Switching Median Filter with Signal Dependent Thresholds

Ryosuke Kubota

, Keisuke Onaga

and Noriaki Suetake

Department of Intelligent System Engineering, Ube National College of Technology,

2-14-1 Tokiwadai, Ube-shi, Yamaguchi, Japan

Advanced Course of Production Systems, Ube National College of Technology,

2-14-1 Tokiwadai, Ube-shi, Yamaguchi, Japan

Graduate School of Science and Engineering, Yamaguchi University,

1677-1 Yoshida, Yamaguchi-shi, Yamaguchi, Japan

Keywords:

Switching Median Filter, Random-valued Impulse Noise, Distribution Distance, Genetic Algorithm.

Abstract:

In this paper, we propose a new switching median ﬁlter with signal dependent thresholds designed by a genetic

algorithm (GA). The switching median ﬁlter detects noise-corrupted pixels based on a threshold. Then it

restores only the detected pixels. The present switching median ﬁlter deals with the random-valued impulse

noises, whose distribution is ideally assumed as a uniform distribution. In the present method, the switching

median ﬁlter, which has two kinds of the thresholds, is introduced. One is switching thresholds to detect

the noise, and the other is selecting thresholds to choose the suitable switching threshold. As the suitable

selecting threshold, a variance of signals is used. Then all of the switching and selecting thresholds of the

proposed switching median ﬁlter are automatically optimized by using GA. To optimize the thresholds with

GA, distribution distance between the assumed and the detected noises is employed as a ﬁtness function. The

validity and effectiveness of the proposed method is veriﬁed by some experiments.

1 INTRODUCTION

Along with developments of digital technologies,

high quality digital images are strongly needed in

many research and application ﬁelds. However, the

digital images are often corrupted by the impulse

noise in the image sensing and/or transmission pro-

cesses. It is thus signiﬁcant to restore the noise in

the image before subsequent processing. In order to

realize a ﬁne restoration of the image corrupted by

the salt-and-pepper impulse noise, various nonlinear

ﬁlters based on the median ﬁlter have been studied

so far (Chen et al., 1999), (Chen and Wu, 2001),

(Akkoul et al., 2010). Especially a switching median

ﬁlter, which was proposed by Sun and Neuvo (Sun

and Neuvo, 1994), has frequently employed and stud-

ied in order to apply it to a random-valued impulse

noise removal in recent years (Suetake, 2002), (Ng

and Ma, 2006), (Zhang et al., 2008), (Suetake et al.,

2011).

The typical switching median ﬁlters include a

noise detector based on a switching threshold, and

carry out the median ﬁltering to only noise-corrupted

pixels. In order to apply the switching median ﬁl-

ter effectively, its switching threshold of the detector

has to be appropriately tuned for the signal of con-

cern. However, it is difﬁcult to determine the optimal

threshold automatically, because the suitable thresh-

old varies depending on a mix of factors, e.g., a prop-

erty of input image, noise content rate, the window

size of the detector and so on.

To cope with this problem, the authors proposed

a distribution distance-based threshold auto-tuning

method for switching median ﬁlter so far (kubota and

Suetake, 2010). In this method, the distribution dis-

tance between a noise model assumed beforehand and

the noise signal detected by the detector was intro-

duced as a distinct indicator for the optimal tuning

of the threshold. This method can free the users

from the tiresome tuning, and has been also applied

to the random-valued impulse noise removal of the

color image (kubota and Suetake, 2011). However,

the switching median ﬁlter with only a ﬁxed switch-

ing threshold has an inherent limitation on the noise

detection performance.

To address this issue, we propose a new switch-

222

Kubota R., Onaga K. and Suetake N..

Switching Median Filter with Signal Dependent Thresholds Designed by using Genetic Algorithm.

DOI: 10.5220/0004851702220227

In Proceedings of the 9th International Conference on Computer Vision Theory and Applications (VISAPP-2014), pages 222-227

ISBN: 978-989-758-003-1

 2014 SCITEPRESS (Science and Technology Publications, Lda.)

ing median ﬁlter with some signal dependent thresh-

olds and their optimal designing method based on

the noise probability distribution. In this paper, the

switching median ﬁlter, which has two kinds of the

thresholds, is introduced. One is switching thresholds

to detect the noise, and the other is selecting thresh-

olds to choose the suitable switching threshold. In the

proposed method, we employ a variance of signals for

the selecting threshold. Thus, the suitable switching

threshold is decided depending on the signal of con-

cern by using the selecting thresholds.

