IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 21, NO. 9, SEPTEMBER 2012 3981

Efficient Image Denoising Method Based on a NewAdaptive Wavelet Packet Thresholding Function

Abdolhossein Fathi and Ahmad Reza Naghsh-Nilchi

Abstract— This paper proposes a statistically optimum adap-tive wavelet packet (WP) thresholding function for image denois-ing based on the generalized Gaussian distribution. It appliescomputationally efficient multilevel WP decomposition to noisyimages to obtain the best tree or optimal wavelet basis, utilizingShannon entropy. It selects an adaptive threshold value which islevel and subband dependent based on analyzing the statisticalparameters of subband coefficients. In the utilized thresholdingfunction, which is based on a maximum a posteriori estimate,the modified version of dominant coefficients was estimatedby optimal linear interpolation between each coefficient andthe mean value of the corresponding subband. Experimentalresults, on several test images under different noise intensityconditions, show that the proposed algorithm, called OLI-Shrink,yields better peak signal noise ratio and superior visual imagequality—measured by universal image quality index—comparedto standard denoising methods, especially in the presence of highnoise intensity. It also outperforms some of the best state-of-the-art wavelet-based denoising techniques.

Index Terms— Adaptive thresholding, image denoising, noisereduction, optimal wavelet basis (OWB).

I. INTRODUCTION

IN MANY applications, image denoising is used to producea good estimate of the original image from noisy states.

Image denoising techniques are necessary to eliminate as muchrandom additive noise as possible while retaining importantimage features, such as edges and texture.

Wavelet transform, because of its signal representation witha high degree of sparseness and its excellent localization prop-erty, has rapidly become an indispensable image processingtool for a variety of applications, including compression [1],[2], gray-level or color image denoising [3]–[7], object track-ing [8], and texture analyzing [9], [10]. In essence, waveletdenoising attempts to remove the noise presented in the imagewhile preserving the image characteristics regardless of itsfrequency content. It involves the following three steps: 1) alinear forward wavelet transform; 2) nonlinear thresholding;and 3) a linear inverse wavelet transform.

A well-known wavelet thresholding (shrinking) algorithm,named WaveShrink, was introduced by Donoho in 1995 as a

Manuscript received January 27, 2011; revised March 14, 2012; acceptedMay 6, 2012. Date of publication May 22, 2012; date of current versionAugust 22, 2012. The associate editor coordinating the review of thismanuscript and approving it for publication was Dr. Farhan A. Baqai.

The authors are with the Department of Computer Engineering,University of Isfahan, Isfahan 81744, Iran (e-mail: fathi@eng.ui.ac.ir;nilchi@eng.ui.ac.ir).

Color versions of one or more of the figures in this paper are availableonline at http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TIP.2012.2200491

powerful tool in denoising signals degraded by additive whitenoise [11]–[13]. WaveShrink is based on the fact that for manyof real-life signals, a limited number of wavelet coefficients inthe lower subbands are well sufficient to reconstruct the orig-inal signal. Usually, the numerical values of these coefficientsare relatively large compared to noise coefficients. Therefore,by eliminating (shrinking) coefficients that are smaller than aspecific value, called threshold, we can nearly eliminate noise,while preserving hard to keep attributes of the original imagesuch as edges. Thus, choosing threshold values is extremelyimportant.

In the literature, various techniques for adaptive selection ofthreshold values [2], [14], [15], and new thresholding methodsincluding fuzzy logic [16], neural networks [17], and waveletpacket (WP) base using Wiener filtering [18] are reported.

In addition, scientists exploit different types of dependenciesbetween the wavelet coefficients, such as intrascale depen-dency [19], [20] (which takes into account the dependencybetween the coefficients in each subband) and interscaledependency [21] (which considers the dependency betweenscales), to improve denoising further. There is also a report of atechnique that combined both inter- and intrascale dependencyinformation [22].

Furthermore, Dabov et al. utilized collaborative filteringalong with grouping of similar blocks in the image [23]. Theydivided the input image into series of blocks. Similar blocksare then labeled as one group. Finally, blocks of each groupare denoised by applying a collaborative Wiener filtering inthe wavelet domain.

Still, learning methods such as support vector machines(SVMs) are used by scientists to classify wavelet coefficientsinto two classes: clean and noisy [24], [25]. Then, thresholdingis applied only on the noisy set of coefficients.

In this paper, a statistical optimization process along withadaptive and subband-dependable methodologies are appliedto both the thresholding function and wavelet transform, inorder to advance the denoising even further. Unlike standardwavelet-based methods, we used WP transform (WPT) alongwith optimal wavelet basis (OWB) for image decomposition.Then, for each wavelet subband, an adaptive and subband-dependent threshold value is calculated based on analyzingthe subband’s statistical parameters. Next, a new thresholdingfunction, called OLI-Shrink, is proposed to shrink small coef-ficients leading to calculate a modified version of dominantcoefficients. The modification is done using optimal linearinterpolation between each coefficient and the mean value ofthe corresponding subband. Compared with other prominent

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3982 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 21, NO. 9, SEPTEMBER 2012

Fig. 1. WP decomposition.

methods, the proposed algorithm has both significantly low-ered the overall mean square error (MSE) and improved thevisual appearance of the denoised image.

II. WP AND OWB

Wavelet transform has gained wide acceptance as a valuabletool for common signal and image processing tasks becauseof the fact that the wavelets are localized in both frequencyand time domains.

