Biomedical Engineering Reference
In-Depth Information
condition
u
(
t
,
x
)
=
u
D
in [0
,
T
]
×
∂,
(11.3)
u
(0
,
x
)
=
u
0
(
x
)in
(11.4)
.
Without loss of generality, we may assume
u
D
=
0. The Perona-Malik function
g
:
IR
0
→
IR
+
is nonincreasing,
g
(0)
=
1, admitting
g
(
s
)
→
0 for
s
→∞
[45].
Usually we use the function
g
(
s
)
=
1
/
(1
+
Ks
2
),
K
≥
0.
G
σ
∈
C
∞
(
IR
d
)isa
smoothing kernel, e.g. the Gauss function
1
(4
πσ
)
d
/
2
e
−|
x
|
2
/
4
σ
G
σ
(
x
)
=
(11.5)
,
which is used in presmoothing of image gradients by the convolution
IR
d
∇
G
σ
(
x
−
ξ
)
I
0
(
ξ
)
d
ξ,
∇
G
σ
∗
I
0
(11.6)
=
with
I
0
being the extension of
I
0
to
IR
d
given by periodic reflection through the
boundary of image domain. The computational domain
is usually a subdo-
main of the image domain; it should include the segmented object. In fact, in
most situations
corresponds to image domain itself. We assume that an initial
state of the segmentation function is bounded, i.e.
u
0
∈
L
∞
(
). For shortening
notations, we will use the abbreviation
g
0
=
g
(
|∇
G
σ
∗
I
0
|
)
.
(11.7)
Due to smoothing properties of convolution, we always have 1
≥
g
0
≥
ν
σ
>
0
[5, 27].
Equation (11.2) is a regularization, in the sense
|∇
u
|≈|∇
u
|
ε
=
2
+|∇
u
|
2
ε
[19], of the segmentation equation suggested in [7-9, 30, 31], namely,
g
0
∇
u
|∇
u
|
u
t
=|∇
u
|∇ ·
(11.8)
.
However, while in [19] the
ε
-regularization was used just as a tool to prove the
existence of a viscosity solution of the level set equation (see also [10, 12]), in
our work
ε
is a modeling parameter. As we will see later, it can help in suitable
denoising and completion of missing boundaries in images. Such regularization
can be interpreted as a mean curvature flow of graphs with respect to a specific
Riemann metric given by the image features [49].
