PerformingNon-Rigid Structure fromMotion - Deformable Surface 3D Reconstruction from Monocular Images - page 72

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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