In the traditional 1-D discrete wavelet transform (DWT), asshown in Fig. 1, the original signal L0 is transformed into low-resolution (Ls) and high-resolution (Hs) subbands, explicitly[L3 | H3 | H2 | H1], which corresponds to a certain choice ofthe basis representation. However, it could be decided at eachstage whether to split only the Ls or the Hs part. Splittingthe Hs parts would result in a representation with respect toanother basis with a full binary tree of possible basis functions,or the WPTs, first introduced by Coifman and Wickerhauser[26]. In this type of transform, the optimal representation basisof the input signal is selected by optimizing a function knownas “cost function” in each subband.

The cost functions may determine the cost value for eachnode and its children in the obtained full binary tree. Thealgorithm starts with computing the cost values from thedeepest level nodes. If the sum of the cost values for twochildren nodes is lower than the cost value of their parent node,then the children are retained, otherwise, they are eliminated.For example, in Fig. 1, four such decisions are made: eithera pair of blocks in the bottom row or the block immediatelyabove them (dotted boxes) may be selected.

This cost value computation process is recursively repeatedup to the tree’s root. The result is a basis that has the leastcost among all the possible bases in this tree, so-called bestbasis or optimal basis [27].

There are many possible choices of cost functions. Amongthem are 1-norm, Shannon entropy, log energy, and so on (seealso [27]). In this paper, Shannon entropy cost function ischosen and implemented. The Shannon entropy of coefficientsof subband S is calculated as follows [27]:

SE(S) = −∑

i

S2i log(S2

i ). (1)

Since an image is a 2-D signal, instead of a binary tree,there exists a quad tree, and of course, the same principle maybe applied. In this case, the original image is transformed into

Fig. 2. Results of different wavelet decompositions. (a) Image decompositionscheme. (b) Traditional DWT. (c) WP decomposition. (d) Obtained OWB.

four pieces, which are normally labeled LL, LH, HL, and HH,as shown in Fig. 2(a). In this figure, both traditional DWT andWP decomposition used in our algorithm for selecting optimalwavelet basis are shown.

III. PROPOSED WAVELET SHRINKING ALGORITHM

In this paper, a new adaptive thresholding function is intro-duced to improve the denoising efficiency. Besides, insteadof using a traditional wavelet transform for input imagedecomposition in mainstream literature, an OWB is employed.The reason behind selecting the OWB packet is its dynamicdecomposition nature in forming the subbands. The thresholdvalue is then picked up based on analyzing the statisticalparameters of each subband coefficient.

The thresholding function, necessary in the enhancementand/or elimination of the wavelet coefficients, is obtained usingBayesian maximum a posteriori (MAP) estimate. Then, theoptimal linear interpolation between each coefficient and themean value of the corresponding subband are used to calculatethe modified version of dominant coefficients. The details arediscussed next.

A. Fast OWB Extraction

The algorithm discussed in Section II uses the bottom-upprocedure to extract the optimal basis from the full WP treeof an input image. This algorithm starts at the deepest level ofthe tree and eliminates quads of nodes that have higher costthan their parent node at each level while working back towardthe root.

Instead of the above highly computational complex algo-rithm, an alternative fast method for extracting OWB, whichwas introduced by Kaur et al. [1], is employed. This methodis a top-down search algorithm for selecting the optimal basis.The algorithm starts at the root and generates the optimal basis

FATHI AND NAGHSH-NILCHI: EFFICIENT IMAGE DENOISING BASED ON ADAPTIVE WP THRESHOLDING FUNCTION 3983

Algorithm 1 Algorithm of Fast OWB Extraction

Step I: Choose L as the maximum number of levels forWP decomposition.

Step II: While the current level (d) of decomposition is lessthan L, for each existing subband (parent node)Si

d (0 ≤ i < 4d − 1), do the following.

1) Compute the subband’s Shannon entropy SE(Sid )

as the cost function.2) Decompose Si

d into four subbands (children nodesLL4i

d+1, LH4i+1d+1 , HL4i+2

d+1 and HH4i+3d+1 )

and compute the Shannon entropyof them: SE(LL4i

d+1), SE(LH4i+1d+1 ), SE(HL4i+2

d+1 )

and, SE(HH4i+3d+1 ).

3) If SE(Sid) < (SE(LL4i

d+1) + SE(LH4i+1d+1 ) +

SE(HL4i+2d+1 ) + SE(HH4i+3

d+1 )), then only retain theparent node and eliminate children nodes, other-wise, retain parent and children nodes.

4) If there are no nodes to split, the process ofextracting the OWB reaches the end.

tree without growing the tree to full depth. In this algorithm,we use Shannon entropy to produce the optimal wavelet basis.The pseudo code of this algorithm is shown in Algorithm 1.

B. Threshold Value Determination

Finding an optimal value for thresholding is not an easytask. A small threshold may let noisy coefficients be admitted,and hence the resultant images remain noisy. A large thresholdsets a larger number of coefficients to zero, which leads tosmoothing of the image and may cause blurring and artifacts,and hence the resultant images may lose some signal values.Therefore, an optimum threshold value, which is adaptable toeach subband characteristics, is desired to maximize the signaland minimize the noise. We picked an optimum thresholdselection algorithm proposed by Chang et al. [2]. In thisalgorithm, an adaptive threshold value λs for each subbandS at level d is calculated as

λs = αd,s

(σ 2

η

σX,s

)(2)

where σ 2η and σ 2

X,s are the variances of noise and clean imagecoefficients in the subband S, respectively. In the originalalgorithm, the term αd,s was set to one, nevertheless, weemploy this value to make the threshold suitable in eachdecomposition level and each of subband. In other words,we set αd,s for a larger threshold values for high-frequencysubbands based on their level of decomposition and theircorresponding subbands.

