Biomedical Engineering Reference
In-Depth Information
stands for the image content model, including any of the color, shape, and texture
features. A is used to balance the force from the front and the force from additional
items, and A
1.
4.3. Courant Friedrich Levy Condition
To ensure the stability when employing the numerical schemes to solve the
PDEs, the full matrix approach requires a time step that satisfies aCourant Friedrich
Levy (CFL) condition with regard to the maximum velocity over the entire do-
main, not simply in response to the speed of the front itself. Analogous with the
underlying wave equation, for an advective speed function F and the first-order
space scheme, a CFL condition requires the front to cross no more than one grid
cell in each time step. Thus, we require that
max
F t
x,
(34)
where the maximum is taken over values for F at all possible grid points, not
simply those corresponding to the zero level set. In practice, one can quickly scan
the range of the values for the speed function F and choose an appropriate time
step accordingly.
In a narrow-band implementation, the time step can be adaptively chosen
according to the maximum velocity field achieved within the narrow band. This
is advantageous when the front speed changes substantially as it moves (such as
in curvature flow). In such problems, the CFL restriction for the velocity field for
all the level sets may be much more stringent than the one for those sets within the
narrow band.
4.4. Augmented Speed Function for Segmentation
Some researchers [9, 42] introduce the stop term based on the image gradient:
1
K I ( x, y )=
.
(35)
1+ |∇
G σ ·
I ( x, y ) |
Here, the gradient can be treated as the difference measurement of gray/brightness
values. To adjust the influence of the image gradient on the speed term, we can
redefine it:
K I ( x, y )=
p
1
|∇
G σ ·
I ( x, y ) |
p =1
,
(36)
1+ |∇
G σ ·
I ( x, y ) |
1+ |∇
G σ ·
I ( x, y ) |
p
1. When p is larger, K I will be smaller, so it will control
the speed to decrease faster. Then the speed term can be written as
F
where the constant p
= K I ( k + F A ) ,
(37)
φ yy φ x 2 φ x φ y φ xy + φ xx φ y
( φ x + φ y ) 3 / 2
where k =
is the curvature of the front at point ( x , y ).
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