Image Processing Reference
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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
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
w i φ i ,
i =
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.
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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