All of the switching and selecting thresholds of

the proposed switching median ﬁlter are automati-

cally designed by using GA, which is a search al-

gorithm based on the mechanism of natural selection

and natural genetics (Goldberg, 1989), (Davis, 1990).

GA solves optimization problems by using individu-

als which are represented by bit-strings or real valued-

genes. In the proposed method, GA tunes all thresh-

olds so as to minimize a distribution distance between

the assumed and the detected noises.

Through the experiments, the effectiveness and

validity of the proposed method are illustrated.

2 NOISE MODEL AND

SWITCHING MEDIAN FILTER

2.1 Random-valued Impulse Noise

In this paper, we consider a monochrome image cor-

rupted with the random-valued impulse noise in a

transmission of the digitalized signal. Impulse noises

are caused by malfunctioning pixels in camera sen-

sors, faulty memory locations in hardware or trans-

mission in a noisy channel.

A signal x(i, j) corrupted with the random-valued

impulse noise is represented by:

x(i, j) =

{

s(i, j), probability 1 − p,

h, probability p,

(1)

where, s(i, j) is the source signal, and takes 256 level

(8 bit) values. p represents a noise occurrence proba-

bility. The noise-corrupted pixel value h takes from 0

to 255, because each bit in s(i, j) inverts randomly.

This model is the most popular and focuses on a

bit error in the digitalized signal transmission. There-

fore, the signals corrupted with the random-valued

impulse noises can be assumed as a uniform distri-

bution.

2.2 Detailed-preserving Median Based

Filter

A detailed-preserving median based ﬁlter is the most

popular switching median ﬁlter (Sun and Neuvo,

1994). In this method, the impulse noises are detected

by considering difference between a pixel value of

concern and a median of its neighboring pixel values.

This method uses a noise position image f

(ε)

(i, j)

deﬁned by:

(ε)

(i, j) =

{

1, |x(i, j) − x

MED

(i, j)| ≥ ε,

0, otherwise,

(2)

where x

MED

(i, j) stands for the output signal at the

pixel (i, j) by the ordinary median ﬁlter. The noise

position image contains coordinate information of the

detected noises. In the noise position image, “1” rep-

resents that the pixel is corrupted by the noise. The

switching median ﬁlter carries out the median ﬁlter-

ing only for the detected pixels by using the noise po-

sition image. Here ε is a threshold.

In the past, various types of the switching median

ﬁlter have been proposed for the effective detection

of the salt-pepper impulse noise. Furthermore, some

of them have been also applied to the detection of the

random-valued impulse noise. However, in the con-

ventional methods, the suitable threshold ε has been

adjusted manually and empirically depending on the

situations, because effective indicators for the ﬁlter

design have not been discussed so far. Additionally,

the switching median ﬁlter with only a ﬁxed thresh-

old has an inherent limitation on the noise detection

performance.

In order to raise the noise detection performance,

it is preferable to change the threshold depending on

the target and its neighboring signals automatically.

3 PROPOSED METHOD

3.1 Counstruction of Proposed

Switching Median Filter

The proposed switching median ﬁlter has two sets of

thresholds. One is the switching thresholds {ε

| m =

1,··· ,N}, and the other is the selecting thresholds

| n = 1, ··· , N − 1}. Each switching threshold is a

value, and it is used in order to judge the target signal

as the source or the noise. The selecting thresholds

work to choose the suitable switching threshold from

the set of the switching thresholds. Thus, the pro-

SwitchingMedianFilterwithSignalDependentThresholdsDesignedbyusingGeneticAlgorithm

223

posed switching median ﬁlter is expressed by:

(ε

)

(i, j) =

{

1, |x(i, j) − x

MED

(i, j)| ≥ ε

0, otherwise,

(3)











, v < v

, v

n−1

≤ v < v

, v

N−1

≤ v,

(4)

where v

is a variance value calculated from x(i, j)

and its neighboring signals in a window. Additionally,

these thresholds satisfy the following conditions:

0 < ε

< ··· < ε

, (5)

and

0 < v

< ··· < v

N−1

. (6)

3.2 Adaptive Threshold Tuning by

Genetic Algorithm

In the proposed switching median ﬁlter, all of

the thresholds, which are {ε

,··· ,ε

} and

,··· ,v

N−1

}, are decided by using GA.

In the proposed method with GA, an in-

dividual I

is constructed by the serially-

concatenated switching and selection thresh-

olds. Thus the k-th individual is represented by

= {ε

,··· ,ε

,··· ,v

kN−1

where 5 ≤ ε

≤ 100 and 10 ≤ v

≤ 7,000.

