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Contents lists available at ScienceDirect

Signal Processing

Signal Processing 90 (2010) 428–439

0165-16

doi:10.1

� Cor

E-m

ejk@gm

gkook@

journal homepage: www.elsevier.com/locate/sigpro

Despeckling of medical ultrasound images using Daubechiescomplex wavelet transform

Ashish Khare a, Manish Khare b, Yongyeon Jeong c, Hongkook Kim d, Moongu Jeon d,�

a Department of Electronics and Communication, University of Allahabad, Allahabad, Indiab Centre for Development of Advanced Computing, Noida, Indiac Department of Radiology, Chonnam National University Hwasun Hospital, Hwasun, Chonnam, Republic of Koread Department of Information and Communications, Gwangju Institute of Science and Technology, Gwangju, Republic of Korea

a r t i c l e i n f o

Article history:

Received 15 October 2008

Received in revised form

3 June 2009

Accepted 9 July 2009Available online 17 July 2009

Keywords:

Medical image denoising

Speckle noise removal

Daubechies complex wavelet transform

Wavelet shrinkage

84/$ - see front matter & 2009 Elsevier B.V. A

016/j.sigpro.2009.07.008

responding author.

ail addresses: ashishkhare@jkinstitute.org (

ail.com (M. Khare), yjeong@chonnam.ac.kr

gist.ac.kr (H. Kim), mgjeon@gist.ac.kr (M. Jeo

a b s t r a c t

The paper presents a novel despeckling method, based on Daubechies complex wavelet

transform, for medical ultrasound images. Daubechies complex wavelet transform is

used due to its approximate shift invariance property and extra information in

imaginary plane of complex wavelet domain when compared to real wavelet domain. A

wavelet shrinkage factor has been derived to estimate the noise-free wavelet

coefficients. The proposed method firstly detects strong edges using imaginary

component of complex scaling coefficients and then applies shrinkage on magnitude

of complex wavelet coefficients in the wavelet domain at non-edge points. The proposed

shrinkage depends on the statistical parameters of complex wavelet coefficients of noisy

image which makes it adaptive in nature. Effectiveness of the proposed method is

compared on the basis of signal to mean square error (SMSE) and signal to noise ratio

(SNR). The experimental results demonstrate that the proposed method outperforms

other conventional despeckling methods as well as wavelet based log transformed and

non-log transformed methods on test images. Application of the proposed method on

real diagnostic ultrasound images has shown a clear improvement over other methods.

& 2009 Elsevier B.V. All rights reserved.

1. Introduction

Among all modalities of medical imaging system,ultrasonography has been considered as one of the mostnon-invasive and powerful technique for imaging organsand soft-tissue structures of the human body. Ultrasono-graphy is often preferred due to its relatively low-cost andalso there are no ionizing radiations. Beside theseadvantages, medical ultrasonographic images are of poorvisibility, resulting from speckle noise [1]. Speckle occursespecially in the images of liver and kidney whose

ll rights reserved.

A. Khare), mkhar-

(Y. Jeong), hon-

n).

underlying structures are too small to be resolved bylarge wavelength [2]. The presence of speckle resultsdegradation in image quality and makes it difficult forhuman interpretation and diagnosis. Thus speckle reduc-tion (despeckling) is an important issue for analysis ofultrasound images. Many algorithms have been developedfor despeckling. They can be broadly classified in thefollowing categories [28]: local statistics, homomorphic,anisotropic diffusion, and the most recent one is waveletfiltering.

The standard example of local statistics filters are theKuan Filter [3], the Frost filter [4], the Lee filter [5], theGamma MAP filter [6], etc. and their variations [6].Although these filters perform well for removal of specklenoise, they have major limitations in preserving sharpfeatures of the original image. Homomorphic filter isapplied in FFT domain. In the homomorphic filtering [29]

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A. Khare et al. / Signal Processing 90 (2010) 428–439 429

the image is denoised in fast Fourier transform (FFT)domain, then the inverse FFT is calculated. Anisotropicdiffusion [30] and its variant [31] are also used for speckleremoval. These are nonlinear filtering techniques forsimultaneously performing contrast enhancement andnoise reduction by using the coefficient of variation.

Uses of discrete wavelet transform (DWT) basedtechniques are the recent trends for speckle removal[7–9,11,14,22,26,29,32,40]. Since speckles are multiplica-tive in nature, so many DWT-based approaches use alogarithmic transform to convert signal-dependentspeckle noise to additive white noise. Then waveletthresholding is used to remove the noise. After thresh-olding, an exponential operation is employed to convertthe log-transformed image back to non-logarithmicformat. These approaches are based on the assumptionthat the mean of log-transformed speckle noise is equal tozero. However, exact value of the mean of log-transformedspeckle noise is not equal to zero and thus requirescorrection to avoid extra distortion in the restored image[9]. There are some wavelet based techniques that avoidlog-transform. Another drawback is the use of down-sampling operation in DWT, which results in shift-variant[10] and makes it difficult to preserve original imagediscontinuities in the wavelet domain [11]. Phaseinformation also plays an important role in ultrasoundimage and DWT is inefficient for providing the phaseinformation. All these shortcomings will be reduced bythe use of complex wavelet transform. Several complexwavelet transforms like the dual tree complex wavelettransform (DTCWT) [10], the projection-based complexwavelet transform [12], the steerable pyramid complexwavelet transform [13], etc. have been proposed. In theabove mentioned complex transforms, real-valued filtersinstead of complex-valued filters are used and due topresence of redundancy, they are computationally costlyas well. For avoiding these shortcomings, the Daubechiescomplex wavelet transform has been used [14–16], whichis approximately shift-invariant and also provides phaseinformation. The Daubechies complex wavelet transform[16] is a natural extension of the concepts of Daubechiesreal-valued wavelet transform. As Daubechies [17]has given real solutions for wavelet equation, Lina [16]has explored the complex solutions for the equation andproved that complex solutions do exist leading complexDaubechies wavelet transform. Complex Daubechies wa-velets can be made symmetric. The symmetry property ofthe filter makes it easy to handle the boundary problemsfor finite length signal [18]. A linear phase filter is requiredto preclude the nonlinear phase distortion and to keep theshape of the signal. However obtaining linear phase forthe complex Daubechies wavelet is difficult. Recently amethod to achieve both symmetry and linear phase oncomplex Daubechies wavelet was proposed [19]. In thepresent paper, we used symmetric and approximatelylinear phase complex Daubechies wavelet filter [20],which is nearly shift-invariant and non-redundant.

