Biomedical Engineering Reference
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Figure 6. Numbering scheme for derivative computation.
3.4. Numerical Implementation
In image segmentation case, the previous discussion in Section 3.2.2 shows
that the speed function usually includes: (1) a curvature-based speed term F ( κ );
(2) a constant speed term ( F A ), where the speed can be expressed as a function
F A ( x, y ) of the location ( x, y ); (3) a stopping criterion g I based on the image gra-
dient to extract the boundary (Eq. (21)). The level set partial differential equation
for this curve evolution is:
∂φ
∂t = g I ( ακ |∇ φ | + βF A ( x, y ) |∇ φ | ) ,
(22)
where α and β are constants; F A
is a constant speed term; g I
is also constant
during the propagation.
Consider the implementation of this PDE within the image domain. First,
define the function φ ( t ) in the narrow band area. We use the Euclidean distance
from point ( x, y ) to the zero level set as the value of the signed distance. The sign
is negative if the point is inside the curve, and positive if ( x, y ) is outside the curve.
All the calculations are carried out only within the narrow band area.
The evolution equation can now be represented as
φ ( t +∆ t )= φ ( t )+ g I ( αT 1 + βT 2 )∆ t.
(23)
First, we describe the finite-difference derivatives required for level set update and
curvature computation in T 1 and T 2 . For the convenience of representation of
derivatives, the neighborhood of φ is specified with the numbering scheme shown
in Figure 6.
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