The image noise is assumed to be an additive Gaussianwhite noise. In some applications of image denoising, thevalue of the input noise variance is known or can be measuredbased on the information other than the corrupted image. Ifthis is not the case, we may estimate it by applying the robust

median estimator on the HH1 subband’s coefficients (Y H H1i, j )

as introduced by Donoho, and so forth [11]

σ̂ 2η =

⎡

⎣median

(∣∣∣Y H H1i, j

∣∣∣)

0.6745

⎤

⎦2

. (3)

We also adapted (3) to estimate the image noise variance.Since the noise is additive, the observation model can bewritten as below

Y si, j = Xs

i, j + ηsi, j (4)

where Y si, j are the noise coefficients of subband S, Xs

i, j arethe coefficients of the clean subband, and ηs

i, j are noise coef-ficients. We assume that Y s

i, j , Xsi, j , and ηs

i, j have generalizedGaussian distribution models. Since the coefficients of theclean image and the noise are independent, we may write

σ 2Y,s = σ 2

X,s + σ 2η (5)

where σ 2Y,s is the variance of coefficients (Yi, j ) in subband S.

From this, the value of σ 2X,s can be derived as

σ 2X,s = max(σ 2

Y,s − σ 2η , 0). (6)

If the value of σ 2X,s is zero, the obtained value of threshold λs

is infinite. Then, we set all the coefficients to zero.As mentioned earlier, the term αd,s in (2) is introduced

to make the threshold value more dependent to each of thedecomposition levels and each of the subbands. Since imageinformation exists more in the low-frequency subband than inthe high-frequency subband and since the probability of theexistence of noise in the high-frequency component is greater,applying a greater threshold value to the high-frequency sub-bands reduces the effect of noise more effectively. Also fortwo successive levels (e.g., levels 1 and 2 in Fig. 1), as thelevel of decomposition is increased, the frequency bandwidthsof the created subbands become more limited. That is, becausethe high-frequency components of L1 need to be larger thanL2 (L1 = L2 + H2), the threshold value for L1 should begreater than L2. Based on these observations, we introducedthe term αd,s to increase the threshold value in high-frequencysubbands based on their level of decomposition and theirpositions at corresponding levels.

To this end, a subband weighting function (SWF) in hori-zontal (SWFH ) and vertical (SWFV ) directions at level L ofthe WP decomposition is introduced. This function should bean increasing function on both directions. We used differentincreasing functions for SWF (see Section IV) and concludedthat better results may be obtained when the SWF is definedas below

SWFH/V (i) = i2

22L for i = 1, 2, . . . , 2L (7)

where i is the index of subbands at the highest level ofdecomposition in horizontal and vertical directions, whendecomposed subbands are arranged in the matrix structure, asshown in Fig. 3. The factor 22L is used to normalize the SWF,because in level L we have 2L subbands in each direction. TheSWF function is sampled at the midpoint of each subband atthe highest decomposition level.

3984 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 21, NO. 9, SEPTEMBER 2012

Fig. 3. SWF used for computing the level and subband-dependent thresholdvalue. In the selected subband, the value of α3,6 is 2.448.

The value of αd,s for each subband s at each level d iscalculated as the sum of SWF values in horizontal (SWFH )and vertical (SWFV ) directions that span by subband s asbelow

αd,s =∑

i⊂s

SWFH (i) +∑

j⊂s

SWFV ( j). (8)

An example of obtaining the value αd,s for the sixth subbandin level 3 is shown in Fig. 3. From this figure, the highlightedsubband in horizontal direction spans the positions 9–12, andin vertical direction spans the positions 5–8. Therefore, thevalue of α3,6 is calculated as below

α3,6 = 7SWFH (9 : 12) + SWFV (5 : 8)

= 0.32+0.4+0.48+0.57+0.25+0.19+0.14+0.098

= 2.448. (9)

Therefore, by employing the term αd,s the proposed thresh-old value calculation becomes level and subband dependent.

C. Thresholding Algorithm

Denoising by wavelet is performed by thresholding algo-rithm, in which coefficients smaller than a specific value, orthreshold, will be canceled. Hard and soft thresholding aretwo popular methods for wavelet thresholding [12]. In hardthresholding algorithm, the wavelet coefficients (Y s

i, j ) less thanthe threshold λs are replaced with zero. That is

δHλS

(Y Si, j ) =

⎧⎨

⎩0,

∣∣∣Y Si, j

∣∣∣ ≤ λS

Y Si, j ,

∣∣∣Y Si, j

∣∣∣ > λS .(10)

In the soft thresholding algorithm, however, the waveletcoefficients (Y s

i, j ) less than the threshold λs are replaced withzero and the others are modified by subtracting the thresholdvalue λs using the following:

δSλS

(Y Si, j ) =

⎧⎨

⎩0,

∣∣∣Y Si, j

∣∣∣ ≤ λS

sign(Y Si, j )(

∣∣∣Y Si, j

∣∣∣ − λS),∣∣∣Y S

i, j

∣∣∣ > λS .(11)

The soft thresholding is more efficient and yields bettervisually pleasing images than hard thresholding, but it doesnot use the optimal value for modification of large coefficients.

