Biomedical Engineering Reference
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segmentation. The key idea is to represent an interface Γ Ω in the image domain
R 3 implicitly as the zero level set of an embedding function φ : R 3 Ω:
Γ= {
x
|
φ ( x )=0 }
,
(1)
and to evolve Γ by propagating the embedding function φ according to an appro-
priate partial differential equation. The first applications of this level set formalism
for the purpose of image segmentation were proposed in [3, 4, 5]. Two key ad-
vantages over explicit interface propagation are the independence of a particular
parameterization and the fact that the implicitly represented boundary Γ can un-
dergo topological changes such as splitting or merging. This makes the framework
well suited for the segmentation of several objects or multiply connected objects.
When segmenting medical images, one commonly has to deal with noise,
and missing or misleading image information. For certain imaging modalities
such as ultrasound or CT, the structures of interest do not differ much from their
background in terms of their intensity distribution (see Figure 1). Therefore, they
can no longer be accurately segmented based on the image information alone. In
recent years, researchers have therefore proposed to enhance the level set method
with statistical shape priors. Given a set of training shapes, one can impose in-
formation about which segmentations are a priori more or less likely. Such prior
shape information was shown to drastically improve segmentation results in the
presence of noise or occlusion [6, 7, 8, 9, 10, 11]. Most of these approaches are
based on the assumption that the training shapes, encoded by their signed distance
function, form a Gaussian distribution. This has two drawbacks: First, the space
of signed distance functions is not a linear space; therefore, the mean shape and
linear combinations of eigenmodes are typically no longer signed distance func-
tions. Second, even if the space were a linear space, it is not clear why the given
set of sample shapes should be distributed according to a Gaussian density. In fact,
as we will demonstrate in this work, they are generally not Gaussian distributed.
Recently, it was proposed to use nonparametric density estimation in the space of
level set functions [8] in order to model nonlinear distributions of training shapes.
(The term nonlinear refers to the fact that the manifold of permissible shapes is not
merely a linear subspace.) While this resolves the above problems, one sacrifices
the efficiency of working in a low-dimensional subspace (formed by the first few
eigenmodes) to a problem of infinite-dimensional optimization.
In the present chapter, we propose a framework for knowledge-driven level
set segmentation that integrates three contributions. 1 First, we propose a statistical
shape prior that combines the efficiency of low-dimensional PCA-based methods
with the accuracy of nonparametric statistical shape models. The key idea is to
perform kernel density estimation in a linear subspace that is sufficiently large to
embed all training data. Second, we propose to estimate pose and translation pa-
rameters in a more data-driven manner. Thirdly, we optimally exploit the intensity
information in the image by using probabilistic intensity models given by kernel
density estimates of previously observed intensity distributions.
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