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__________________________________978-1-4244-9034-/10/$26.00©2010 IEEE 173

A Comparative Study of Thresholding Techniques for Image Denoising

Vaibhav Nigam1, Sajal Luthra2, Smriti BhatnagarElectronics and Communication Department

3

Jaypee Institute of Information TechnologyNoida, India

Email: 1vnigam2007@yahoo.co.in, 2sajalluthra@gmail.com, 3smriti.bhatnagar@jiit.ac.in

Abstract— This paper is a comparative study of four different thresholding techniques for image denoising in wavelet domain. The methods compared are Visual Shrink, Normal or Bayesian Shrink, Neighbor Shrink and Modified Neighbor Shrink. Visual and Normal Shrink are independent of window size whereas other two shrinks are dependent on window size. To benchmark against the best possible denoising technique four techniques have been compared. In this paper we have presented a more practical, implementation oriented work for the extensive theoretical and algorithmic work presented in the literature.

Keywords- Threshold, Wavelet, Visual Shrink, Bayesian Shrink, Neighbor Shrink, Modified Neighbor Shrink

I. INTRODUCTION

The image de-noising naturally corrupted by noise is a classical problem in the field of signal or image processing. Additive random noise can easily be removed using simple threshold methods. An image is often corrupted by noise in its acquition and transmission. Many further uses of these images require that the noise will be (partially) removed - for aesthetic purposes as in artistic work or marketing, or for practical purposes such as computer vision. Image de-noising is used to remove the additive noise while retaining as much as possible the important signal features. De-noising of natural images corrupted by Gaussian noise using wavelet techniques is very effective because of its ability to capture the energy of a signal in few energy transform values.Wavelet transform is good at energy compaction, the small coefficient are more likely due to noise and large coefficient due to important signal feature. It is because of this fact that noise can be separated from the original signal efficiently.

II. WHY WAVELETS

Spatial filters have long been used as the traditional means of removing noise from images and signals. These filters usually smooth the data to reduce the noise, but, in the process, also blur the data. A different class of methods exploits the decomposition of the data into the wavelet basis and shrinks the wavelet coefficients in order to denoise the data. These techniques use wavelets to transform the data into a different basis, where "large" coefficients correspond to the signal, and "small" ones represent mostly noise. The denoised data is obtained by inverse-transforming the suitably thresholded, or shrunk, coefficients. They perform well across a variety of images and noise levels. In this technique we can calculate and apply thresholds globally, in

a level dependent manner or in a subband dependent manner.

III. WAVELET TRANSFORM

In wavelet analysis the use of a fully scalable modulated window solves the signal-cutting problem. The window is shifted along the signal and for every position the spectrum is calculated. Then this process is repeated many times with a slightly shorter (or longer) window for every new cycle. In the end the result will be a collection of time-frequency representations of the signal, all with different resolutions. Wavelet functions are based on small waves, called wavelets of limited duration. The Discrete Wavelet Transform (DWT) of image signals produces a non-redundant image representation, which provides better spatial and spectral localization of image formation, compared with other multi scale representations.

IV. IMAGE DENOISING ALGORITHM

Let f (NxN) be any image corrupted by independent and identically distributed (i.i.d), white Gaussian noise with zero �� ��� �������� �������� �� �� � ����� ������ original image. The goal is to estimate the signal from noisy observations such that peak signal to noise ratio (PSNR) is maximum. (1)

Let its wavelet transform be, (2)

The wavelet thresholding denoising method processes each coefficient of F from the detail subbands with a soft threshold function to obtain where is matrix of thresholded wavelet coefficient. The denoised estimate is its inverse transform.

The algorithm used for denoising is very simple to implement and computationally efficient. The steps are:

� Decompose the image into several scales.� For each wavelet coefficient apply soft

thresholding.� Reconstruct the image with the altered wavelet

coefficients.

V. THRESHOLDING

Thresholding is a simple non-linear technique, which operates on one wavelet coefficient at a time. In its most

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basic form, each coefficient is thresholded by comparing against threshold, if the coefficient is smaller than threshold it is set to zero; otherwise it is kept or modified. Replacing the small noisy coefficients by zero and inverse wavelet transform on the result may lead to reconstruction with the essential signal characteristics and with less noise. Fig. 1 and Fig. 2 show decomposition and reconstruction of image respectively.

Fig. 1

Fig. 2

A. Threshold SelectionOptimal Thresholding occurs when the thresholding

parameter is set to noise level i.e. ���ncoefficients to be included in the regression model satisfy the hard thresholding rule with T being set to known noise ���� � � �!�n, unwanted noise enters the estimate. � � �"�n, information belonging to signal will be destroyed.