In order to overcome this limitation, we introduce a newthresholding algorithm (OLI-Shrink) that uses optimal linearinterpolation between each coefficient and corresponding sub-band mean in the modification of dominant coefficients

δO L IλS

(Y Si, j ) =

⎧⎨

⎩0,

∣∣∣Y Si, j

∣∣∣ ≤ λS

Y Si, j − β(Y S

i, j − μS),∣∣∣Y S

i, j

∣∣∣ > λS

(12)

where μs is the mean value of the coefficient of subband s;and β is computed as follows:

β = σ 2η

(σ 2X,s + σ 2

η )∼= σ 2

η

σ 2Y,s

. (13)

This efficient thresholding function is obtained by com-bining the WaveShrink technique with the Bayesian MAPestimation of a signal x from a noisy version of it (y). Byusing the Bayes rule, we can obtain the posterior probabilitydensity function (pdf) of x as below [28]

fX |Y (x |y)= 1

fY (y)fY |X (y|x). fX (x)= 1

fY (y)fη(y − x). fX (x)

(14)where y is the observed value and η is an additive Gaussianwhite noise that is modeled as (4). Since the Gaussian pdfcan be used for estimating the wide variety of natural images(see [2], [29]–[32]) and that the wavelet coefficients in asubband of a natural image can be summarized adequatelyby a Gaussian distribution, we assume that x and η haveGaussian pdfs

fX (x) = N(x, μx , σ 2x ) = 1√

2πσxexp

{ −1

2σ 2x(x − μx )

2}

fη(n) = N(n, μη, σ 2η ) = 1√

2πση

exp

{−1

2σ 2η

(n − μη)2

}.

(15)

Substitution of the pdfs of x and η in (14) yields

fX |Y (x |y) = 1

fY (y).

1√2πση

exp

{−1

2σ 2η

(y − x − μη)2

}

× 1√2πσx

exp

{ −1

2σ 2x(x − μx )

2}

= 1

fY (y).

1

2πσxση

× exp

{−σ 2

x (y − x − μη)2 + σ 2

η (x − μx )2

2σ 2x σ 2

η

}.

(16)

To obtain the MAP estimate, we set the derivative of thelog-likelihood function (ln( fX |Y (x |y))) with respect to x tozero as

∂[ln( fX |Y (x |y))]∂x

= 2σ 2x (y − x − μη) − 2σ 2

η (x − μx)

2σ 2x σ 2

η

= 0.

(17)

FATHI AND NAGHSH-NILCHI: EFFICIENT IMAGE DENOISING BASED ON ADAPTIVE WP THRESHOLDING FUNCTION 3985

Algorithm 2 Proposed Denoising Algorithm

Step 1: Perform WP decomposition to obtain OWB of anoisy image up to four levels (L = 4) by usingShannon entropy.

Step 2: Estimate the noise variance using (4).

Step 3: For each subband S in level d , compute the thresh-old value and statistical parameters:

1) the subband’s variance (σ 2Y,s);

2) the subband’s mean (μs);3) estimate the variance of clean image using (6);4) the term αd,s using (8);5) the term β using (13);6) the threshold value using (2).

Step 4: Threshold all subband’s coefficients using the pro-posed thresholding technique given in (12).

Step 5: Perform the inverse WPT to reconstruct thedenoised image.

From (17) the MAP estimate is given by

x̂ = σ 2X

σ 2X + σ 2

η

(y − μη) + σ 2η

σ 2X + σ 2

η

μX . (18)

Since the noise is supposed to be independent identicallydistributed Gaussian with zero mean (μη = 0) and additivenoise, the mean value of x is

μX = μy − μη = μy = 1

M N

M∑

i=1

N∑

j=1

Yi, j . (19)

In addition, from (13) we can write

σ 2X

σ 2X + σ 2

η

= 1 − σ 2η

σ 2X + σ 2

η

= 1 − β. (20)

Thus, x is estimated by a weighted linear interpolation ofthe unconditional mean of x and the observed value y

x̂ = σ 2X

σ 2X + σ 2

η

(y − μη) + σ 2η

σ 2X + σ 2

η

μX

= (1 − β)(y − μη) + βμX

= y − β(y − μY ). (21)

This optimal linear interpolation between each coefficientand corresponding subband’s mean is combined with waveletthresholding algorithm to yield our proposed thresholdingfunction expressed in (12). Now, based on these analyses, oursimple to implement and efficient denoising algorithm may bedescribed as in Algorithm 2.

IV. EXPERIMENTAL RESULTS

The performance of the proposed noise reduction algorithmis measured using quantitative performance measures such aspeak signal noise ratio (PSNR) and in terms of the visualquality of the images using universal image quality index(UIQI) [33]. The PSNR is given by

PSNR (X, X̂) = 10 log10

(2552

MSE

)d B (22)

Fig. 4. Natural, medical, and textural test images (Dataset 1) that are usedin the first section of our experiments.

where X is the original image and X̂ is the denoised imageand the MSE between the original and denoised images isgiven as

MSE = 1

M N

M∑

i=1

N∑

j=1

(X (i, j) − X̂(i. j))2 (23)

where M and N are the width and height of the image,respectively. Also the UIQI is given by [33]

UIQI (X, X̂) = 4σX,X̂ μX μX̂

(σ 2X + σ 2

X̂)(μ2

X + μ2X̂)

(24)

where σX,X̂ is the covariance of X and X̂ . Also μX and μX̂ are

the mean values of X and X̂ , and σX and σX̂ are the variancesof X and X̂ , respectively.

