APPLICATION OF SELF-QUOTIENT ε- FILTER TO IMPULSE
NOISE CORRUPTED IMAGE
Mitsuharu Matsumoto
The Education and Research Center for Frontier Science, The University of Electro-communications
1-5-1, Chofugaoka, Chofu-shi, Tokyo, 182-8585, Japan
Keywords:
Feature extraction, Self-quotient ε-filter, ε-filter, Impulse noise.
Abstract:
This paper introduces an application of self-quotient ε-filter (SQEF) to impulse noise corrupted images. SQEF
is an improved self-quotient filter (SQF) to extract the image feature from noise corrupted image. Although
SQF is a simple nonlinear filter to extract the feature from the image, it cannot extract the feature from the
noise corrupted image. On the other hand, SQEF can extract the feature not only when it is applied to the clear
image but also when it is applied to the noise corrupted image. In this paper, we especially focus on feature
extraction from impulse noise corrupted image, and investigate the effectiveness of self-quotient ε-filter to
impulse noise corrupted images.
1 INTRODUCTION
Self-quotient filter (SQF) is a simple nonlinear filter
to extract the feature from an image (Wang et al.,
2004). It needs only an image, and can extract in-
trinsic lighting invariant property of an image while
removing extrinsic factor corresponding to the light-
ing. Feature extraction by SQF is simpler than that
based on multi-scale smoothing (Gooch et al., 2004).
SQF can extract the outline of the objects independent
of shadow region. However, as it assumes that the im-
age does not include noise, it can not extract the shape
and texture when the noise damages the image. The
noise influence becomes large due to the self-quotient
effect of SQF.
To solve the problem, we have introduced an ad-
vanced self-quotient filter labeled self-quotient ε-filter
(SQEF) (Matsumoto, 2010). The proposed filter is
based on the idea of SQF and ε-filter (Arakawa et al.,
2002). ε-filter is a simple edge-preserving nonlinear
filter. Although many studies have been reported to
reduce the small amplitude noise while preserving the
edge (Himayat and Kassam, 1993; Tomasi and Man-
duchi, 1998), it is considered that ε-filter is a promis-
ing approach due to its simple design. It does not need
to have the signal and noise models in advance. It is
easy to be designed and the calculation cost is small
because it requires only switching and linear opera-
tion. We can extract the feature from noisy facial im-
ages clearly by defining SQEF as the ratio of two dif-
ferent ε-filters. This paper focuses on the application
of SQEF to impulse noise corrupted images.
In next section, we describe the algorithm and the
feature of SQEF. Experimental results are shown to
clarify the effectiveness of the proposed method for
impulse noise corrupted image in Sec.3. Conclusions
are given in Sec.4.
2 SELF-QUOTIENT ε- FILTER
Let us define x(i
1
,i
2
) as the image intensity at the
point i = (i
1
,i
2
) in the image. The aim of self-quotient
filter is to separate the intrinsic property and the ex-
trinsic factor, and to remove the extrinsic factor. To
solve the problem, self-quotient filter assumes that a
smoothed version of an image has approximately the
same illumination as the original one. In self-quotient
filter, we first calculate the following equation:
z(i
1
,i
2
) =
x(i
1
,i
2
)
F[x(i
1
,i
2
)]
, (1)
where x(i
1
,i
2
) is the original image and F is the
smoothing function. z(i
1
,i
2
) is then binarized to ex-
136
Matsumoto M. (2010).
APPLICATION OF SELF-QUOTIENT - FILTER TO IMPULSE NOISE CORRUPTED IMAGE.
In Proceedings of the International Conference on Computer Vision Theory and Applications, pages 136-139
DOI: 10.5220/0002886001360139
Copyright
c
 SciTePress

(a) Original image from
Yale image database (file
name: yaleB01 P00A-
005E-10.pgm)
(b) Noisy image
(c) Filter output when we
used original image
(d) Filter output when we
used noisy image
Figure 1: Self-quotient image from original image and im-
pulse noise corrupted image.
tract face features from the images. When the quan-
tization bit of the image is 8 bits, z(i
1
,i
2
) is binarized
as follows:
Q(i
1
,i
2
) =