In the optimization process of GA, crossover, mu-

tation and selection operators are applied to the popu-

lation. The proposed tuning method employs a distri-

bution distance between a noise model assumed be-

forehand and the noise detected by the detector as a

ﬁtness function. The distribution distance is a good

indicator for the evaluation of the noise detecting per-

formance, and its validity for the single threshold

adjusting has been veriﬁed in (kubota and Suetake,

2010). When the distribution distance is small, it is

thus judged as a ﬁne noise detection is achieved. The

ﬁtness function with its modiﬁcation to the proposed

switching median ﬁlter is described below.

The sets of the suitable thresholds {ε

∗

| m =

1,··· ,N} and {v

∗

| m = 1,··· ,N − 1}, which maxi-

mize the ﬁtness function F, is searched. The ﬁtness

function F

of k-th individual is represented by:

1 +D(H

d,(Ω

)

. (7)

D(H

d,(Ω

)

) is the distribution distance (L

norm)

and is represented by:

D(H

d,(Ω

)

) =

255

∑

ℓ=0

d,(Ω

)

(ℓ) −H

(ℓ)|, (8)

where H

d,(Ω

)

stands for a probability density function

of the noises detected with the set of the thresholds

Ω

= {ε

| m = 1,··· ,N, and n = 1, ··· , N −

1}. The probability density function is then obtained

from a normalized histogram of a set of x(i, j) which

satisﬁes f

(ε

)

(i, j) = 1. H

represents the probability

density function of an assumed random-valued im-

pulse noise, i.e., a uniform distribution, because the

assumption on the distribution of the noise is imposed

by the model of Eq. (1). H

(ℓ) is

256

for arbitrary ℓ

here, because H

is a uniform distribution.

The proposed method has three features from the

viewpoint of the usefulness, although the proposed

method is very simple. One is that the proposed

method can determine the suitable and speciﬁc thresh-

olds for various input images and/or their noise oc-

currence probability automatically, if the noise cor-

rupted to input image is the random-valued impulse

noise. Another is that the proposed method can ad-

just some kinds of thresholds by evaluating only a

distribution distance. The other is that the proposed

method does not require desirable output images for

the adjusting, because this method works only with

the internally-assumed noise model and the ﬁlter out-

puts. Therefore, it can be said that the proposed

method is very effective in the situation where the

MSE evaluation cannot be performed. Furthermore,

the proposed method can be used without any changes

of algorithm even if the noise occurrence probability

is changed.

4 EXPERIMENTAL RESULTS

The effectiveness and validity of the proposed method

are veriﬁed by experiments employing a digital im-

age as shown in Figure 1. Figure 1 constitutes of

512×512 pixels. In the experiments, the input images

are corrupted by the random-valued impulse noise

with p = 0.05. The enlarged image of input image

is shown in Figures 1(c).

In the experiment, we used the switching me-

dian ﬁlter with 3 switching thresholds and 2 selecting

threshold, i.e., N = 3. The window size of the switch-

ing median ﬁlter is 3 × 3. The noise removal per-

formance of the proposed method is compared with

that of the ordinary switching median ﬁlter, which

has only single switching threshold. Furthermore, the

noise removal performance of the proposed tuning

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224

(a)

(b)

(c)

Figure 1: Test image: (a) Original noise-free image, (b)

Enlarged image of a part of (a), (c) Enlarged input image.

method is also compared to those of a mean square

error (MSE)-based tuning method and L

norm-based

method. In this experiment, MSE is used as an index

for the quantitative evaluation, although the evalua-

tion based on MSE can not be applied in the practical

situation. The MSE is calculated from the original

noise-free image shown in Figure 1(a) and the output

image restored by the switching median ﬁlter. The

MSE becomes small when the restored image is sim-

ilar to the original one.

In the MSE-based method, each threshold of the

Table 1: Step size and range of each threshold.

Threshold Step Lower Upper

1 15 49

1 50 79

1 80 100

1,000 1,000 4,000

1,000 6,000 9,000

switching median ﬁlter is decided so as to minimize

the MSE between the original noise-free image and

the output image obtained by the switching median

ﬁlter. In the same manner, each threshold of the

switching median ﬁlter by using the L

norm-based

method is decided so as to minimize the L

norm be-

tween distribution distances of the internally-assumed

noise and the noise detected by the switching median

ﬁlter. To decide the thresholds based on the MSE and

the L

norm and to reduce the computational costs,

a spatial segmentation search is used. The step size

and the lower and upper limits is each thresholds are

shown in Table 1. From Table 1, we can estimate that

the number of trials is 336,000 in order to obtain the

best combination of all the thresholds. In this experi-

ment, the computation time of 1 trial needs 0.142 sec-

ond.

