Comparison of Wavelet thresholding for image denoising using different shrinkage

7/29/2019 Comparison of Wavelet thresholding for image denoising using different shrinkage

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International Journal of EmergingTrends & Technology in Computer Science(IJETTCS)Web Site: www.ijettcs.org Email: [email protected], [email protected]

Volume 2, Issue 1, J anuary – February 2013 ISSN 2278-6856

Volume 2, Issue 1 January - February 2013 Page 57

ABSTRACT- Transformation of digital image becomes a

major method of communication in the modern age, but theimage obtained after transmission the data tends to get noisy

and thereby the further processing does not lead to good

results. Hence, Pre processing of an image is very essential.

The pre processing being worked upon is the denoising of

images. The received image needs processing before it can be

used in applications. Image denoising involves the

manipulation of the image data to produce a high quality

image and wavelet transforms have been applied. For

denoising the image different noise models including

Gaussian noise, salt and pepper noise and speckle noise.

Selection of the denoising algorithm is application dependent.

In this paper, a comparative analysis of different

combinations of the suggested threshold values and

thresholding techniques has been carried out very efficiently.

This has been done in order to find more possible

combinations that can lead to the best denoising technique

and compare the results in term of PSNR and MSE.

Keywords— wavelet thresholding, Sure Shrink, BivariateShrink, Bays Shrink and Block Shrink

1. INTRODUCTION

In the past two decades, many noise reduction techniqueshave been developed for removing noise and retainingedge details. Most of the standard algorithms use a

defined filter window to estimate the local noise varianceof a noise image and perform the individual uniquefiltering process. The result is generally a greatly reducednoise level in areas that are homogeneous. But the imageis either blurred or over smoothed due to losses in detailin non-homogenous areas like edges or lines. This createsa barrier for sensing images to classify, interpret andanalyze the image accurately especially in sensitiveapplications like medical field. The primary goal of noise reduction is to remove thenoise without losing much detail contained in an image. To achieve this goal, we make use of a mathematical

function known as the wavelet transform to localize animage into different frequency components or useful subbands and effectively reduce the noise in the sub bandsaccording to the local statistics within the bands. The

main advantage of the wavelet transform is that the image

fidelity after reconstruction is visually lossless. Thewavelet de-noising scheme thresholds the waveletcoefficients arising from the wavelet transform. Thewavelet transform yields a large number of smallcoefficients and a small number of large coefficients.Wavelets are especially well suited for studying non-stationary signals and the most successful applications of wavelets have been in compression, detection anddenoising. The method consists of applying the DWT tothe original data, thresholding the detailed waveletcoefficients and inverse transforming the set of thresholded coefficients to obtain the denoised signal.

Given a noisy signal y =x +n; where x is the desiredsignal and n is independent and identically distributed(i.i.d) Gaussian noiseN (0, σ2), y is first decomposed intoa set of wavelet coefficients w =W[y] consisting of the

desired coefficient θ and noise coefficient n. By applyinga suitable threshold value T to the wavelet coefficients,the desired Coefficient θ=T[w] can be obtained; lastly aninverse transform on the desired coefficient θ will

generate the denoise signal x =W T[θ].

Figure 1: Block Diagram for DWT based denoisingframework

In the experiments, soft thresholding has been used overhard thresholding because it gives more visually pleasantimages as compared to hard thresholding; reason beingthe latter is discontinuous and yields abrupt artifacts in

the recovered images especially when the noise energy issignificant. Flowchart for Image Denoising Algorithmusing Wavelet Transform is shown in figure 2.

1.1 SURE SHRINK

Comparison of Wavelet thresholding for image

denoising using different shrinkageNamrata Dewangan1, Devanand Bhonsle2

1M.E. ScholarShri Shankara Charya Group of Institution, Junwani, Bhilai,

2Sr. Assistant ProfessorShri Shankara Charya Group of Institution, J unwani, Bhilai






A threshold chooser based on Stein’s Unbiased RiskEstimator (SURE) was proposed by Donoho and Johnstone and is called as Sure Shrink. This method

specifies a threshold value for each resolution level j inthe wavelet transform which is referred to as leveldependent threshold. The goal of Sure Shrink is tominimize the mean squared error, defined as,

(1)

Where Z(X,Y ) is the estimate of the signal, S(X,Y ) is theoriginal signal without noise and n is the size of thesignal. The Sure Shrink threshold t* is defined as

(2)

Where t denotes the value that minimizes Stein’sUnbiased Risk Estimator, σ is the noise variance and an

estimate of the noise level σ was defined based on the

median absolute deviation given by

(3)

and n is the size of the image.