255 (if z(i
1
,i
2
) > Th)
0 (if z(i
1
,i
2
) ≤ Th)
, (2)
where Th represents a threshold value.
Due to the process of Eq.1, the texture and edge
can be extracted because the original image is divided
by the smoothed image. However, self-quotient fil-
ter assumes that the image does not include the noise.
When we consider the noise corrupted image, the
noise is reduced in the smoothed images F[x(i
1
,i
2
)],
while the original image x(i
1
,i
2
) includes the noise.
As a result, the influence from the noise in SQF is
emphasized very much due to the self-quotient effect
of SQF in Eq.1.
We show some examples to clarify the handling
problem. Figure 1 shows the filter outputs of SQF
when we used the original image and noisy im-
age. Figures 1(a) and 1(b) show the original image
and noisy image with 30% impulsive noise, respec-
tively. The original image is cut from Yale face image
database (file name: yaleB01 P00A-005E-10.pgm)
(Georghiades et al., 2001). Figures 1(c) and 1(d) show
the filter outputs of the original image and noisy im-
age, respectively. As shown in Fig.1, SQF works well
if the image is not damaged with noise. However,
it cannot extract the feature from the image clearly
when the image is damaged with noise. Our aim is
to propose an improved SQF to handle the image in-
cluding noise.
A simple idea to solve the noise influence in self-
quotient filter is to use two smoothed filters instead of
original image as follows:
z(i
1
,i
2
) =
F
1
[x(i
1
,i
2
)]
F
2
[x(i
1
,i
2
)]
. (3)
F
1
and F
2
should be different because the output al-
ways becomes 1 if F
1
and F
2
are the same smoothed
filter.
However, even if we design SQF by using two dif-
ferent smoothed filters, not only the noise is smoothed
but also the texture and shape are blurred. As the blur
level of one smoothed filter is different from the other,
it is also difficult to handle impulsive noise. Hence,
we need to employ alternative filters, which can re-
duce the small amplitude noise effectively, while pre-
serving the texture and shape information instead of
simple smoothed filter. The alternative filters should
be simple to keep the simplicity of self-quotient filter.
Based on the above prospects, SQEF is designed
as follows:
z(i
1
,i
2
) =
Φ
ε
1
[x(i
1
,i
2
)]
Φ
ε
2
[x(i
1
,i
2
)]
, (4)
where Φ
ε
represents ε-filter described as follows:
y(i
1
,i
2
) =
Φ
ε
[x(i
1
,i
2
)] = x(i
1
,i
2
)+ (5)
K
∑
j
1
=−K
K
∑
j
2
=−K
a( j
1
, j
2
)F(x(i
1
+ j
1
,i
2
+ j
2
) − x(i
1
,i
2
)),
where a( j
1
, j
2
) represents the filter coefficient.
a( j
1
, j
2
) is usually constrained as follows:
K
∑
j
1
=−K
K
∑
j
2
=−K
a( j
1
, j
2
) = 1. (6)
F(x) is the nonlinear function described as follows:
|F(x)| ≤ ε : −∞ ≤ x ≤ ∞, (7)
where ε is a constant number constrained as follows.
0 ≤ ε. (8)
It should be noted that calculation cost of ε-filter is
small because it requires only switching and linear
operation. See Refs.(Arakawa et al., 2002; Arakawa
and Okada, 2005) if the reader would like to know the
details about ε-filter.
z(i
1
,i
2
) is then binarized as follows:
Q(i
1
,i
2
) =