In the GA of the proposed method, the number of

genes in each individual is 5 (3 switching and 2 se-

lecting thresholds). The population size is 100. The

range of each threshold is the same to that in Table 1.

The window size for the calculation of the variance

(selecting threshold) is 5 × 5. The crossover, muta-

tion and selection methods are the BLX-α crossover

(Eshleman and Schaffer, 1993), uniform mutation and

roulette wheel selection with the elitism, respectively.

Furthermore, the crossover and mutation probabilities

are 0,3 and 0.05, respectively. The number of gener-

ations for the search is 1,000. Thus, the number of

trials is 100,000 in order to ﬁnish the search in the

proposed method, and less than those of the MSE and

the L

-based methods.

Table 2 shows the tuning results of the MSE, L

and the proposed method.

From Table 2, it is observed that the noise restora-

tion performance is improved slightly when the num-

ber of the switching thresholds increases. Besides,

-based and the proposed methods seem to be good,

although the obtained thresholds are not perfectly

same to that by the MSE-based method. From these

results, it is conﬁrmed that the distribution distance

between the assumed and the detected noise is a good

indicator for the multiple thresholds tuning. Further-

more, the distribution distance-based threshold ad-

justing method seems to be highly promise a secure

performance even if the detector has some thresholds.

SwitchingMedianFilterwithSignalDependentThresholdsDesignedbyusingGeneticAlgorithm

225

Table 2: Experimental results by the MSE-, the L

-based optimal tuning and the proposed tuning methods.

Method ε

∗

MSE L

MSE-based method 27 - - - - 9.0 0.16

-based method 21 - - - - 9.7 0.13

MSE-based method 25 50 80 2,000 6,000 8.7 0.15

-based method 22 54 85 3,000 6,000 9.4 0.12

Proposed method 12 22 83 60 5,862 9.4 0.12

(a)

(b)

(c)

Figure 2: Noise restored images, (a) MSE-based method,

(b) L

-based method, (c) Proposed method.

Figure 2 shows the parts of output images restored

by the switching median ﬁltering with the signal de-

pendent multiple thresholds obtained based on the

MSE, L

-based and the proposed methods. It is ob-

served that the restoration images by the L

-based and

the proposed method are the almost same to those by

the MSE-based optimal tuning. Furthermore, the pro-

posed method seems to be effective in the situation

where the MSE evaluation cannot be performed (i.e.,

the desirable ﬁltered image cannot be obtained be-

forehand).

Figure 3(a) shows a transition of the smallest L

norm of the best individual in the searching process

with the GA. From Figure 3(a), it can be observed that

the better thresholds are found as the generations pro-

gresses by the proposed method. Additionally, Figure

3(b) shows the MSE calculated by the best individual

at each generation. From Figure 3(b), it can be seen

that the good thresholds which is almost the best solu-

tion has been found arround 50-th generation. Thus,

the number of trials in order to obtain the good thresh-

olds in the proposed method is 5,000, which is calcu-

lated by the number of generation and the population

size. On the other hand, the number of trials of the

spatial segmentation search is 336,000 in order to ob-

tain the best combination of all the thresholds. There-

fore, the proposed method can ﬁnd the appropriate

and reasonable thresholds faster than the L − 1-based

spatial segmentation search.

From these results, the effectiveness and validity

of the proposed method are conﬁrmed.

5 CONCLUSION

In this paper, we proposed a new switching median

ﬁlter with signal dependent thresholds and their opti-

mal designing method by using GA.

The thresholds of the proposed switching median

ﬁlter are automatically optimized by using GA. To op-

timize the thresholds with GA, the distribution dis-

tance between the assumed and the detected noises is

used as the ﬁtness function. The effectiveness and va-

lidity of the proposed method were illustrated.

Future works are to improve the searching perfor-

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0 200 400 600 800 1000

0.123

0.124

0.125

0.126

0.127

0.128

0.129

Generations

L1 Norm

(a)

0 200 400 600 800 1000

Generations

MSE

(b)

Figure 3: Optimizing transitions of L

norm by the pro-

posed method, (a) L

norm of the best individual, (b) MSE

calculated by the best individual.

mance of GA and to establish other noise models for

various types of the noises.

ACKNOWLEDGEMENTS

This work was supported by the Ministry of Educa-

tion, Science, Sports and Culture, Grant-in-Aid for

Young Scientists (B), 24700237.

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