Biomedical Engineering Reference
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the negative curvature only. Based on the different definition of the function, the
movement of the front or surface is showing a different pattern. For the first case,
all the parts of the contour are moving, while the parts with positive curvature are
moving inward and the parts with negative curvature are moving outward. For
the second case, only the concave parts (parts with negative curvature) are moving
outward.
Figure 4 depicts propagation of the closed curve in 4 steps, which proves the
theorem. At the same time, it also proves the ability of the level set method to
deal with curvature-related problems. It is straightforward for higher-dimensional
applications around curvature. In the following images, propagation of a closed
3D surface is also shown. An important application could be to help unwrap the
fold parts of the original 3D image.
The level set methods exploit the fact that curves moving under their curvature
smooth out and disappear. If we take pixel values as a topographic map from an
image, the graylevel value is the height of the surface at that point. Let each closed
curve move under the curvature. Then very small ones, like spikes of noise, will
disappear quickly. The real boundaries still remain sharp, and they will not blur.
They just move according to their curvature (Figure 3).
The extension of this noise removal application is valuable for image seg-
mentation. By leaving the longer edges with smaller curvatures, the regions of
interests are highly enhanced.
3.2. Image Segmentation Using the Fast Marching Method
The Narrow-Band Method is able to combine the relative geometric factors
onto the front propagation. Thus, more complex speed functions F could be used
and the front line could be moving more accurately. Positive speed functions F
that depend on position and vary widely from point to point are best framed as
boundary value problems and approximated through the use of Fast Marching
Methods. With the help of the min heap method, Fast Marching Methods are
extremely more computationally efficient than the level set method. The boundary
value formulation does not need a time step and its approximation is not subject
to CFL conditions, so the implementation is simpler. Note that the domain of
dependence of a hyperbolic partial differential equation (PDE) for a given point in
the problem domain is that portion of the problem domain that influences the value
of the solution at the given point. Similarly, the domain of dependence of an explicit
finite-difference scheme for a given mesh point is the set of mesh points that affect
the value of the approximate solution at the given mesh point. The CFL condition,
named for its originators — Courant, Friedrichs, and Lewy — requires that the
domain of dependence of the PDE must lie within the domain of dependence of
the finite-difference scheme for each mesh point of an explicit finite-difference
scheme for a hyperbolic PDE. Any explicit finite-difference scheme that violates
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