255 (if z(i
1
,i
2
) > Th)
0 (if z(i
1
,i
2
) ≤ Th)
, (9)
APPLICATION OF SELF-QUOTIENT W- FILTER TO IMPULSE NOISE CORRUPTED IMAGE
137

where Th represents a threshold value.
In ε-filter, ε is an essential parameter to reduce the
noise appropriately. If ε is set to an excessively large
value, the ε-filter becomes the same as linear filter.
On the other hand, if ε is set to 0, it does nothing to
reduce the noise anymore, that is, the filter output be-
comes the input image itself. Hence, SQF is a subset
of SQEF. When we take into account the design of
SQF, numerator in Eq.4 should become similar to the
original image, while denominator in Eq.4 should be-
come an smoothed image.
By setting ε adequately, we can effectively reduce
small amplitude noise while preserving shape and tex-
ture information. Hence, the optimized ε value is uti-
lized as numerator in Eq.4. It should be noted that the
optimized ε can be obtained automatically by using
signal-noise decorrelation criterion (Matsumoto and
Hashimoto, 2009). On the other hand, a sufficient
large ε should be used as the denominator in Eq.4 to
emphasize the feature of the image.
When we apply SQEF to impulse noise corrupted
image, it is considered that both ε-filters in SQEF
keep the impulse noise in the image unlike when two
smoothed filters are employed. Hence, when one fil-
ter output in SQEF is divided by the other filter in
SQEF, the impulse noise effect is reduced by the self-
quotient effects.
3 EXPERIMENTS
To evaluate the filter characteristics of SQEF, we con-
ducted the evaluation experiments using various types
of facial images. Some facial images are selected
from Yale image database (Georghiades et al., 2001)
and facial parts are cut from them. The image size is
256 pixels × 256 pixels. We added 10%, 20% and
30% impulse noise to images, respectively. Through-
out the experiments, the filter coefficient a
i
is set to
1/(2K + 1)
2
to make it uniform weight. To test the
robustness of the the proposed method concerning the
window size, the window size was changed from 3 ×
3 to 9 × 9. We show the results when the window size
was set to 5 × 5 as examples. Similar results could be
obtained throughout all the experiments regardless of
the window size.
Figures 2, 3, and 4 show the experimental results
when we used the image with impulse noise whose
percentage was 10%, 20% and 30%, respectively.
Figures 2(a), 3(a) and 4(a) show the original image
for comparison.
Figs.2(b), 3(b) and 4(b) show the input image with
impulse noise whose percentage is 10%, 20% and
30%, respectively. Figs. 2(c), 3(c), and 4(c) show the
(a) Original image (b) Image with 10% im-
pulse noise.
(c) Filter outputs when
SQF was used.
(d) Filter outputs when
SQEF was used.
Figure 2: Experimental results when 10% impulse noise is
added.
(a) Original image (b) Image with 20% im-
pulse noise.
(c) Filter outputs when
SQF was used.
(d) Filter outputs when
SQEF was used.
Figure 3: Experimental results when 20% impulse noise is
added.
filter outputs of SQF when we used the input image
with noise whose percentage is 10%, 20% and 30%,
respectively. Figs. 2(d), 3(d), and 4(d) show the fil-
ter outputs of SQEF when we used the input image
with noise whose percentage is 10%, 20% and 30%,
respectively. As shown in Figs.2, 3 and 4, SQEF can
extract the shape and texture information with reduc-
ing the noise, while SQF can not extract the feature
clearly.
VISAPP 2010 - International Conference on Computer Vision Theory and Applications
138

(a) Original image (b) Image with 30% im-
pulse noise.
(c) Filter outputs when
SQF was used.
(d) Filter outputs when
SQEF was used.
Figure 4: Experimental results when 30% impulse noise is
added.
4 CONCLUSIONS
In this paper, we introduced the applications of a self-
quotient ε-filter to impulse noise corrupted image.
We showed some experimental results and confirmed
the effectiveness of the proposed method. The algo-
rithm is simple and calculation cost is small. It can
extract face features from face images with impulse
noise. Future works include application of the pro-
posed method to face recognition. We also would like
to apply it to medical images to extract disease site.
ACKNOWLEDGEMENTS
This research was supported by the research grant
of Support Center for Advanced Telecommunications
Technology Research (SCAT), by the research grant
of Foundation for the Fusion of Science and Tech-
nology, by Special Coordination Funds for Promoting
Science and Technology, and by the Ministry of Edu-
cation, Science, Sports and Culture, Grant-in-Aid for
Young Scientists (B), 20700168, 2008.
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