that can be obtained

from a rigid factorization algorithm. One advantage of this shape parameterization is that the basis

is completely determined by the rest shape. Therefore, there is no need to optimize it. Some of

the corresponding basis shapes are depicted in Fig.
6.11
. One drawback of this model is that, if the

values of the coefficients are left unconstrained, it can produce unrealistic shapes. On the other hand,

when the coefficients are initialized correctly and are appropriately bounded, the resulting technique

allows for the reconstruction of complex shapes, as depicted by Fig.
6.12
.

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