Procedia Technology 6 ( 2012 ) 10 – 15

2212-0173 © 2012 The Authors. Published by Elsevier Ltd. Selection and/or peer-review under responsibility of the Department of Computer Science & Engineering, National Institute of Technology Rourkeladoi: 10.1016/j.protcy.2012.10.002

2nd International Conference on Communication, Computing & Security [ICCCS-2012]

A New Soft-Thresholding Image Denoising Method Mantosh Biswas* and Hari Om

Indian School of Mines, Dhanbad-826004, Jharkand, India

Abstract

In this paper, we propose a new method of noise removal from an image corrupted with Gaussian noise using soft-thresholding. There are two types of thresholding: Soft and Hard thresholding. The Universal thresholding method i.e. VisuShrink is based on the Hard-thresholding and it is not appropriate for Soft-thresholding. Our proposed method is simple and adaptive since the estimation of thresholding parameters depends on the data of wavelet coefficients. According to the experimental results, this proposed method has higher Peak Signal-to-Noise Ratio (PSNR) and visual effects than the VisuShrink. © 2012 The Authors. Published by Elsevier Ltd. Selection and/or peer-review under responsibility of the Department of Computer Science & Engineering, National Institute of Technology Rourkela Keywords: Image Denoising; Threshold Rule; Peak Signal-to-Noise Ratio (PSNR);

1. Introduction

Estimating a signal that is corrupted by additive noise has been of interest to many researchers for practical as well as theoretical reasons. To recover original signal from the noisy signal is the main challenge. We want the recovered signal to be as close as possible to the original signal. Traditionally, the denoising techniques are linear such as Wiener filter. Recently, nonlinear techniques that are based on wavelet transforms have become popular [1]. Wavelet transforms play a major role in image compression and image denoising [2]. Weaver et al. are the researchers of the earliest denoising papers using wavelet [3]. In this paper, they have showed that the noise could be significantly reduced without reducing the edge sharpness by using wavelet thresholding. Donoho and Johnstone have proved several important theoretical results such as wavelet shrinkage i.e. nearly

* Corresponding author. Tel.: +91 7209103160, +91-326-2235473; fax: 91-326-2296563. E-mail address: mantoshb@gmail.com

Available online at www.sciencedirect.com

© 2012 The Authors. Published by Elsevier Ltd. Selection and/or peer-review under responsibility of the Department of Computer

Science & Engineering, National Institute of Technology Rourkela Open access under CC BY-NC-ND license.

Open access under CC BY-NC-ND license.

11 Mantosh Biswas and Hari Om / Procedia Technology 6 ( 2012 ) 10 – 15

optimal in minimax sense and have better convergence rate [4-5]. Coifman and Donoho point out that this algorithm exhibits visual artifacts: Gibbs phenomena in the neighbourhood of discontinuities. This problem has

k Estimator (SURE), that minimizes the mean squared error [6]. One of the most popular methods namely, BayesShrink has been discussed by Chang et al. in which the threshold is derived using the Bayesian method [7-8]. BayesShrink method is a subband-dependent which means that the thresholding is done at each subband in the wavelet decomposition. It is also known as smoothness adaptive. Other works done in the field of wavelet thresholding is discussed in [9-12]. In this paper, we propose a new thresholding method using soft-thresholding. We demonstrate that our proposed method outperforms the traditional ones in terms of PSNR; thus improving the denoised results significantly. Simulation results are also given to show the efficacy of our proposed method. The rest of the paper is organized as follows. Section 2 explains some basic concepts used in existing threshold methods. Section 3 describes our proposed denoising method. Experimental results and analysis are given in section 4. Finally, our concluding remarks are given in section 5.

2. Denoising Techniques with Existing Threshold

Consider an original signal si,j of size MxM and noisy signal ni,j. Add the Gaussian noise to original signal in order to get the noisy signal fi,j , i.e., fi,j =si,j + ni,j (1) Apply the wavelet transform to (1) to get the wavelet coefficients Fi,j. Modify the wavelet coefficients Fi,j using the soft thresholding and then take inverse-wavelet transform to get the denoised image f [5]. Fij t, if Fij ijf = Fij + t, if Fij -t (2) 0, if |Fij| < t where t is the threshold value. In their landmark paper, Donoho and Johnstone have discussed a simple but powerful wavelet-based denoising scheme called VisuShrink [4]. The results of VisuShrink are very smooth with a pleasant visual appearance. However, it is known that VisuShrink tends to over-smooth the signal, thereby losing some details (e.g. sharp edges) of the original signal that results in the increased estimation error. VisuShrink uses the Universal threshold, T, which is proportional to the standard deviation of the noise, is defined as [4]:

T = Mlog2 (3) where 2 represents the noise variance, which is defined as: 2 = [(median| fij |) / 0.06745]2 (4) where fij HH1 subband thresholding.

3. Proposed Denoising Method

Our proposed method consists of estimation of new thresholding function followed by denoising process.

