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IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 6, NO. 6, JUNE 1997 879
Correspondence
A Noise-Filtering Method Using
a Local Information Measure
Azeddine Beghdadi and Ammar Khellaf
Abstract— A nonlinear-noise filtering method based on the entropy
concept is developed and compared to the well-known median filter and
to the center weighted median filter (CWM). The performance of the
proposed method is evaluated through subjective and objective criteria.
It is shown that this method performs better than the classical median
for different types of noise and can perform better than the CWM filter
in some cases.
I. INTRODUCTION
There are two basic approaches for noise filtering, namely, spatial
methods and frequency methods. Most of the spatial smoothing
processes as the mean and the median filters, which are widely used
[1]–[5], generally tend to remove noise without explicitly identifying
it. It is, though, possible to filter selectively the noise signal by
comparing the gradient grey level in a neighborhood to a fixed
threshold [2], [6]. The frequency smoothing methods [7], [8] remove
the noise by designing a frequency filter and by adapting a cut-
off frequency when the noise components are decorrelated from the
useful signal in the frequency domain. Unfortunately, these methods
are time consuming and depend on the cut-off frequency and the
filter function behavior. Furthermore, they may produce artificial
frequencies in the processed image. This correspondence introduces a
new nonlinear filter based on the entropy concept. Since the pioneer
work of Frieden [9], the use of entropy [10], [11] in image analysis
has attracted a great number of researchers, especially in image
reconstruction [12], [13] and segmentation [14], [15]. In our previous
work [16] that followed the idea of Shiozaki [17], which consists in
defining a local entropy and thus performing a local treatment, we
have shown that by amplifying the local entropy of the contrast, one
can get a contrasted image. It has also been suggested that a noise-
filtering treatment can be obtained by decreasing the entropy of the
local contrast in a given neighborhood. In this correspondence, we
develop this point in detail in Section II, and in Section III give
some examples to show the effectiveness of the proposed filtering
method. A comparison with filters of comparable complexity—the
well-known median filter [3] and the CWM filter [18], [19]—followed
by a general discussion is also given. The proposed method is
not compared to the weighted median (WM) filter, which is more
complex, since it requires the optimization of the weights using some
error criterion under certain constraints [19], [21]. Finally, Section IV
is devoted to the conclusion and perspective.
Manuscript received February 17, 1995; revised June 20, 1996. The
associate editor coordinating the review of this manuscript and approving
it for publication was Prof. Moncef Gabbouj.
A. Beghdadi is with the Laboratoire des Propri´et´es M´ecaniques et Ther-
modynamiques des Mat´
eriaux, C.N.R.S. LP 9001, Institut Galil´
ee, Universit´
e
Paris Nord, 93 430 Villetaneuse, France (e-mail: bab@lpmtm.univ-paris13.fr).
A. Khellaf is with the Groupe d’Analyse d’Images Biologiques, CNAM,
Universit´
e Paris V, 75 015 Paris, France.
Publisher Item Identifier S 1057-7149(97)03735-4.
II. NOISE FILTERING—A NEW APPROACH
In the present approach, we propose a method of identifying
the noise by using the local contrast entropy. A picture element is
considered as noise when the associated local contrast is very different
from those of its neighboring pixels. Therefore, a local contrast
threshold allowing this discrimination is defined according to the local
contrast entropy. Thus, a pixel is identified as noise according to its
contribution weight to the local contrast entropy. Given a pixel ,
center of a window , of grey level , the associated contrast is
defined according to the Weber–Fechner law by
(1)
where is the mean grey level of the surround region of the center
pixel in the window and is the gradient level. In contrast to
other contrast definitions [6], [20], this quantity gives the same local
contrast for grey levels situated at the same distance from the mean
grey level. One can associate to the local contrast the probability
or (2)
where is the window size. Once the probability of the local contrast
is defined, one can estimate the probability to find a noise point. In
fact, a zero contrast zone, i.e., a homogeneous region, corresponds to
a zero probability. That means that the probability to find a noise point
in a homogeneous region is equal to zero. In information context, we
say that the degree of uncertainty to find a noisy point is minimum
and is equal to zero. Therefore, to give a measure of the degree of
uncertainty that a point in a region is a noise, we associate to the
given region a local contrast entropy defined as follows:
(3)
Therefore, a noise pixel or isolated point heavily contributes to the
contrast entropy since its probability is high. The basic idea of the
proposed technique is to modify the grey level of the pixel according
to its contrast value or its contrast probability. Then, the grey level
of the current pixel is transformed with respect to a threshold
contrast probability corresponding to a window where all the
local contrasts are equally distributed; this results in a maximum
entropy. From (2), one can show that this case occurs when all the
local contrasts are equal and different from zero. It corresponds to a
region composed of two homogeneous subregions of the same area,
for example, a well-contrasted sharp symmetrical edge. Thus, a pixel
is considered as noise signal when the corresponding probability
is greater than or equal to the critical value , where is
the size of the analysis window. We give to the pixel the mean or
better the median grey level if the associated probability is greater
than or equal to . In the following we quantitatively justify the
choice of this critical value for the contrast probability.