Figure 2: Flowchart for Image Denoising Algorithmusing Wavelet Transform

It is smoothness adaptive, which means that if theunknown function contains abrupt changes or boundariesin the image, the reconstructed image also does.[5,11]

1.2 BAY ES SHRINK The goal of this method is to minimize the Bayesian risk,and hence its name, Bayes Shrink. The Bayes threshold,tB, is defined as

(4)

Where is the noise variance and is the signal

variance without noise. From the definition of additivenoise we haveW(x, y) =s(x, y) +n(x, y). Since the noiseand the signal are independent of each other, it can bestated that

(5)

Can be computed as shown below

(6)

The variance of the signal, is computed as

(7)

With and , the Bayes threshold is computed from

Equation (4). Using this threshold the wavelet coefficients

are threshold at each band.[17]

1.3 BIVARIATE SHRINKAGE

New shrinkage function which depends on bothcoefficient and its parent yield improved results forwavelet based image denoising. Here, we modify theBayesian estimation problem as to take into account thestatistical dependency between a coefficient and itsparent. Let w2 represent the parent of w1 (w2 is thewavelet coefficient at the same position as w1, but at thenext coarser scale.) [20]

Theny1=w1+n1 y2=w2+n2 (8)

Where y1 and y2 are noisy observations of w1 and w2 andn1 and n2 are noise samples. Then we can write

Y =w +ny=(y1, y2)w=(w1, w2)n=(n1, n2) (9)

Standard MAP estimator for w given corrupted y is

(10)






This equation can be written as

(11)

(12)

According to bays rule allows estimation of coefficientcan be found by probability densities of noise and priordensity of wavelet coefficient.We assume noise is Gaussian then we can write noise as

(13)

Joint of wavelet coefficients

(14)

We know from equation (9)

(15)Let us define f (w) =log (pw (w))

Then using equation (13) and (14)

(16) This equation is equivalent to solving followingequations

(17)

(18)

Where and represents the derivatives of

with respect to w1 and w2 respectively. We know

can be written a

(19)

From this

.

(20)

From equations (18), (19), (20) and (21) MAP estimator

can be written as

(21)

1.4 BLOCK SHRINK

Block Shrink is a completely data-driven blockthresholding approach and is also easy to implement. Itcan decide the optimal block size and threshold for everywavelet subband by minimizing Stein’s unbiased riskestimate (SURE). The block thresholding simultaneouslykeeps or kills all the coefficients in groups rather thanindividually, enjoys a number of advantages over theconventional term-by-term thresholding. The blockthresholding increases the estimation precision byutilizing the information about the neighbor waveletcoefficients. The local block thresholding methods allhave the fixed block size and threshold and samethresholding rule is applied to all resolution levelsregardless of the distribution of the waveletcoefficients.[21]

Figure 3: 2×2 Block partition for a Wavelet subband

As shown in Figure 3, there is a number of subbandproduced when we perform wavelet decomposition on animage. For every subband, we need to divide it into a lotof square blocks. Block Shrink can select the optimalblock size and threshold for the given subband byminimizing Stein’s unbiased risk estimate. Experimentalresults show that Block Shrink outperforms significantlythe classic Sure Shrink by the term-by-term thresholdingand Neigh Shrink with the fixed overlapping block sizeand threshold proposed by Chen et al. Experimentalresults showed that the PSNRs which Block Shrink

yielded were substantially higher than those that SureShrink and Neigh Shrink did. As a matter of fact, BlockShrink enjoys the advantages of Sure Shrink and NeighShrink and gets rid of there.






2. RESULT This paper presents a comparative analysis of differentshrinkage for image denoising techniques using wavelettransforms. We have experimented with four different

thresholding methods (Bays Shrink, Sure shrink,Bivariate shrink, Block Shrink) using the different noise(Gaussian noise, salt and paper noise and speckle noise)and report the results for the 512×512 standard testimages Figure 4. Our results are measured by the PSNRand MSE. Result for an image shown below affected bygaussian noise having different variances are given in the Table 1 and 2.

Figure 4: Standard Test Image

Table 1: Gaussian noise, PSNR

Thresholding

Technique

Sure

Shrink

Bays

Shrink

Bivariate

Shrinkage

Block

Shrink

Variance

(σ)

σ= 0.01 27.41 24.35 68.44 30.48

σ= 0.02 28.32 24.93 68.46 29.46

σ= 0.03 29.48 25.62 68.48 28.00

σ= 0.04 30.47 26.22 68.56 26.60

σ= 0.05 31.64 27.07 68.59 25.18

Table 2: Gaussian noise, MSE

Thresholding

Technique

Sure

Shrink

Bays

Shrink

Bivariate

Shrinkage

Block

Shrink

Variance (σ)

σ= 0.01 117.9 238.7 0.009 58.16

σ= 0.02 95.52 208.4 0.009 73.62

σ= 0.03 73.27 177.8 0.0092 107.8

σ=0.04 58.28 154.9 0.009 142.0

σ= 0.05 44.54 127.5

6

0.0089 196.8

4

3. CONCLUSIONImage denoising, using wavelet techniques are effectivebecause of its ability to capture the energy of signal in afew high transform values, when natural image iscorrupted by Gaussian noise. Wavelet thresholding is anidea in which is removed by killing coefficient relative tosome threshold. Out of various thresholding techniquessoft-thresholding is most popular. This paper presents acomparative analysis of various image denoisingthresholding techniques (Sure Shrink, Bays Shrink,

Bivariate Shrinkage, Block Shrink) using wavelettransforms. A lot of combinations have been applied inorder to find the best method that can be followed fordenoising intensity images. From the PSNR and MSEvalues as shown in tables, it is clear that BivariateShrinkage giving better results under different noisevariance conditions for all of the images. TheComparative graph for PSNR and MSE are given below

Figure 5: Comparative Graph for PSNR for thresholding

techniques for Gaussian noise having different variances






Figure 6: Comparative Graph for MSE for thresholding

techniques for Gaussian noise having different variances

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Comparison of Wavelet thresholding for image denoising using different shrinkage