Te-Jen Su1 Jui-Chuan Cheng2 Yang-De Sun3
1College of Information Technology Kun Shan University Tainan 710, Taiwan, R.O.C.
2,3Department of Electronic Engineering National Kaohsiung University of Applied Sciences
Kaohsiung 807, Taiwan, R.O.C.
1[email protected] 2[email protected] 3[email protected]
Abstract—This paper proposes a novel method for designing templates of Cellular Neural Network (CNN) for color image noise removal. The control of CNN systems is achieved via Particle Swarm Optimization (PSO) with Time-Varying Acceleration Coefficients (PSO-TVAC). Based on PSO-TVAC method, the proposed approach can automatically update the parameters of the templates of CNN to optimize them for diminishing noise interference in polluted image. The demonstrated examples are compared favorably with other available methods, which illustrate the better performance of the proposed PSO-TVAC-CNN methodology.
Keywords- Cellular Neural Network; Color Image Noise Removal; Particle Swarm Optimization with Time-Varying Acceleration Coefficients
I. INTRODUCTION
Particle swarm optimization is a population based stochastic optimization technique developed by Dr. Eberhart and Dr. Kennedy in 1995 [1, 2, 9–11], inspired by social behavior of bird flocking or fish schooling. It is easily implemented in most programming languages and has proven both very effective and quick for a diverse set of optimization problems. However, local convergence problem and slow later convergence problem are the two critical shortcomings of PSO that limit its applications [3]. A Particle Swarm Optimization with Time-Varying Acceleration Coefficients (PSO-TVAC) is presented in this paper, which allows to effectively overcome the two mentioned problems [12].
A novel class of information processing system called cellular neural networks was proposed by L.O. Chua and L.
Yang in 1988 [6, 7]. CNN is characterized by the parallel computing of simple processing cells locally interconnected.
It has been widely used for image processing, pattern recognition, signal processing, etc. In recent years, the problems of CNN templates design for image processing have received considerable attention [4, 5].
A new method that combines the discrete-time cellular neural network template learning method with an adaptive particle swarm optimization, and applies to gray image noise cancelation was developed [8]. The approach is able to find the template values easily without complex mathematic
computing processes but also to obtain the balance of convergence speed and convergence accuracy. This work is extended from the previous study [8]; we attempt to apply the technique of gray image noise cancellation to color image noise cancellation by separating the color image into three Gray-Scale RGB elements.
The rest of this paper is organized as follows: in Section 2, the Particle Swarm Optimization techniques, while in Section 3, the Cellular Neural Network is discussed. In Section 4, the CNN based on PSO-TVAC template learning for images noise cancellation is presented. Examples are given in Section 5 to demonstrate the proposed methodology.
Finally, conclusion is drawn in Section 6.
II. PARTICLE SWARM OPTIMIZATION WITH TIME -VARYING ACCELERATION COEFFICIENTS (PSO-TVAC)
In PSO, suppose that the search space is D-dimensional, and then the ith particle is represented as Xi = (xi1,xi2,…,xiD).
The velocity (rate of the position change) of this particle is denoted as Vi = (vi1,vi2,…,viD). The best previous position of the ith particle is represented as Pi = (pi1,pi2,…,piD) . In other words, Pi involves the best previous position, which Xi has visited (the best position called pbest). The index of the best particle among all the particles in the swarm is defined as the symbol g (called gbest). The particles are manipulated according to the equations 1 and 2. In its canonical form, Particle Swarm Optimization is modeled as follows [6-8]:
) ( ) ( )
( ) ( ) ( ) 1
( id 1 1 id id 2 2 gd id
id t wv t crand p x crand p x
v (1)
) 1 ( ) ( ) 1
(t x t v t
xid id id (2) where
vid (t+1) : velocity of particle i at iteration t+1 vid (t) : velocity of particle i at iteration t xid (t+1) : position of particle i at iteration t+1 xid (t) : position of particle i at iteration t c1 : cognitive parameter
c2 : social parameter
rand()1 : random number uniform distribution U(0,1) rand()2 : random number uniform distribution U(0,1) pid : pbest position of particle i
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pgd : gbest position of swarm w : inertia weight
The objective of PSO-TVAC is to enhance the global search in the early part of the optimization and to encourage the particles to converge toward the global optimum at the end of the search. With a large cognitive parameter and small social parameter at the beginning, particles are allowed to move around the search space, instead of moving toward the population best. However, a small cognitive parameter and a large social parameter allow the particles to converge to the global optimum in the latter part of the optimization. Under this development, the cognitive parameter c1 starts with a high value c1max and linearly decreases to c1min. Whereas the social parameter c2 starts with a low value c2min and linearly increases to c2max. This modification can be mathematically represented as follows:
1min current number of iterations.
