A NOISE REMOVAL MODEL WITH
ANISOTROPIC DIFFUSION BASED ON VISUAL GRADIENT
Li Shi-Fei, Wang Ping and Shen Zhen-Kang
ATR Lab, National University of Defense Technology, Deya Road, Changsha, P. R. China
Keywords: Diffusion filter, Visual gradient, Noise removal.
Abstract: In recent years considerable amount of researchers have been devoted to anisotropic diffusion method and
achieved a series of important development. However, human visual system which perceived and
interpreted images has been paid little attention to in all these models. In this paper, we define a visual
gradient, which is looked as a generalization of the image gradient. After that we substitute the visual
gradient for the image gradient in the anisotropic diffusion model to keep to some extent consistent with
human visual system for the first time. Finally numerical results show the proposed method’s performance.
1 INTRODUCTION
Since Perona (Perona and Malik, 1990) introduced
the anisotropic diffusion to image processing and
proposed a multiscale smoothing edge detection
model first, a considerable amount of researchers
have been devoted to theoretical and practical
understanding of this and related methods. The idea
of anisotropic diffusion is that if the gradient of a
point is large the smoothing process will be low and
therefore the exact localization of the edges will be
preserved. Accordingly, anisotropic diffusion has a
good property of eliminating noise while preserving
high frequency components.
However, it has some disadvantages. For
example, Perona’s model is ill-posed and unstable
and it is difficult to confirm the model’s parameters.
In the recent years there are many modified versions
of Perona’s model have been presented. Nordström
(Nordström,
1990) proposed a biased diffusion to
regulate the ill-posed nature of the function. Catté et
al. (Catté et al.,
1992) have given a thorough
derivation of the process and proposed a stable
scheme for implementation, eliminating the problem
of choosing the number of required iterations. Other
researches have been achieved by such as Alvarez et
al. (Alvarez et al., 1
992), You (Yu-Li, Y and
Wenyuan 1996) and Barcelos et al. (Barcelos et al. ,
2003).
As we all know, all images are eventually
perceived and interpreted by the HVS (Human
Visual System). But these modified models consider
little the influence of human visual system. In fact,
Perona’s anisotropic diffusion model bases on the
gradient of the image which is not integrated the
information of human visual system. So we define
the visual gradient by using the properties of HVS
and develop an anisotropic diffusion model based on
visual gradient to remove image noise without losing
the boundaries or edges.
2 ANISOTROPIC DIFFUSION
Let u be the representation of the reconstructed
image. This representation can be defined as a
function of
2
Ω⊂ RR
that associate the
pixel
2
(, )xyR to its grey level image
(, )uxy
;
Ω
is the image support (generally, a
rectangle). Perona (
1990) substituted the standard
heat equation by the following anisotropic diffusion
equation (for the sake of briefness, in all models we
omit boundary and initial conditions):
(( ) )
t
udivfuu
=
∇∇
(1)
where
f
is a non-increasing smooth function such
that
() 0fs , (0) 1f
=
and lim ( ) 0
s
fs
→∞
= . The
idea is that if the gradient is large, then the diffusion
will be low, therefore the exact localization of the
edges will be preserved.
However, this model still has several theoretical
and practical difficulties. For instance, if the image
61
Shi-Fei L., Ping W. and Zhen-Kang S. (2009).
A NOISE REMOVAL MODEL WITH ANISOTROPIC DIFFUSION BASED ON VISUAL GRADIENT.
In Proceedings of the Fourth International Conference on Computer Vision Theory and Applications, pages 61-64
DOI: 10.5220/0001653700610064
Copyright
c
SciTePress
is very noisy, the gradient will be very large; as a
result the function will be close to zero at almost
every point. Consequently almost all noise will
remain when we use this model smooth the gravely
noisy image.
Aiming to this operator’s theoretical and
practical difficulties, considerable amount of
researchers proposed many modified models. We list
some typical modified models and study their
properties.
Catté (
1992) proposed and studied the following
model:
(( ) )
t
udivgGuu
σ
=∇ (2)
Where
Gu
σ
∇∗ denotes a convolution of the image
at time
t with a Gaussian kernel of scale
σ
which is
to be given a priori. This model alleviates
Perona’s
model’s ill-posedness, but it inducts a new
parameter
σ
.Moreover proper selection of this
parameter is critical to the success of the proposed
anisotropic diffusion.
