Biomedical Engineering Reference
In-Depth Information
geometry. The latter represents an object shape and the former puts constraints
on how the shape may vary over space and time. Deformable models-based seg-
mentation approaches have many limitations, including their sensitivity to initial-
ization, the handling of the topological changes (curve breaking and merging)
during the course of evolution, and their dependence on a large number of tun-
ing parameters. The level set-based segmentation techniques offer a solution to
some of the limitations of the classical deformable models [49, 50, 51, 52]. These
techniques, for which the evolving curve is represented as the zero level set of
a higher-dimensional function, handle efficiently the topological changes. In ad-
dition, curve initialization can be either manual or automatic, and needs not be
close to the desired solution. However, an accurate initial estimation of each class
parameters is needed to ensure accurate segmentation.
In this work, we use our level set based-adaptive multimodal image segmenta-
tion approach [53, 54] to segment the images into WM, GM, and CSF. In [53], we
have proposed a novel and robust level set-based segmentation approach suitable
for both 2D and 3D medical applications. We build a statistical Gaussian model
with adaptive parameters for each region to be segmented, and we explicitly embed
these models into the partial differential equations (PDEs) governing the curves
evolution. The parameters of each region are initially estimated using the Stochas-
tic ExpectationMaximization (SEM) algorithm and are automatically re-estimated
at each iteration step. Our method differs from the one proposed in [55] in the
sense that it is more suitable for multimodal images and allows for re-estimation
of the probability density functions representing each class. The robustness and
accuracy of our technique were demonstrated through our work on MR images
and angiography [53]. Figure 3 shows the results of extracting the WM from a
stack of MR images using our evolution model, while Figure 4 shows some results
of our segmentation technique when applied to T1-weighted MRI scans to extract
the WM, GM, and CSF.
3.3. Distance Map and Shape Description
Distance map
D
(
x
), also known as a distance transform, assigns to each point
in a given image (2D or 3D) a minimal distance from a locus of points (usually
object boundaries) as shown in Figure 5.
The exact computation of
D
(
x
)) is very time consuming, especially for large
datasets. Therefore,
D
(
x
) can be discretely approximated using the Chamfer
metric [56], or continuously approximated by solving a special first order, nonlinear
partial differential equation, known as the eikonal equation. Several methods have
been proposed to solve the eikonal equation [57, 58, 59, 60, 61, 62], the most stable
and consistent of which is the fast marching method (FMM), which is applicable
to both Cartesian [61, 63] and triangulated surfaces [64, 65]. The FMM combines
entropy satisfying upwind schemes and fast sorting techniques to find the solution
in one pass algorithm. Although the FMM is the most stable and consistent method