UIQI is a real number between 0 and 1 inclusive. The bestvalue UIQI = 1 is achieved if and only if X = X̂ . UIQI isdesigned by modeling image distortion in combination withloss of correlation, luminance, and contrast distortion and iseasy to calculate [33].

In the first part of our experiments, we evaluated the effectof different parts of the proposed method on denoising perfor-mance. In these experiments, eight images (see Fig. 4) contam-inated by Gaussian white noise at different standard deviations:σ = 5, 10, 15, 20, 30, 40, and 50 are used. Daubechies waveletwith eight vanishing moments (Db8) is employed to decom-pose the input image into four wavelet levels. The obtainedresults were collected on an average of 20 independent runs.

At first, to obtain a proper SWF, we evaluated the per-formance of the proposed method when different increasingfunctions, as below, were used for SWF

SWFi2 (i) = i2

22L

SWF2i (i) = 2i

SWFN2i (i) = 2i

24L

SWFlog(i) = log2(i)

2L

SWFL ,d(i) = 2L−d

SWF1(i) = 1. (25)

3986 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 21, NO. 9, SEPTEMBER 2012

Fig. 5. Obtained PSNR results on Dataset 1 for different SWF functions.

Fig. 6. Obtained PSNR results on Dataset 1 for different thresholdingfunctions.

The obtained PSNR values for different SWF functions aredepicted in Fig. 5. The results show that all SWF functionshave good performance and with the exception of functionSWF = 2i , their performances are similar. The proposed SWFshows slightly better performance.

Next, the performance of the proposed thresholding func-tion is evaluated by comparing it with the soft and hardthresholding methods. The obtained PSNR results for thesethresholding functions are shown in Fig. 6, indicating that theproposed thresholding function has better performance thanothers, especially at low SNRs.

Furthermore, we evaluated the effect of the decompositionmethod. We applied the proposed method using differentwavelet transform strategies. In this paper, we used DWT andWPT along with OWB for decomposing the input images. Theobtained PSNR results are shown in Fig. 7, indicating thatthe proposed decomposition method shows a slightly betterperformance.

In the second part of the evaluation experiments, the pro-posed algorithm is compared with traditional average filteringand standard wavelet-based image denoising methods such asVisuShrink [11], SureShrink [12], and BayesShrink [2] underdifferent noise intensities. In this experiment, the same set ofimages of Fig. 4, with the same noise contaminations wereused. We again used Daubechies wavelet (Db8) to decomposethe input images into four levels.

The obtained results for PSNR and UIQI values of alldenoising methods at different noise levels are summarized in

Fig. 7. Obtained PSNR results on Dataset 1 for different decompositionmethods.

Fig. 8. Comparison of the obtained PSNR values of different denoisingmethods on Dataset 1.

Fig. 9. Comparison of the obtained UIQI values of different denoisingmethods on Dataset 1.

Tables I and II, respectively. Furthermore, the average value ofPSNR and UIQI are illustrated in Figs. 8 and 9, respectively.These results are collected on an average of 20 independentruns on the images.

From the obtained results shown in Table I and Fig. 8, weconclude that the proposed denoising algorithm outperformsthe better PSNR value compared to the other methods for allnoise intensity situations. When looking closer at the results,we observe that our method outperforms:

1) SureShrink [12] by +6.2 dB on average;2) VisuShrink [11] by more than +5.7 dB on average;3) average filtering by more than +4.2 dB on average;4) BayesShrink [2] by more than +1.9 dB on average.Moreover, Table II and Fig. 9 show that the proposed

method outperforms other methods with higher UIQI values.

FATHI AND NAGHSH-NILCHI: EFFICIENT IMAGE DENOISING BASED ON ADAPTIVE WP THRESHOLDING FUNCTION 3987

TABLE I

OBTAINED PSNR RESULTS (dB) OF DIFFERENT METHODS ON DATASET 1

Noise σ 5 10 15 20 30 40 50 5 10 15 20 30 40 50Method Lena HouseAverage filter 35.9 31.3 28.1 25. 5 22.5 20.2 18.4 33.2 30.2 27.6 25.5 22.3 20.1 18.3

VisuShrink [11] 34.3 28.2 24.6 22.1 18.7 16.4 14.6 33.8 28.2 24.7 22.2 18.8 16.4 14.7

SureShrink [12] 25.1 25.1 25.1 25.1 25.0 24.8 24.6 21.2 21.2 21.1 21.1 21.1 21.1 20.8

BayesShrink [2] 35.6 30.9 28.4 26.9 25.2 22.9 22.1 34.0 29.8 27.2 25.5 22.9 21.4 19.5

Proposed method 36.2 32.5 30.9 29.8 28.5 27.2 26.5 33.1 30.6 28.8 27.4 25.2 24.4 23.6Method Clown BoatAverage filter 37.2 31.8 28.6 26.2 22.9 20.6 18.8 36.2 31.5 28.3 26.1 22.7 20.4 18.7

VisuShrink [11] 34.8 28.3 24.8 22.5 19.2 16.9 15.1 34.5 28.3 24.8 22.3 19.1 16.7 15.1

SureShrink [12] 24.4 24.4 24.4 24.3 24.1 23.8 23.4 21.6 21.6 21.5 21.5 21.4 21.3 21.1

