Digital Signal Processing Reference
In-Depth Information
is always to diminish the filtering effect in image regions not affected by the
noise process.
40
-
50
12.3
Anisotropic Diffusion
A powerful filtering technique, called anisotropic diffusion (AD), has been
introduced by Perona and Malik (P-M)
29
,
53
in order to selectively enhance
image contrast and reduce noise using a modified heat diffusion equation
and the concepts of scale space.
54
The main concept of
AD
is based on modification of the isotropic diffusion
equation (Equation 12.12), with the aim of inhibiting the smoothing across
image edges. This modification is done by introducing a conductivity function
that encourages intraregion smoothing over interregion smoothing.
Since the introduction of the P-M method, a wide variety of techniques have
been elaborated including multiscale approaches, extensions to vector-valued
imaging,
55
,
56
multigrid methods,
57
mathematical morphology-inspired tech-
niques, and many others.
56
,
58-65
Diffusion is a transport process that tends to level out concentration dif-
ferences, and in this way it leads to equalization of the spatial concentration
differences. The elementary law of diffusion states that flux density
is di-
rected against the gradient of concentration
F
in a given medium
=−
c
∇
F
,
where
c
is the diffusion coefficient. If we use the continuity equation
∂
F
∂
∂
F
∂
+∇=
0
,
we obtain
=∇
[
c
∇
F
]
.
(12.11)
t
t
If
F
denotes a real-valued function representing the digital image, the
equation of linear and isotropic diffusion is
(
x, y, t
)
c
∂
,
∂
(
)
2
F
(
)
+
∂
2
F
(
)
F
x, y, t
x, y, t
x, y, t
=
(12.12)
x
2
y
2
∂
t
∂
∂
where
x, y
are the image coordinates,
t
denotes time, and
c
is the conductivity
coefficient.
Perona and Malik suggested that conductivity coefficient
c
should be de-
pendent on the image structure, and therefore they proposed the following
partial derivative equation (PDE):
∂
F
(
x, y, t
)
=∇
[
c
(
x, y, t
)
∇
F
(
x, y, t
)
]
.
(12.13)
∂
t
is a monotonically decreasing function
of the image gradient magnitude and usually contains a free parameter
K
,
The conductivity coefficient
c
(
x, y, t
)
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