B. Estimation of Thresholding Parameters 1) Visual Shrink [3][5]: Threshold T can be calculated

using the formulae,

This method performs well under a number of applications because wavelet transform has the compaction property of having only a small number of large coefficients. All the rest wavelet coefficients are very small. This algorithm offers the advantages of smoothness and adaptation. However, it exhibits visual artifacts.

2) Bayesian or Normal Shrink [4][5] : It calculates the threshold value (TN), which is adaptive to different subband characteristics.

# �$ � �%�� ������ & �� %��'�� ��% (�� �% scale using the following equation:

Lk is the length of the subband which is estimated from the subband HH1, using the formula

is the standard deviation of the subband under consideration

3) Neighbor Shrink [3]: Let d (i,j) denote the wavelet coefficients of interest

d(i,j) = d(i,j)* B(i,j)where

B(i,j) = (1 – T2/ S2(i,j))+and

S2 � )�2(i,j)

4) Modified Neighbor Shrink [3]: It is same as Neighbor shrink except

B(i,j) = (1 – (3/4)*T2/S2(i,j))+

C. Evaluation CriteriaThe objective quality of reconstructed image is measured

by:

where mse is mean square error between original(x) and denoised image( )

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Fig. 3 (from left to right): Original Image, Noised Image, Denoised Visual, Denoised Bayesian, Denoised Neighbor and Denoised Modified Neighbor using Haar Wavelet

VI. RESULT

The experiments are conducted on all natural gray scale test image Lena of size 512 × 512 a� ��((��� ���� ���� � =10, 15, 20, 25. The wavelet transform employs Haar and Daubechies wavelet at four scales of decomposition. To assess the performance of Bayesian Shrink, it is compared with Visual Shrink, Neighbor Shrink and Modified Neighbor Shrink.

Table I and Table II shows the values of PSNR obtained after applying Haar Wavelet and Daubechies 2 Wavelet respectively on Lena.

TABLE I. PSNR VALUES OF (I) VISUAL SHRINK (II) BAYESIAN SHRINK (III) NEIGHBOR SHRINK AND (IV) MODIFIED NEIGHBOR SHRINK FOR

LENA FOR HAAR WAVELET

Noisy Visual Shrink

Bayesian Shrink

Neighbor Shrink (3x3)

Modified Neighbor Shrink (3x3)

Lena���� 28.1493 28.1102 32.2820 33.1951 32.9537���* 24.6215 26.5392 30.0021 30.9716 30.4328��+� 22.1010 25.4760 28.3490 29.3907 28.5759��,� 20.1629 24.6244 27.0884 28.1993 27.1588

TABLE II. PSNR VALUES OF (I) VISUAL SHRINK (II) BAYESIAN SHRINK (III) NEIGHBOR SHRINK AND (IV) MODIFIED NEIGHBOR SHRINK FOR

LENA FOR DAUBECHIES2 WAVELET

Noisy Visual Shrink

Bayesian Shrink

Neighbor Shrink (3x3)

ModifiedNeighbor Shrink (3x3)

Lena���� 28.1493 28.9396 32.8905 33.8439 33.4761���* 24.6215 27.3098 30.5939 31.6837 30.9368��+� 22.1010 26.1311 28.8787 30.0915 29.0227��,� 20.1629 25.1649 27.6743 28.9310 27.6670

VII. CONCLUSION

In this paper, global, level and subband adaptive thresholding techniques are compared to address the issue of image recovery from its noisy counterpart. This paper efficiently implements the Daubechies and Haar wavelet transform and compares them for image denoising. The results obtained are compared from the PSNR point of view between input image and the reconstructed output image. The image denoising algorithm uses soft thresholding [1] to provide smoothness and better edge preservation at the same time. The Daubechies wavelet has proved to be more efficient than Haar wavelet. Neighbor Shrink has proved to be best thresholding technique among the applied techniques.

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Fig. 4 (from left to right): Original Image, Noised Image, Denoised Visual, Denoised Bayesian, Denoised Neighbor and Denoised Modified Neighbor using Daubechies Wavelet

REFERENCES

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[4]. L. Kaur, S. Gupta, R.C. Chauhan, “Image Denoising using Wavelet Thresholding”, 3rd Indian Conference on Computer Vision, Graphics and Image Processing, ICVGIP, Volume 22 Number 14 (2002).

[5]. M. C. Motwani, M. C. Gadiya, R.C. Motwani, F. C. Harris, Jr., “Survey of Image Denoising Techniques”, Proceedings of GSPx, Santa Clara, CA, Sep. 2004.

[6]. R.C. Gonzalez, R. E. Woods,[2001] Digital Image Processing,Second Edition, Prentice Hall.

[7]. S. Ravishankar, B.V.Uma, “Application of Stationary Wavelet Transform for improvement of Rate-Reach performance in ADSL interference environment”, Journal of Wavelet Theory and Applications ISSN 0973-6336, Volume 2 Number 1 (2008), pp. 83–92.