Biomedical Engineering Reference
In-Depth Information
incorporated the global shape information into segmentation by using an elliptic
Fourier decomposition of the boundary and placing a Gaussian prior on the Fourier
coefficient. Chan et al. [35] and Leventon et al. [29] introduced the shape priors
into geodesic active models. To improve the robustness of level set-based methods
to noisy and incomplete data, Rousson and Paragios combined a shape influence
term [36].
3.
LEVEL SET METHODS AND IMAGE SEGMENTATION
3.1. Formulations
The level set method, developed by Osher and Sethian, is a zero equivalent
surface method [18]. Its basic idea is to change the movement track of a planar
curve into the movement track of a three-dimensional surface. Though this conver-
sion complicates the solution, it exhibits many other advantages: parameter-free
representation, topological flexibility, and the ability to handle local deformations
are its main strengths. These properties make the level set framework a suitable
choice for describing cardiac structure.
The classical level set boundary is defined as a zero level set of an implicit
representation Φ of the evolving curve. In detail, we now evolve the implicit
level set function Φ(
x, y, t
), to denote the evolution of a curve Γ(
t
) in its normal
direction with a speed
F
(
x, y
). At time
t
, the zero level set (
x, y
)
|
φ
(
x, y, t
)=0
describes the evolved contour Γ(
t
), namely,
Φ(Γ(
t
)
,t
)=0
.
(1)
When we differentiate this equation with respect to
t
and use the chain rule, we
have
∂φ
∂t
+
∇
∂
∂t
·
φ
.
(2)
Because
F
denotes the speed of the curve in its normal direction, it can be char-
acterized as
∂
∂t
·
N
=
F,
(3)
where
N
represents the outwards normal, and can be obtained by
∇
φ
N
=
.
(4)
|∇
φ
|
Substitute Eqs. (3) and (4) into (2), and we can get the evolution equation for
φ
as
Φ
t
+
F
|∇
Φ
|
=0
.
(5)