III. CELLULAR NEURAL NETWORK
A two-dimensional CNN array is considered in which the cell dynamics are described by the following nonlinear ordinary differential equation with linear and nonlinear terms [13–16]: of the specified CNN cell, respectively. A(i,j;k,1) is called the feedback cloning template, B(i,j;k,1) is called the feedforward or input control template, Dij;kl are nonlinear terms applied for
v
( v
is a generalized difference). The state and output vary in time, the input is static (time independent), and the CNN is single-layer with nearest neighbor linear.In this paper, Dij;kl is the generalized nonlinear term applied to vukl(t)xij(t) , the voltage difference of either the input or state values. The nonlinear template D is as follows:
(7)
where d0, d1, d2 ,are the parameters in the nonlinear template D. K should be set very close to this value in an attempt to separate the noise effects and the image structure.
IV. CNNBASED ON PSO-TVACTEMPLATE LEARNING
A digital image is composed of pixels which can be thought of as small dots on the screen and may be represented as m-by-n matrices. MATLAB is a matrix processing language for a wide range of applications. The color image in MATLAB is described as a two dimensional matrix in uint8 format. In order to apply the gray image noise cancellation from the precious study [8], we separate the color image into RGB elements. These three Gray-Scale images will be blend into a colorful image after finishing noise cancellation processes proposed. The process followed to perform noise cancellation is shown in Fig. 1
PSO-TVAC-CNN
Figure 1. Block diagram of the image separation and blending
In this case, PSO-TVAC is employed to design templates of CNN for canceling the noise interference in Gray-Scale images. The templates are designed as following pattern structures, respectively:
(8)
where a0, a1, a2 ,b0, b1, b2 ,d0, d1, d2, I, K are elements of the swarm, in order to satisfy output saturation effectively, we set a2 =0 , b0 =0, b1 =0, b2 =0, I=0, xij(0)= uij(t). The training image is corrupted by the salt and pepper noise shown in Fig. 2.
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Figure 2. The training images (a) Input image to the CNN (b) Desired output image of CNN.
The following equation is used as an objective function (error function); the block diagram is shown in Fig. 3.
k
i
d
c i P i
P Error
1
)2
( )
( (9)
where k denotes the total pixel of the picture, Pc(i) is the value of the ith pixel of the input image and Pd(i) stands for the pixel of the desired output image. Each resulting image is compared with the desired image which should be obtained in the end. The comparison allows to compute the value of the error function, and consequently obtain the best template.
Figure 3. Block diagram of the objective function.
The process for implementing the PSO-TVAC based on CNN is shown as Fig. 4.
Start
Initialize each particle
random position
random velocity
Cellular Neural Network Simulator
If a criterion is met?
Update the position and velocity of the particle
Determine the and gbest
Next particles
pbesti
Get the optimal templates Stop
No
Yes Start
Initialize each particle
random position
random velocity
Cellular Neural Network Simulator
If a criterion is met?
Update the position and velocity of the particle
Determine the and gbest
Next particles
pbesti
Get the optimal templates Stop
No
Yes
Figure 4. Flow chart of PSO-TVAC-CNN.
V. EXAMPLES AND RESULTS
In this section, we present the examples polluted by different percentage of noise density interference and using our proposed method PSO-TVAC-CNN compares with PSO-CNN [5] for gray and color image noise cancellation respectively.
A. Examples 1
Consider a 256×256 LENA Gary-Scale image Fig. 5(a) which is polluted by the salt and pepper noise 10%, 20%, 30% in Fig. 5(b) - Fig. 5(d), respectively. The parameters of the proposed method PSO-TVAC-CNN and PSO-CNN are set as indicated in Table 1 and Table 2, respectively. The self-adapting inertia weight w is defined in [17].
TABLE I. PSO-TVACPARAMETERS SETTING
The number of swarm size 12
The maximum position Xmax 10
The maximum velocity Vmax 1
Acceleration coefficient c1max 2.5 Acceleration coefficient c1min 0.5 Acceleration coefficient c2max 2.5 Acceleration coefficient c2min 0.5
Inertia weight W 0.8
Iterations 500
TABLE II. PSOPARAMETERS SETTING
The number of swarm size 12
The maximum position Xmax 10
The maximum velocity Vmax 1
Acceleration coefficient c1 2.05 Acceleration coefficient c2 2.05
Inertia weight W 0.8
Iterations 500
Figure 5. (a) Original LENA gray-Scale image (b) The contaminated image with 10% noise (c) The contaminated image with 20% noise (d) The
contaminated image with 30% noise.
5(a) 5(b)
5(c) 5(d) (a) (b)
PSO-TVAC
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According to these parameters, the consequences of approximated optimal templates A, D, and threshold K were found by the PSO-TVAC after a few iterations.