BayesShrink [2] 37.1 32.9 29.8 28.4 25.1 21.9 21.2 36.2 31.3 28.9 26.9 24.1 21.8 21.6

Proposed method 39.6 35.5 33.4 31.5 29.2 27.5 25.7 38.4 33.8 31.6 29.9 28.0 26.7 25.7Method Medical 1 Medical 2Average filter 37.2 31.8 28.4 25.9 22.6 20.2 18.5 37.4 31.9 28.7 26.4 23.1 20.8 19.1

VisuShrink [11] 32.5 30.9 30.0 29.5 28.6 27.8 26.9 31.7 29.1 27.9 27.1 25.8 24.7 23.7

SureShrink [12] 30.5 30.3 30.1 29.8 29.0 28.2 27.2 26.6 26.6 26.5 26.3 25.9 25.1 24.2

BayesShrink [2] 38.3 34.9 32.8 31.3 29.6 28.0 27.1 38.3 34.6 32.5 30.9 28.3 25.1 23.1

Proposed method 39.6 36.0 33.9 32.7 30.8 28.7 27.2 39.1 35.5 33.1 31.9 29.3 27.1 25.2Method Texture 1 Texture 2Average filter 35.6 31.2 28.1 25.8 22.5 20.4 18.6 35.7 31.3 28.2 25.9 22.7 20.5 18.9

VisuShrink [11] 27.4 25.6 24.5 23.6 22.7 22.2 21.8 28.2 26.7 25.7 25.1 24.3 23.8 23.3

SureShrink [12] 22.6 22.6 22.5 22.5 22.4 22.2 21.9 24.7 24.7 24.6 24.6 24.4 24.0 23.7

BayesShrink [2] 35.9 31.6 29.7 28.4 26.4 24.9 23.8 35.9 31.7 29.4 28.6 26.6 25.4 24.4

Proposed method 36.1 32.6 30.8 29.3 27.6 26.2 24.7 36.5 32.4 30.4 29.2 27.4 26.3 25.0

TABLE II

OBTAINED UIQI RESULTS (dB) OF DIFFERENT METHODS

Noise σ 5 10 15 20 30 40 50 5 10 15 20 30 40 50Method Lena House

Average filter 0.9867 0.9843 0.9804 0.9751 0.9600 0.9407 0.9169 0.9943 0.9884 0.9790 0.9662 0.9312 0.8877 0.8381

VisuShrink [11] 0.9933 0.9731 0.9406 0.8997 0.8016 0.6998 0.6029 0.9953 0.9824 0.9614 0.9327 0.8612 0.7780 0.6974

SureShrink [12] 0.9393 0.9393 0.9392 0.9389 0.9371 0.9348 0.9306 0.8999 0.8994 0.8984 0.8973 0.8948 0.8905 0.8841

BayesShrink [2] 0.9950 0.9854 0.9738 0.9637 0.9469 0.9111 0.8914 0.9954 0.9884 0.9776 0.9670 0.9392 0.9140 0.8688

Proposed method 0.9953 0.9898 0.9847 0.9799 0.9717 0.9632 0.9546 0.9938 0.9884 0.9823 0.9756 0.9618 0.9492 0.9353Method Clown Boat

Average Filter 0.9963 0.9956 0.9943 0.9926 0.9876 0.9803 0.9706 0.9981 0.9945 0.9887 0.9808 0.9593 0.9323 0.9006

VisuShrink [11] 0.9981 0.9918 0.9818 0.9689 0.9349 0.8929 0.8448 0.9974 0.9888 0.9750 0.9570 0.9103 0.8543 0.7958

SureShrink [12] 0.9813 0.9811 0.9808 0.9805 0.9791 0.9766 0.9721 0.9549 0.9548 0.9545 0.9536 0.9518 0.9497 0.9454

BayesShrink [2] 0.9987 0.9965 0.9935 0.9897 0.9814 0.9635 0.9471 0.9982 0.9944 0.9903 0.9842 0.9702 0.9491 0.9460

Proposed method 0.9990 0.9980 0.9969 0.9958 0.9929 0.9888 0.9833 0.9988 0.9968 0.9945 0.9920 0.9866 0.9801 0.9738Method Medical 1 Medical 2

Average filter 0.9976 0.9916 0.9820 0.9686 0.9324 0.8873 0.8361 0.9981 0.9933 0.9858 0.9757 0.9496 0.9144 0.8742

VisuShrink [11] 0.9928 0.9897 0.9871 0.9853 0.9818 0.9777 0.9723 0.9928 0.9867 0.9822 0.9786 0.9708 0.9623 0.9519

SureShrink [12] 0.9885 0.9881 0.9874 0.9864 0.9836 0.9797 0.9741 0.9761 0.9760 0.9753 0.9743 0.9710 0.9659 0.9576

BayesShrink [2] 0.9981 0.9959 0.9933 0.9905 0.9857 0.9791 0.9732 0.9984 0.9963 0.9940 0.9914 0.9841 0.9665 0.9469

Proposed method 0.9987 0.9969 0.9948 0.9927 0.9880 0.9825 0.9758 0.9988 0.9971 0.9952 0.9929 0.9867 0.9791 0.9688method Texture 1 Texture 2

Average filter 0.9968 0.9914 0.9826 0.9709 0.9394 0.9019 0.8581 0.9981 0.9948 0.9893 0.9820 0.9628 0.9390 0.9090

