The 2016 International Conference on Computer Science, Technology and Application (CSTA2016) were held in Changsha, China on March 18–20, 2016. The main objective of the joint conference is to provide a platform for researchers, academics and industrial professionals to present their research findings in the fields of computer science and technology.The CSTA2016 received more than 150 submissions, but only 67 articles were selected to be included in this proceedings, which are organized into 6 chapters; covering Image and Signal Processing, Computer Network, Algorithm and Simulation, Data Mining and Cloud Computing, Computer Systems and Application, Mathematics and Management.
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Image denoising based on a new thresholding function
Jun-Ke Kou*
Department of Applied Mathematics, Beijing University of Technology, Beijing, 100124, China E-mail: [email protected]
Yan-Ling Fu
Department of Information Engineering, Henan Finance and Taxation College, Zhengzhou, Henan, 451464, China E-mail: [email protected]
In this paper, a new continuous thresholding function is proposed. This new thresholding function does not only retain the advantages of conventional standard thresholding functions, but also overcome their shortcomings. The larger (than λ) coefficients are often taken as effective signal while the smaller (than λ) are noise. The proposed method of removing noise from the disturbed signal is to keep the larger coefficients and suppress the smaller coefficients. The simulation results also show that the proposed function can achieve better denoising performance than traditional methods.
Images are usually disturbed by noise. What’s more, the larger noise may be cover the original images. So, image denoising is a important step in the image processing application.
With the development of wavelet analysis, wavelet analysis has better abilities to describe the local characteristics of image in time-frequency domain. Owing to this property of wavelet, many simple and effective wavelet thresholding methods are taken by [1-2]. The most useful hard and soft thresholding functions are analysed by [3]. A wavelet-based denoising method with principal component analysis is given by [4]. The reference [5] introduced a 3D extension of the wavelet nanlysis based bilateral filtering for rician noise removal. A nonlinear suboptimal detector was suggested by [6], this detector consists of a nonlinear wavelet denoising filter, followed by a replica correlator. Silva, Dutra and Rodrigo [7] proposed an adaptive threshold denoising method based on edge strength. [8] discussed How much influence of the choice of mother wavelet functions in image denoising. Nasri and Pour [9] introduced the adaptive thresholding function into wavelet domain by using the thresholding neural network. Recently, A pre-processing hard thresholding algorithm is proposed by [10]. From those results, we can see that the thresholding function is a critical factor in image denoising. In this paper, we construct a new thresholding function. This function overcome the disadvantage of conventional thresholding functions in theory. What’s more, the simulation results also indicate the new thresholding function has better denosing effect.
2.Wavelet-Based Noise Reduction
In this section, we will simply introduce the basis of signal denoising problem. Taking the original signal is x. Then the disturbed signal y is
where n is the noise. In practice, we always assume the signal is a finite data. This means
The aim of the noise reduction is obtain an estimated signal
from the disturbed signal y. In other words, the propose of this problem is to remove the noise in y and make the estimate
as close to x as possible. The objective of noise reduction is to estimate the real signal x from y to minimize the MSE risk.
Noise reduction by wavelet can be described as following. First, one obtains the empirical wavelet coefficients by discrete wavelet transform (DWT). Second, detailed coefficients are revised based on a thresholding function and an appropriate threshold value. Finally, using inverse wavelet transform for the modified coefficients, the reconstructed image is achieved.
In noise reduction, the thresholding function is very important. The most commonly-used thresholding functions are the soft-thresholding function and the hard-thresholding function, which are shown in Fig.1. The standard soft-thresholding function is defined as
where λ > 0 is the threshold. The hard-thresholding function is defined as
Fig. 1. soft and hard thresholding function (λ = 5).
It is well known that the energy of the signal is usually focused on a few coefficients while the energy of noise is spread among all coefficients in the wavelet domain. So, the larger (than λ) coefficients are often taken as effective signal while the smaller (than λ) are noise. If we want to remove the noise from the disturbed signal, we should kept the larger and suppress the smaller.
Although the smaller coefficients are frequent noise, they may be some important high-frequency components. Therefore, the conventional soft and hard thresholding function can possibly remove some important components when the smaller (than λ) coefficients are modified to zero. On the other hand, because of the discontinuity, hard thresholding function may result in ringing artifact, pseudo-gibbs and image visual distortion.
3.A New Thresholding Function
In order to overcome the standard thresholding functions’ weakness and improve the denoising performance, a new high-order differentiable thresholding function is constructed as follow:
Fig. 2 New thresholding function (λ = 5).
It is easy to see from (5) and Fig.2 that this thresholding function is always continuous. It is similar with the standard hard-thresholding function when |y| → +∞. On the other hand, it shrinks the wavelet coefficient to a smaller value but not to zero with |y| ≤ λ. This mean it will keep some important high-frequency components (those may be the detail information of image). Thus, this new thresholding function is not only keep the advantage of the conventional standard soft (hard) thresholding function, but also avoid the bad influence of the conventional thresholding function.
4.Experimental Results
To analyze the denoising effect of the proposed approach, standard images (facets, wbarb, woman) have been used. This images are initially disturbed by Gaussian random noise with different standard deviation of 10, 15, 20. To investigate the effectiveness of the algorithm, commonly-used performance Peak Signal to Noise Ratio (PSNR) has been used. In the proposed approach, the threshold value λ = σ
, M, N is the size of the image, σ is the noise standard deviation. The performance of image denoising by hard thresholding function, soft thresholding function and our proposed method are shown in Table 1. From Table 1, it is easy to see that PSNR of the image is decreasing when the standard deviation increasing. This means the image is More seriously corrupted by noise when the standard deviation increasing. On the other hand, we can see that our proposed approach has a higher PSNR than other image denoising methods. In other words, the proposed approach is very close to the optimal solutions. The noise can be removed effectively.
Table 1. Comparison of the different methods.
5.Summary
In this paper, a new thresholding function has been proposed for image denoising. As a result, the numerical results show that the proposed method has better denoising effect.
References
1.D. L. Donoho, L. M. Johnstone. Adaptating to unknown smoothness via wavelet shrinkage. Journal of the American Statistical Association. 90(432): 1200-1224, 1995.
2.D. L. Donoho. Denosing by soft thresholding. IEEE Transactions on information theorey. 41(3): 613-627, 1995.
3.D. L. Donoho, L. M. Johnstone. Ideal spatial adaptation by wavelet shrinkage. Biometrika. 81(3): 425-455, 1998.
4.R. G. Yang, M. G. Ren. Wavelet denoising using principal component analysis. Expert systems with applications. 38(1): 1073-1076...
Table of contents
Cover
Halftitle
Title
Copyright
Preface
Committees
Keynote Speech
Contents
Chapter 1 Image and Signal Processing
Chapter 2 Computer Network and Information Security
