Biomedical Engineering Reference
In-Depth Information
of Eq. (2), where
T
represents a time when a segmentation result is achieved. The
Perona-Malik function
g
IR
0
→
IR
+
is nonincreasing,
g
(0)=1, admitting
:
g
(
s
)
→
0 for
s
→∞
[21]. Usually we use the function
g
(
s
)=1
/
(1 +
Ks
2
),
C
∞
(
IR
d
K
≥
0.
G
σ
∈
) is a smoothing kernel, e.g., the Gauss function
1
(4
πσ
)
d/
2
e
−|x|
2
/
4
σ
,
G
σ
(
x
)=
(6)
which is used in pre-smoothing of image gradients by the convolution
ξ
)
I
0
(
ξ
)
dξ,
I
0
=
∇
G
σ
∗
∇
G
σ
(
x
−
(7)
IR
d
with
I
0
the extension of
I
0
to
IR
d
given by periodic reflection through the boundary
of the image domain. The computational domain Ω is usually a subdomain of
the image domain, and it should include the segmented object. In fact, in most
situations Ω corresponds to the image domain itself. Due to the properties of
function
g
and the smoothing effect of convolution, we always have 1
≥
g
0
≥
ν
σ
>
0 [22, 24]. In [51, 53], the existence of a viscosity solution [54] of the
curvature driven level set equation [11], i.e., Eq. (1) with
g
0
≡
1, was proven. For
analytical results on Eqs. (1) and (2), respectively, we refer the reader to [47, 45]
and [49, 55], respectively.
3.
SEMI-IMPLICIT 3D CO-VOLUME SCHEME
Our computational method for solving the subjective surface segmentation
equation (2) uses an efficient and unconditionally stable semi-implicit time dis-
cretization, first introduced for solving level set-like problems in [16], and a three-
dimensional complementary volume spatial discretization introduced in [56] for
image processing applications. In this section we present the serial algorithm of
our method, and in the next section we introduce its parallel version suitable for
a massively parallel computer architecture using the message passing interface
standard.
For time discretization of nonlinear diffusion equations there are basically
three possibilities: implicit, semi-implicit, or explicit schemes. For spatial dis-
cretization usually finite differences [13, 14], finite volumes [57, 58, 59, 26], or
finite-element methods [60, 61, 62, 63, 64, 30, 24] are used. The co-volume tech-
nique (also called the complementary volume or finite volume-element method)
is a combination of the finite-element and finite-volume methods. The discrete
equations are derived using the
finite-volume methodology
, i.e., integrating an
equation into the so-called control (complementary, finite) volume. Very often the