A. Noise Model—How to Identify the Undesirable Information
It could be noticed that the proposed method is data dependent.
It becomes thus difficult to perform a complete analytical analysis
without assuming some a priori knowledge of the signal and the
noise. In the following, for the sake of simplicity we choose a size of
1057–7149/97$10.00 1997 IEEE
880 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 6, NO. 6, JUNE 1997
(a) (b) (c)
(d) (e) (f)
Fig. 1. (a) Original image. (b) Noisy image (Gaussian and impulsive noise). (c) Median filter. (d) CWM filter (weight . (e) CWM
((f) Our method.
3 3 pixels for the analysis window and we consider some typical
cases of spatial grey-level distribution in the window. Let us now
analyze the following decision rule: is a noise grey level iff
(4)
First Case—A Spot Point: This is the easiest case. It consists of a
well-localized spot point with a grey level surrounded by its eight
neighbors of grey level such that . It is easy to show
that . In other words, the degree of uncertainty to identify
this point as noise is zero.
Second Case—A Homogeneous Region: An example of a homo-
geneous and noise free region consists of 3 3 pixels having almost
the same grey level, say . It can be easily shown that the contrast
entropy associated to this case is zero or very low and thus .
In this case we are sure, with a degree of uncertainty equal to zero,
that there is no noise in the given window.
Third Case—Critical Situation: This case corresponds to a sharp
transition or a ramp. Indeed, in such a situation, the number of pixels
having a grey level greater than the mean grey level is the same as
that of the complementary set. It results in a contrast probability equal
to . It is easy to establish this result from (3). If is the window
size, then , and the considered pixel is not a noise point.
Fourth Case—A Transition Region: To simplify the analysis, let
us consider a window of size , where pixels have nearly the same
grey level and pixels have the grey level . Let be the grey
level associated to the central pixel. One can easily obtain the mean
grey level and the local contrasts , and corresponding,
respectively, to the central point, a pixel of the first class, and a pixel
belonging to the second class. It is easy to show that if the grey level
of the two regions 1 and 2 are identical, then it results in a zero
probability. It corresponds to , and . This case
has been already considered (a spot point).
A similar analysis for the case when yields
the following decision rule. The central point of grey level is
considered as noise if
or (5)
Now, let us consider the case where is different from and
is not equal to . The size of the window is .
A similar computation leads to
(6)
It can be noticed that the first side of the inequality is nothing else
than the difference between the grey level of the center pixel and the
mean grey level of the surround pixels. The term on the right side
of the relation is proportional to the interclass variance. Indeed, the
interclass variance is
(7)
Therefore, condition (6) can be written
(8)
In summary, conditions (5) and (8), which are equivalent to
condition (4), state that a pixel is considered as a noise element
if its grey level is far from those of its neighbors. This distance is
measured through statistical parameters. This is debatable but so are
other similar noise testing models.
Fifth Case—Actual Noisy Window: This case corresponds to the
most encountered configuration in nontextured images. It consists in
a noisy point embedded in a quasihomogeneous region as shown
below.
The grey levels of the surrounding points can be written in the form
, where is the mean grey level and is a small
IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 6, NO. 6, JUNE 1997 881
(a) (b) (c)
(d) (e) (f)
Fig. 2. (a) Original image. (b) Noisy image (structured noise). (c) Median filter. (d) CWM filter (weight (e) CWM ( (f) Our method.
value. In this case, is different from the mean grey level, that is
can be written as follows: where is much greater
than . Using the contrast definition and the associated probability,
one can easily show that the central point gives the major contribution
to the local contrast entropy. The associated probability is greater than
the threshold value . Indeed, from (3) the probability of the
center pixel is greater than if the following condition is satisfied:
(9)
The condition (9), equivalent to the inequality ,is
intuitively appealing. Indeed, this condition states that the examined
pixel is considered as a noise point if the corresponding gradient
grey level is greater than the average gradient level computed in its
neighboring.
III. EXPERIMENTAL RESULTS
To test the efficiency of the proposed method, two types of additive
noise have been considered. The first one is a mixture of a zero
mean Gaussian noise (with ) and a impulsive noise (with
probabilty ), and the second one is a structured noise. The
method is compared to the classical median filter and the CWM filter
with 5 5 square window. The degraded signal with the additive
structured noise is generated following the expressions:
, where is the
original grey level, the indicator function, the interference noise
given by: and the
desired maximum noise amplitude. In the experiments, the following
values are chosen for the noise parameters:
and . With these values the fraction
of corrupted pixels is 18%. The size of the test images is 256 256
pixels quantized with 256 grey levels. For subjective comparison only
subjective criteria, namely, the visual perception quality, are used. For
objective comparison the well-known normalized mean square error
(NMSE) and the mean absolute error (MAE) are used as in [18],
[19] and [21].