From a geometric point of view, Alvarez et al.
(1
992) modify the diffusion operator in a way that
the diffusion process becomes more intense along
the edges and less intense along the perpendicular
direction of the edges:
()()
t
u
ugGu udiv
u
σ
=∇∗
(3)
After that ,there are other modified model In (
Yu-
Li,1996; Barcelos et al.,2003. etc.) Following the
idea in (
Perona, 1990; Nordström, 1990; Alvarez, et al.,
1992), Barcelos et al. (2003) proposed the well-
balanced equation:
()(1)()
t
u
ugudiv guI
u
λ
=∇
(4)
All these modified models have the same difficulty
of selection of Gauss kernel
G
σ
.As a matter of the
fact, using
Gu
σ
∇∗is a regularization process
which complicates the model and increases
calculation. Furthermore their studies may be more
reasonable if they had paid attention to the
properties of human visual system.
3 ANISOTROPIC DIFFUSION
BASED ON HVS
3.1 Properties of the HVS
As information carriers, all images are eventually
perceived and interpreted by human visual system.
As a result, human vision psychology and
psychophysics play an important role in the
successful communication of image information.
This is an important new area that needs to be
further explored, this paper try to have a first attempt
of integrating human visual information into
anisotropic diffusion model.
(i,j)
(i+1,j+1)
(i,j+1)
(i-1,j)
(i+1,j)
(i+1,j-1)
(i,j-1)
(i-1,j-1) (i-1,j+1)
Figure 1: point (, )ij and its eight neighbors.
Weber’s law (Pratt,1991) is the famous portrait
of the function of human visual system in spatial
domain. This law reveals the universal influence of
the background stimulus
u on human’s sensitivity to
the intensity increment
g
δ
, or the so-called WPD
(
Weber Perceptive Different) which is denoted as
w
. It
claims that the fraction
/gg
δ
is a constant at a
great range of luminance:
/g g const
(5)
But when the background luminance is very strong
or week the fraction is not a constant. In other
words, the WPD
w is a function of the background
luminance
g .The expression is as following (Lihua,
2005):
2
20 12
088
88
( ) 0.002( 88) 8 88 138
7( 138)
8 138 255
255 138
g
g
wg g g
g
g
≤<
=−+
+<
(6)
VISAPP 2009 - International Conference on Computer Vision Theory and Applications
62
In(6) a small neighborhood around each pixel,
i.e. a pixel
33× (or55× et al.) region is considered
(see Figure 1). The region consists of the central
pixel
(, )ij which gray level is denoted using
(, )gij and a surround consisting of the eight
neighboring with a mean gray level, we use the
mean gray value, which is denoted using
(, )gij, of
the surrounding eight neighboring points standing
for the background luminance:
1, 0 1, 0
1
(, ) ( , )
8
llkk
gij gi l j k
=− =−
=++
∑∑
(7)
0 50 100 150 200 250 300
8
10
12
14
16
18
20
Figure 2: the Weber perceptive different function.
3.2 Visual Gradient
As we all know, anisotropic diffusion model bases
on the Gradient of the image. In order to using the
information of human visual system, we integrate
expression (13) into the gradient and define visual
gradient:
() / ()
v
Gg gwg
=⋅ (8)
where
g
denotes the gradient of the image and
()wg
is the WPD in the background g ,
α
is an
adjustable parameters.
From Figure 2 we know that at the background
luminance 88, the WPD is the minimum, that is,
human visual system is most sensitive to the
gradient, so the corresponding visual gradient must
be greater than the gradient. However, at the
background luminance 255, human visual system’s
sensitivity is weakest, so the corresponding visual
gradient must be greater than the gradient.
In fact, in the definition of visual gradient if
() 1wg
α
⋅≡
, the visual gradient degenerates to
general gradient. Therefore the visual gradient is
essentially a generalization of image gradient.
3.3 New Model and its Properties
Base on the definition of the visual gradient which is
closer to human visual system than gradient, we
substitute the visual gradient for gradient in
anisotropic diffusion model’s diffusion function and
get the new model based on the visual gradient:
(( ()) )
() ()
tv
v
udivfGgg
Gg wg g
α
=
=
⋅⋅
(9)
where ()
v
Gg, g denote correspondingly the visual
gradient, gray level of an arbitrary point and
g
denotes its eight neighbors’ mean gray level. In the
discrete numerical implementation, they are
substituted by
((,))
v
Ggij , (, )gij and (, )gij .