VisuShrink [11] 0.9785 0.9671 0.9569 0.9470 0.9341 0.9248 0.9165 0.9893 0.9845 0.9808 0.9777 0.9730 0.9689 0.9644

SureShrink [12] 0.9328 0.9326 0.9322 0.9315 0.9292 0.9255 0.9203 0.9755 0.9753 0.9751 0.9746 0.9731 0.9706 0.9667

BayesShrink [2] 0.9971 0.9923 0.9880 0.9837 0.9736 0.9635 0.9517 0.9982 0.9952 0.9919 0.9895 0.9843 0.9792 0.9728

Proposed method 0.9973 0.9938 0.9903 0.9868 0.9793 0.9715 0.9621 0.9982 0.9960 0.9937 0.9914 0.9871 0.9823 0.9764

Note that at high-intensity noise, all PSNR values, visualperception, and quality measure UIQI of our method arehigher.

In Figs. 10 and 11, the resulting images of SureShrink,VisuShrink, BayesShrink, average filtering, and the proposedmethod for better comparison are shown. Also some resulting

images of the proposed method at different noise levels areshown in Fig. 12.

In the third section of our experiments we comparedour image denoising algorithm with some of the beststate-of-the-art techniques: Saeedi and Moradi’s FuzzyShrink[16], Shui et al.’s WP-Shrink [18], Chen et al.’s NeighShrink

3988 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 21, NO. 9, SEPTEMBER 2012

Fig. 10. Denoised Lena image corrupted by Gaussian noise of standarddeviation of 30. (a) Noisy image. (b) Average filtering with PSNR = 26.7and UIQI = 0.96. (c) VisuShrink with PSNR = 18.7 and UIQI = 0.80.(d) SureShrink with PSNR = 25.0 and UIQI = 0.93. (e) BayesShrink withPSNR = 25.3 and UIQI = 0.94. (f) Proposed method with PSNR = 28.3 andUIQI = 0.97.

Fig. 11. Denoised image corrupted by Gaussian noise of standard deviationof 30. (a) Noisy image. (b) Average Filtering with PSNR = 23.1 and UIQI= 0.95. (c) VisuShrink with PSNR = 25.8 and UIQI = 0.97. (d) SureShrinkwith PSNR = 25.9 and UIQI = 0.97. (e) BayesShrink with PSNR = 28.3and UIQI = 0.98. (f) Proposed method with PSNR = 29.1 and UIQI = 0.99.

Fig. 12. Denoising results of the proposed method. Left: Noisy images withnoise standard deviations 50, 50, and 20 from top to bottom, respectively.Right: Denoised images with obtained PSNR values of 26.8, 27.5, and 29.4from top to bottom, respectively.

[19], Blu and Luisier’s Sure-Let [20], Sendur and Selesnick’sBi-Shrink [21], Portilla et al.’s GSM-Shrink [22], Dabov etal.’s BM3D-Shrink [23], and Wang et al.’s UDWT-SVM [24].

In this experiment, the results of other methods were directlyimported from their published papers listed in the references orfrom [16]. For a fair comparison, all conditions in the proposedmethod and Saeedi and Moradi’s FuzzyShrink [16] set to be

a b c

d e f

Fig. 13. Obtained results of different methods on Boat image. (a) Noise freeimage. (b) Noisy image with standard deviation 25 dB. (c) BM3D-Shrinkresult. (d) GSM-Shrink result. (e) WP-Shrink result. (f) Obtained result of theproposed method.

Fig. 14. Test images along with some obtained results of the proposed methodin the second experiment. Left: Noise-free test images. Center: Noisy imageswith standard deviations 20, 30, 15, and 20 from top to bottom, respectively.Right: Denoised images with obtained PSNR values of 31.48, 28.39, 32.09,and 30.04 from top to bottom, respectively.

the same. As a result, a critically sampled orthonormal waveletwith eight vanishing moments (sym8) over four decompositionlevels was used. Also the standard grayscale test images:Barbara, Boat, Cameraman, and Goldhill were selected asthe experimental dataset. These images were contaminatedby Gaussian white noise at six different standard deviations:σ = 5, 10, 15, 20, 30, and 50. To compare different methods

FATHI AND NAGHSH-NILCHI: EFFICIENT IMAGE DENOISING BASED ON ADAPTIVE WP THRESHOLDING FUNCTION 3989

TABLE III

OBTAINED PSNR RESULTS (dB) OF DIFFERENT METHODS

Noise σ 5 10 15 20 30 50 5 10 15 20 30 50

Method Barbara (512 × 512) Boat (512 × 512)

FuzzyShrink [16] ‡ 37.75 33.99 31.81 30.31 28.11 25.31 36.89 33.67 31.75 30.24 28.46 25.45

NeighShrink [19] ‡ 32.69 29.04 26.87 25.24 23.33 21.91 31.94 29.22 27.50 26.33 24.76 22.87

Sure-Let [20] ‡ 36.71 32.18 29.66 27.98 25.83 23.70 35.02 32.55 30.72 29.46 27.63 25.39

Bi-Shrink [21] ‡ 36.75 33.17 30.85 29.13 26.47 22.88 35.92 32.97 30.71 29.03 26.27 22.84

GSM-Shrink [22] ‡ 37.62 33.66 31.31 29.66 27.38 24.70 36.86 33.38 31.48 30.01 28.09 25.90