The contaminated image with 10% noise:
2495
The contaminated image with 20% noise:
2488
The contaminated image with 30% noise:
2721
Similarly according to above parameters setting, the PSO found the consequences of approximated optimal templates A, D and threshold K as following:
The contaminated image with 10% noise:
2460
The contaminated image with 20% noise:
1440
The contaminated image with 30% noise:
4617
The error after a few iterations for PSO-TVAC-CNN and PSO-CNN are shown in Fig. 6-8. Table 3 shows the PSNR [18] of the image noise cancellation with both cases.
0 50 100 150 200 250 300 350 400 450 500
200 PSO-CNN and PSO-TVAC-CNN Training
Iteration
Error(pixel)
PSO PSO-TVAC
Figure 6. PSO-TVAC-CNN and PSO-CNN Training for Gray image with 10% noise
200 PSO-CNN and PSO-TVAC-CNN Training
Iteration
Error(pixel)
PSO PSO-TVAC
Figure 7. PSO-TVAC-CNN and PSO-CNN Training for Gray image with 20% noise
200 PSO-CNN and PSO-TVAC-CNN Training
Iteration
Error(pixel)
PSO PSO-TVAC
Figure 8. PSO-TVAC-CNN and PSO-CNN Training for Gray image with 30% noise
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TABLE III. PSNR OF THE GRAY IMAGE (256×256LENA) NOISE CANCELLATION
Salt and Pepper
Contaminate d Image
PSO-TVAC-CNN PSO-CNN
10% Noise 15.1659 dB 34.3663 dB 33.3802 dB 20% Noise 12.0989 dB 31.9834 dB 27.3987 dB 30% Noise 10.3124 dB 29.8721 dB 26.9026 dB
Figure 9. Results by using PSO-TVAC-CNN algorithm for the Gary-Scale image with noise of (a) 10%, (b)20%, (c)30%
Figure 10. Results by using PSO-CNN algorithm for the Gary-Scale image with noise of (a) 10%, (b)20%, (c)30%
Using the above templates, the output images processing by PSO-TVAC-CNN and PSO-CNN are shown in Fig. 9(a) - 9(c) and Fig. 10(a) - 10(c) respectively. By comparing Fig.
6 - 8, Table 3 and Fig. 9(a) – 9(c) with Fig. 10(a) – 10(c) respectively, our proposed method PSO-TVAC-CNN could restrain from noise of the polluted image more effectively than PSO-CNN.
Next we apply the same optimal templates found by PSO-TVAC-CNN to the Color images to prove the better performance.
B. Examples 2
In order to demonstrate that the optimal template has the same performance when processing color images, we have performed similar tests with the color version of the Lean image (Fig. 11(a)). The 256×256 LENA color image in Fig.
11(b) - 11(d) which is polluted by the salt and pepper noise 10%, 20%, 30%, respectively.
11(a) 11(b)
11(c) 11(d)
Figure 11. (a) Original LENA color image (b) The contaminated image with 10% noise (c) The contaminated image with 20% noise (d) The
contaminated image with 30% noise.
By using the proposed method PSO-TVAC-CNN and median filter, the results for the output images obtained in Fig. 12(a) - 12(c) and 13(a) - 13(c). Table 4 show the PSNR of the color image noise cancellation with both cases.
TABLE IV. PSNR OF THE COLOR IMAGE (256×256LENA) NOISE CANCELLATION
Salt and
Pepper Contaminated
Image PSO-TVAC-CNN Median filter 10% Noise 15.1767 dB 34.0704 dB 30.6477 dB
20% Noise 12.1143 dB 30.9633 dB 27.4666 dB
9(a) 9(b)
9(c)
10(a) 10(b)
10(c)
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30% Noise 10.3164 dB 28.3820 dB 22.9037 dB
Figure 12. Results by using PSO-TVAC-CNN algorithm for the color image with noise of (a) 10%, (b)20%, (c)30%.
13(a) 13(b)
13(c)
Figure 13. Results by using median filter algorithm for the color image with noise of (a) 10%, (b)20%, (c)30%.
By comparing Fig. 12(a) - 12(c) with Fig. 13(a) - 13(c), and Table 4, our proposed method (PSO-TVAC-CNN)
could restrain from noise of the polluted image more effectively than PSO-CNN.
VI. CONCLUSION
In this paper, we have presented a Cellular Neural Network templates learning method that combined Particle Swarm Optimization with Time-Varying Acceleration Coefficients, applied to color image noise cancellation.
Template learning is a crucial step in cellular neural network technology. The implementation of PSO-TVAC-CNN is a contribution to the modern heuristics research in the image processing area. From the demonstrated examples, the proposed algorithm shows the better performance of the noise cancellation for color image than PSO-CNN.
In the future research, we hope to improve the adaptive templates training to repair the color images that are polluted by the higher density of miscellaneous noises. For real applications, the proposed method may be implemented and fabricated on FPGA or VLSI technology.
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