Diffusion function is a non-increasing smooth
function such that
() 0fs , (0) 1f = and
lim ( ) 0
s
fs
→∞
=
.Two choices suggested by Perona
are
()
2
() exp /
f
ssk
=−
(10)
and
2
() 1/(1 (/ ))
f
ssk=+ (11)
where
k is a constant to be tuned for a particular
application.
In the proposed method, we use visual gradient
rather than image gradient controlling the
anisotropic diffusion model’s diffusion coefficient.
Ours idea is that in the different background the
gradient is also different and the WPD must obey
some rule. Triggered by this idea, we use the
expression (6) generalizing the gradient to keep to
some extent consistent with human visual system.
4 EXPERIMENTAL RESULTS
We use Perona’s discrete scheme(Perona, 1990) and
obtain the formulation:
[
1
,,
((,))
nn n
ij ij v N S
uu fGij uu
λ
+
=
+∇+
]
,
n
EW
ij
uu+∇ + (12)
where
01/4
λ
<
< for the numerical scheme to be
stable and
, 1,, , 1,,
,
Nij i j ij Sij i j ij
uu u uu u
−+
=−=−
,,1, ,,1,
,
E ij ij ij W ij ij ij
uu u uu u
+−
=−=−
The conduction function is updated at every
iteration as the function of the visual gradient(10)
(11):
(
)( )
2
(, ) exp (, )/
nn
vv
f
Gij Gijk
⎡⎤
=−
⎢⎥
⎣⎦
A NOISE REMOVAL MODEL WITH ANISOTROPIC DIFFUSION BASED ON VISUAL GRADIENT
63
or
2
((,))1/(1((,)/))
nn
vv
f
Gij Gijk=+
(a) (b)
(c) (d)
Figure 3: different noise removal models’ experimental
results. (a) blurred image by Gauss noised with zero mean
and 0.01 variance, (b) filtered by Perona’s model with
iteration 14,(c) smoothed by ALM model with iteration
25,(d)smoothed by the proposed model with iteration 8.
(a) (b)
(c) (d)
Figure 4: different noise removal models’ experimental
results.(a) noised image by salt & pepper noise which
density is 0.05, (b) smoothed by Perona’s model with
iteration 17, (c) filtered by ALM model with iteration 45,
(d) smoothed by the proposed model with iteration 10.
In the experiments, we applied Perona’s model,
ALM model (Alvarez, et al., 1
992) and the proposed
model to smooth the cameraman image with zero
mean and 0.02 variance noise (Figure 3). In Figure 4
we use salt & pepper noise which density is 0.05
blurred the origin image. Among the different
filtered images the reconstructed image using the
proposed model keep to more extent consistent with
human visual system.
5 CONCLUSIONS
In this work, an anisotropic diffusion model for
image smoothing based on the visual gradient is
presented. Our model uses a visual gradient which is
a generalization of the image gradient. Numerical
results show the proposed method’s performance.
As a tentative study of integrating HVS
information into anisotropic diffusion model for the
first time, the proposed model’s performance is
expected to be improved in the further researches.
6 REFERENCES
Perona, P., Malik, J., 1990. Scale-space and edge detection
using anisotropic diffusion, Pattern Analysis and
Machine Intelligence, IEEE Transactions on, vol. 12,
pp. 629-639.
Nordström, K. N, 1990. Biased anisotropic diffusion: a
unif-ied regularization and diffusion approach to edge
detection, Image and Vision Computing, pp. 318 –
327.
Catté, F., Lions, P. L., Morel, J. M. and Coll, T., 1992.
Image Selective Smoothing and Edge-Detection by
Nonlinear Diffusion, Siam Journal on Numerical
Analysis, vol. 29, pp. 182-193.
Alvarez, L., Lions, P. L. and Morel, J. M, 1992. Image
Selective Smoothing and Edge-Detection by Nonlinear
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Yu-Li, Y. Wenyuan, X., 1996. A. Tannenbaum, and M.
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removal and edge detection, Image Processing, IEEE
Transactions on, vol. 12, pp. 751-763,
Pratt.W.K., 1991. Digital image processing, John Wiley &
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nd
edition.
Lihua, G. Jianhua, L. and Shutang,Y,2005. Adaptive
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