UDWT-SVM [24] † - 32.93 - 28.98 26.84 24.26 - 33.30 - 29.79 27.74 25.65

WP-Shrink [18] † - 34.15 32.00 30.50 - - - 33.52 31.70 30.38 - -

BM3D-Shrink [23] 38.31 34.98 33.11 31.78 29.81 27.17 37.28 33.82 32.14 30.88 29.12 26.64

Proposed method 37.95 34.84 33.19 31.78 29.82 26.11 38.18 34.17 32.01 30.55 29.12 25.91

Method Cameraman (256 × 256) Goldhill (512 × 512)

FuzzyShrink [16] ‡ 37.77 33.63 31.17 29.55 27.22 24.21 36.56 33.29 31.37 30.11 28.47 26.23

NeighShrink [19] ‡ 32.18 28.59 26.79 25.52 23.65 21.43 31.61 29.17 27.49 26.86 25.48 23.82

Sure-Let [20] ‡ 35.70 32.21 30.12 28.52 26.51 23.47 35.80 32.58 30.71 29.51 27.89 25.82

Bi-Shrink [21] ‡ 36.61 32.33 29.96 28.03 25.27 21.83 35.28 32.20 30.14 28.45 26.07 22.66

GSM-Shrink [22] ‡ 37.40 33.03 30.67 29.16 27.17 24.81 36.77 33.04 31.15 29.90 28.27 26.48

BM3D-Shrink [23] 38.29 34.18 31.92 30.31 28.64 25.84 37.52 33.62 31.86 30.72 29.16 27.08

Proposed method 39.21 34.70 32.12 30.31 28.09 25.18 37.52 33.53 31.89 30.75 28.77 26.82

‡These results were published in [16] and were directly imported from it.

†These results were directly imported from their paper and they did not use Cameraman and Goldhill images in theirexperiments.

visually, the results of some methods on Boat image are shownin Fig. 13, and the test images along with some resultantimages of the proposed method in different noise levels areshown in Fig. 14. The obtained PSNR results from all runs areshown in Table III. These results were collected on an averageof 20 independent runs.

Table III shows that the PSNR value of the proposed denois-ing algorithm outperforms or has a comparable performancecompared to other methods at various noise levels. Whenlooking closer at the results, we conclude that our methodon average outperforms:

1) Chen et al.’s NeighShrink [19] by more than +4.94 dB;2) Sendur and Selesnick’s Bi-Shrink [21] by more than

+2.38 dB;3) Blu and Luisier’s Sure-Let [20] by more than +1.97 dB;4) Wang et al.’s UDWT-SVM [24] by more than +1.8 dB;5) Portilla et al.’s GSM-Shrink [22] by more than +1.04

dB;6) Saeedi and Moradi’s FuzzyShrink [16] by more than

+0.82 dB;7) Shui et al.’s WP-Shrink [18] by more than +0.71 dB.The worst case is to compare with Dabov et al.’s BM3D-

Shrink [23], which shows a vaguely better result of 0.05 dB onaverage. However, the proposed method is categorically morevisually pleasant (see Fig. 13).

Finally, the computational cost of the proposed method isnoted. Without any optimization of its MATLAB code, it willtake around 8.5 s to process one 512 × 512 image in threelevels of decomposition on a PC with a Pentium-IV 3.2-GHzCPU and a 2.0-GB RAM.

V. CONCLUSION

In this paper, we proposed an efficient algorithm foradaptive noise reduction which combines the optimal linearinterpolation and adaptive thresholding methods in the WPTdomain. Experiments were conducted on different test images,which were corrupted by various noise levels, to assess theperformance of the proposed algorithm, named OLI-Shrink,in comparison with standard and some of the best state-of-the-art wavelet-based denoising methods. The analysis of theresults indicated that the proposed method outperforms all butone of the methods and is categorically more visually pleasantthan all of the other methods. In addition, the computationalcost of the proposed method is modest; and so it is suitable formany image processing applications, such as medical imageanalyzing systems, noisy texture analyzing systems, displaysystems, and digital multimedia broadcasting. It is furthersuggested that the proposed algorithm may be extended tocolor images and video framework, which may further improvevideo denoising.

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Abdolhossein Fathi was born in Delfan, Iran, in1978. He received the B.Sc. degree in computerengineering from the Iran University of Science andTechnology, Tehran, Iran, in 2001, and the M.Sc.degree in computer architecture from the SharifUniversity of Technology, Tehran, Iran, in 2003.He is currently pursuing the Ph.D. degree from theComputer Engineering Department, University ofIsfahan, Isfahan, Iran.

His current research interests include signal andimage processing, computer vision, pattern recogni-

tion, and medical image analysis.

Ahmad Reza Naghsh-Nilchi received the B.S.,M.S., and Ph.D. degrees from the University of Utah,Salt Lake City, all in electrical engineering.

He is an Associate Professor with the Universityof Isfahan, Isfahan, Iran, and has been the Chairmanof the Computer Engineering Department for threeterms. He is the author or co-author of severaljournal articles and conference papers, and has writ-ten a section of a book. He has collaborated withinternationally known institutions and peers, and wasa Research Scholar with the National University of

Ireland, Galway, Ireland, in 2011, and with the University of California,Irvine, in 2012. His current research interests include medical image andsignal processing as well as intensive computing.

Dr. Naghsh-Nilchi is the Editor-in-Chief of the Iranian Journal of Engi-neering Sciences. He was listed in Who’s Who in the World in 2011.