Fig. 1(a) shows a digitized image of mandrill. This image presents
a typically difficult case for filtering purposes. In fact, many interest-
ing small structures have size and tone values comparable to those of
the noise. Thus, filtering such an image seems to be a rather difficult
task. Fig. 1(b) shows the image of Fig. 1(a) after adding a Gaussian
and impulsive noise. Fig. 1(c) is the result of applying a 5 5
median filter and Fig. 1(d) and (e) corresponds to the CWM filter
with, respectively, a weight of 3 and 5. Fig. 1(f) displays the result
obtained with the proposed method. Through these results, it can be
noticed that the median filter smooths out the noise as well as the
image details. This results in a blurred image. This undesirable effect
is less important when using the CWM filter. Whereas, our method
cleans the image without blurring the contours. Furthermore, Table
I clearly shows that the proposed method performs better than the
median filter and the CWM with the lower weight. However, for
the high weight, the CWM yields lower errors (NMSE and MAE)
and, thus, objectively performs better than our method. However, a
simple visual comparison clearly shows that the proposed method
preserves better the image contrast and details than the CWM filter.
This disagreement with the quantitative comparison is essentially due
to the fact that the NMSE and MAE measures cannot distinguish
between a few large deviations and many small ones. Consequently,
one has to develop other quantitative measures taking into account
the visual criteria to compare the obtained results. Unfortunately, to
our knowledge, it is difficult to find such measures at present time.
The second comparison concerns the structured noise. One can
observe in Fig. 2(b) a periodical structure with bands of the same
orientation and size. This noise is easily distinguishable from the
actual structure of the image. Thus, it is easy to follow the filter
effects on the noise. This noise can obviously be smoothed out by
using frequency filtering as described in [7] and [8]. But the frequency
methods require orthogonal transformations to decorrelate the image
components. This approach is time consuming and complicated.
882 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 6, NO. 6, JUNE 1997
TABLE I
OBJECTIVE COMPARISON—NMSE’S AND MAE’s ASSOCIATED WITH
THE DIFFERENT FILTERS APPLIED TO THE DEGRADED IMAGE
OF FIG. 1(b) (ZERO MEAN GAUSSIAN NOISE WITH 215
AND IMPULSIVE NOISE WITH THE PROBABILITY )
TABLE II
OBJECTIVE COMPARISON—NMSE’S AND MAE’S
ASSOCIATED WITH THE DIFFERENT FILTERS APPLIED TO
THE DEGRADED IMAGE OF FIG.2(b) (STRUCTURED NOISE)
Furthermore, the result depend on the filter function behavior which
may create artificial frequencies at the output.
Fig. 2 shows the obtained results corresponding to the median
filters and to the proposed method for a 5 5 window size. The
superiority of our technique is clearly demonstrated on this example.
The median is successful in eliminating the grid, but it blurs the
image, and an ondulation effect due to the grid appears in the
processed image of Fig. 2(c). In contrast, the CWM does not blur
the image but the interference noise is not completely removed.
Whereas, our method [see Fig. 2(f)] removes a large fraction of
the noise without sensibly modifying the contours and other details.
Furthermore, it is noticed that the classical median filter performs
better than the CWM filter in smoothing out this structured noise. This
result is not surprising since it was shown by Ko and Lee [12] that the
CWM filter tends to preserve lines and more details at the expense
of less noise suppression. Table II confirms these comparison results.
Indeed, the proposed method yields the lowest NMSE and MAE.
IV. CONCLUSION
A simple method for noise filtering has been presented and
compared to the well-known median filter and the CWM filter. The
obtained results on actual images corrupted by two types of noises
confirm the superiority of the proposed technique over the well-
known median filter and the CWM filter. The usefulness of the
entropy concept for image enhancement purposes is demonstrated.
This superiority is justified by the fact that the proposed method is
simple and successful in smoothing out different noises. It was shown
that for noise smoothing, the CWM can objectively perform better
than the proposed method for some weights. But the central weight
should be carefully selected depending on both the characteristics
of the original image and the added noise. In the proposed method,
such constraints do not exist. This makes the proposed method more
flexible than the median filters, as it does not necessitate sorting
the data. The derivation of a detailled analytical analysis taking into
account the data-dependent character of the method is under study.
Furthermore, the separability of the filter will be considered in a near
future, making thus the method faster.
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