Image Processing Reference
x 10 6
3 x 10 6
Figure 4.9: (a) Singular values of the linear system of Eq. 4.1 written in terms of the 243 vertex coordinates
of a mesh. As mentioned in Chapter 3 , the number of singular values close to zero is the number
of vertices. (b) Describing the shape with 50 PCA modes helps constraining the corresponding linear
system. However, there are still a number of near zero eigenvalues.
Since the vectors s i are computed as eigenvectors of a covariance matrix, following standard practice
in modal analysis, it then makes sense to solve
MS Mx 0
λ r L0
in the least squares sense, where L is a diagonal matrix whose elements are the inverse values of the
eigenvalues associated to the eigenvectors, and λ r is a regularization weight. This favors the modes
that correspond to the lowest-frequency deformations and therefore further enforces smoothness.
In practice, the linear system of Eq. 4.8 is less poorly conditioned than the one of Eq. 4.1 ,
but, as depicted by Fig. 4.9 , its matrix still has a number of near zero singular values, indicating
that there are several smooth shapes that all yield virtually the same projection. As a consequence,
additional constraints still need to be imposed for the problem to become well-posed. We will see in
Section 4.2.3 that forcing geodesic distances to be preserved across the surface in one way of doing
this. Another is to exploit additional sources of image information, as discussed below.
In Salzmann et al. [ 2008a ], it was proposed to treat the small singular values of Eq. 4.8 as
if they were exactly zero and write potential solutions as linear combinations of the corresponding
singular vectors. In other words, the mode weights can be written as
β i m i ,
where the m i are the singular vectors associated to the smallest singular values of the matrix of
Eq. 4.8 . The unknowns become the weights β i . Each set of weights produces a different 3D surface
that projects at approximately the correct place in the input image. Therefore, additional information
must be brought to bear to choose the best possible values of β i .