Image Processing Reference

In-Depth Information

best of our knowledge no similar nonlinear model has yet been used in a continuous optimization

framework for automatic 3D surface shape recovery from noisy image measurements.

In short, the nonlinear FEM models are more accurate but very complex and, in the end, only

adapted to very specific applications. One recurring problem is their very high dimensionality, which

makes fitting to noisy data problematic. Modal analysis has emerged as one potential solution to

this problem. Given a surface represented by an
N
v
-vertex triangulated mesh, it reduces the number

of degrees of freedom by coupling the motion of the vertices into deformation modes obtained by

solving the generalized eigenproblem

ω
2
M
φ,

K
φ

=

(2.2)

where
K
and
M
are the stiffness and mass matrices of Eq.
2.1
. The individual
φ
i
and
ω
i
are the modes

and their corresponding frequencies. The displacement of the mesh vertices can then be written as

3
N
v

u

=

w
i
φ
i
,

(2.3)

i
=

1

where
w
i
is the amplitude assigned to mode
i
. The values
w
i
therefore parameterize the deforma-

tion. In theory, there are 3
N
v
modes and thus parameters. In practice, the lower-frequency modes

have far more influence on the global surface shape than the higher-frequency ones. It is therefore

a valid approximation to discard the latter and to retain only a comparatively small number of the

former. Fitting a surface parameterized in this way to image data thus becomes a much lower-

dimensional problem. Initially introduced in the field of Computer Vision for image segmentation

purposes
Pentland and Sclaroff
[
1991
],
Pentland
[
1990
], modal analysis was also successfully ap-

plied to medical imaging
Nastar and Ayache
[
1996
], and non-rigid motion tracking
Tao and Huang

[
1998
].

While computationally efficient, modal analysis, as usually applied in our field, assumes a

constant stiffness matrix, which implies geometrically and materially linear deformations. This un-

fortunately never is the case, since it is only true for barely visible deformations. Such models are

therefore only rough approximations of the true nonlinear behavior.

2.2

LEARNEDDEFORMATIONMODELS

The physics-based approach is very attractive because it aims at modeling the true behavior of an

object. However, as discussed above, it is very difficult to come up with accurate models. This is both

because key physical parameters are often unknown and because there are pervasive nonlinear effects

that are very complicated to handle. Doing so would involve computationally expensive algorithms

that can get trapped into undesirable local minima. Furthermore, given the usual noisiness of image

data, it is not even entirely clear that this expense would truly result in improved accuracy.

As a result, learning models from training data was proposed as an alternative. Rather than

trying to guess unknown physical parameters, shape statistics are inferred from available examples

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