12 Mantosh Biswas and Hari Om / Procedia Technology 6 ( 2012 ) 10 – 15

3.1. Estimation of Proposed New Thresholding

Finding optimized value of the thresholding is a major problem. A small threshold will surpass all the noisy coefficients. Therefore, the denoised signal is still noisy. Conversely, a large threshold value makes more number of coefficients as zero, which leads to smooth signal and destroys the details that may cause blurs and artefacts. So, we try to find out optimum threshold technique. Our method is adaptive to different subband characteristics by analyzing the parameters of the wavelet coefficients as follows [11]: G(q) =

jiijF

, (5)

for q=0, 1, 2, the Fij ely.

S = M

rGr

)(2

0 (6)

where M = M /2k, here k = 1, 2 ositions. Threshold factor, P = e {(T S) / (T + S)} (7)

We calculate a new threshold value TNew as follows: TNew (8) Once we have estimated the new threshold parameter, we apply it for denoising the noisy signal.

3.2. Our thresholding algorithm can be summarized as follow:

(i) Perform the Jth decompositions on 2-D discrete wavelet transform (DWT) for a noisy image f to get noisy wavelet coefficients F.

(ii) Estimate the noise variance 2 using (4). (iii) Calculate the threshold TNew at each high subband, and apply soft-threshold to the wavelet coefficients.

(iv) Perform inverse discrete 2-D wavelet transform to get reconstructed image f .

(a) (b) (c) Fig. 1. Original test images with 512×512 pixels: (a) Cameraman; (b) Barbara; (c) Lena

13 Mantosh Biswas and Hari Om / Procedia Technology 6 ( 2012 ) 10 – 15

4. Experiment Results and Analysis

We have performed experiments on different images using our proposed method. The results of our proposed method have been compared with that of the VisuShrink denoising thresholding technique. In our

ompositions [13]. The experiments are conducted on the following test images: Cameraman, Barbara and Lena (refer Fig. 1) of size 512×512 at different noise levels: 10, 20, 30, and 50. The quality of test images is measured in terms of PSNR. The experimental results of our proposed method are depicted in Table 1, Figs. 2-4, and Fig. 5.

Table 1. Numerical results (i.e. PSNR in db) for Cameraman, Barbra, and Lena

Image Name

Noise level Methods

VisuShrink Proposed

Cameraman

Barbara

Lena

10 20 30 50

10 20 30 50

10 20 30 50

27.42 28.63 24.77 26.48 23.41 24.95 21.97 23.12

24.70 24.80 22.78 23.67 21.90 22.18 21.02 21.64

28.34 30.77 26.09 28.42 24.82 26.48 23.35 23.79

(a) (b) (c) (d) Fig. 2. Cameraman Image: (a) Original, (b) Noisy image with noise level 20, (c) Denoising using VisuShrink; (d) Denoising using proposed method. (a) (b) (c) (d) Fig. 3. Barbara Image: (a) Original, (b) Noisy image with noise level 20, (c) Denoising using VisuShrink; (d) Denoising using proposed method.

14 Mantosh Biswas and Hari Om / Procedia Technology 6 ( 2012 ) 10 – 15

(a) (b) (c) (d) Fig. 4. Lena Image: (a) Original, (b) Noisy image with noise level 20, (c) Denoising using VisuShrink, (d) Denoising using proposed method.

10 20 30 40 5021

22

23

24

25

26

27

28

29

Noise Level

PSNR

(db)

Cameraman Image

VisuShrink

Proposed Method

(a)

10 20 30 40 5021

21.5

22

22.5

23

23.5

24

24.5

25

Noise Level

PSNR

(db)

Barbara Image

VisuShrink

Proposed Method

(b)

10 20 30 40 5023

24

25

26

27

28

29

30

31

Noise Level

PSNR

(db)

Lena Image

VisuShrink

Proposed Method

(c)

Fig. 5. PSNR gains vs. noise levels of Proposed and VisuShrink methods with images: (a) Cameraman; (b) Barbara; (c) Lena

15 Mantosh Biswas and Hari Om / Procedia Technology 6 ( 2012 ) 10 – 15

In Table 1, we have shown the PSNR gains to each test images (refer Fig.1) for our proposed and

VisuShrink techniques. The PSNR gains of our proposed method are higher than that of the VisuShrink for all noise levels (refer Figs. 5(a)-(c), Table 1). The first image i.e. (a) represents the original one, and the second image i.e. (b) represents the noisy one with noise level 20 in Figs. 2-4. In Figs. 2-4, the third ones i.e. (c) are denoised images using VisuShrink and the fourth ones i.e. (d) are the denoised images using our proposed method. It is evident from these figures that the above denoised images using our proposed method have better visual quality than that using VisuShrink. From the above results and analysis, we can say that our method outperforms over the VisuShrink method.

5. Concluding Remarks

In this paper, we have proposed a new thresholding method that reduces noise significantly from a noisy image. Furthermore, this method improves considerably the visual quality of the noisy image.

Acknowledgements

The authors express their sincere thanks to Prof. S. Chand for his invaluable comments and suggestions.

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