Biomedical Engineering Reference
In-Depth Information
Uniform density
Gaussian density
Kernel density
Figure 2. Schematic plots of different density estimates within a subspace. Darker shading
indicates areas of high probability density for the respective models. The kernel density esti-
mator adapts to the training data more flexibly since it does not rely on specific assumptions
about the shape of the distribution.
The space of signed distance functions is known to not be a linear space.
Therefore, neither the mean shape φ 0 nor a linear combination of eigen-
modes as in (3) will in general be a signed distance function. As a con-
sequence, the functions φ ( x )
favored by the uniform or the Gaussian
distribution cannot be expected to be signed distance functions. The kernel
density estimator (6), on the other hand, favors shape vectors α , which
are in the vicinity of the sample shape vectors α i . By construction, these
vectors correspond to signed distance functions. In fact, in the limit of
infinite sample size, the distribution inferred by the kernel density es-
timator (6) converges toward a distribution on the manifold of signed
distance functions.
Figure 2 shows schematic plots of the three methods for a set of sample data
spanning a two-dimensional subspace in
R 3 . The kernel density estimator clearly
captures the distributionmost accurately. As we shall see in Section 5, constraining
a level set-based segmentation process by this nonparametric shape prior will allow
to compute accurate segmentations even for rather challenging image modalities.
Figure 3 shows a 3D projection of the estimated shape density computed for
a set of silhouettes of a walking person. The bottom row shows shape morphing
by sampling along geodesics of the uniform and the kernel density. These indicate
that the kernel estimator captures the distribution of valid shapes more accurately.
In analogy to shape learning, we make use of kernel density estimation to learn
the conditional probability for the intensity function I in (4) from examples. A
similar precomputation of intensity distributions by means of mixture models was
proposed in [14]. Given a set of presegmented training images, the kernel density
estimate of the intensity distributions p in and p out of object and background are
given by the corresponding smoothed intensity histograms. This has two advan-
tages. First, the kernel density estimator does not rely on specific assumptions
about the shape of the distribution. Figure 1 shows that the intensity distributions
for ultrasound and CT images are not well approximated by Gaussian or Lapla-
 
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