As compared a vast literature for denoising using real-valued wavelet transform, only a small literature isavailable for denoising using complex wavelet transform.Clonda et al. [20], Romberg [33] and Jalobeanu [34]

restored images by using hidden markov model ofcomplex wavelet transform and complex packets, respec-tively. Lina and MacGibbon [35] applied shrinkagefunctions using Bayesian method on real images. Achim[36] proposed a shrinkage function for removal of additivenoise in dual tree complex wavelet domain, for whichbivariate a-stable distribution of complex wavelet coeffi-cients has been assumed.

Most of the available literature assumes the log-transformed multiplicative noise to behave as additive noiseand follow a Gaussian distribution in wavelet domain. ThisGaussian assumption is over simplified and unnatural [37].Recently some researchers [9,11,22,37] have proposed otherdistributions for log-transformed speckle image. Michailo-vich [37] assumed that the speckle noise in wavelet domainis likely to obey the Fisher–Tippet distribution. Bhuiyan et al.[32] modeled the wavelet coefficients of log-transformedimage with a symmetric normal inverse Gaussian distribu-tion, while those of log-transformed noise are assumed to beGaussian distributed. Achim [36] have introduced theconcept of the bivariate a-stable distribution for denoisingin complex wavelet domain, but this proposed distributionand noise removal technique works well only for Gaussianadditive noise. This is major limitation of the proposedmethod by Achim [36]. In the present work we have shownthat the distribution of log-transformed image is a mixtureof two Gaussian distributions and based on this assumptiona wavelet-shrinkage parameter has been derived.

The parameters of shrinkage function are calculatedthrough the observed image statistics. This has made themethod highly adaptive. While most of the denoisingmethods smooth the image at all points, the proposedmethod preserves the strong edge points during thedenoising process. These edge points are detectedadaptively using imaginary component of complex scalingcoefficients and apply shrinkage at non-edge points. Thisis very helpful for preserving the contrast of objects in 2-Dimages. The proposed method has been compared withthe median filter, the wiener filter and standard speckleremoval methods like the Kuan, the Frost, the Lee and theGamma MAP filters. The proposed method is alsocompared with other wavelet based speckle removalmethods like the statistical log-transformed waveletspeckle noise removal [8], non-log transformed general-ized likelihood ratio based wavelet domain noise removal(GenLik) [9] method, and total least square denoisingalgorithm (TLS-denoise) [41]. By means of experimentalresults it has been shown that the present method yieldsfar better results than others [3,4,8,9]. There are severalvisual and numerical image quality measurements existsin the literature [38,39]. For the image quality perfor-mance measure we used signal to mean square error(SMSE) and signal to noise ratio (SNR), as they are bettermeasurements for speckle noise.

The rest of paper is organized as follows: Section 2describes the basic concepts of complex Daubechieswavelet transform and its properties. In Section 3, thedetails of the proposed algorithm, including the methodfor edge detection in complex wavelet domain and theshrinkage function are provided. In Section 4, the resultsof the proposed method for speckle removal are shown

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Fig. 1. (a) Original signal and the shifted signal by one sample, (b) high-pass wavelet coefficient of the original signal and the shifted signal using real db4

wavelet, and (c) magnitude of high-pass complex wavelet coefficients of the original signal and the shifted signal using SDW6 wavelet.

A. Khare et al. / Signal Processing 90 (2010) 428–439430

and compared to other methods, and finally conclusionsare given in section 5.

2. Daubechies complex wavelet transform

In the present work we used complex Daubechieswavelets that are endowed with symmetry property[16,42].

2.1. Construction of Daubechies complex wavelet

The basic equation of the multiresolution analysis isthe scaling equation:

fðtÞ ¼ 2X

n

anfð2t � nÞ (1)

where f(t) is scaling function and ans are coefficients. Theans can be real as well as complex valued and

Pan ¼ 1.

Daubechies’s wavelet bases {cj,k(t)} in one dimensionare defined through the above scaling function andmultiresolution analysis of L2ð<Þ [17].

Daubechies has given the general solution as follows:Let hðzÞ ¼ 1

2ð1þ zÞ. Consider the following polynomial:

pMðzÞ ¼XMk¼0

ð�1Þk2M þ 1

k

� �hðzÞ2M�2khð�zÞ2k (2)

that satisfies pMðzÞ � pMð�zÞ ¼ z.If we denote Zn, n ¼ 1,2,y,M, the set of roots of pM(z)

inside the unit circle (|Zn|o1), then any selection R among

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the roots of pM(z) defines an admissible trigonometricpolynomial

FðzÞ ¼ hðzÞ1þMYm2R

z�1 � Zm

1� Zm

� ��YneR

z�1 � Zn�1

1� Zn�1

!(3)

that satisfies the constraints of multiresolution analysis[17]. The solution of Eqs. (2) and (3) will lead toDaubechies scaling functions. During the formation ofgeneral solution, Daubechies considered an to bereal-valued only. Relaxing the condition for an to be onlyreal-valued will lead to the Daubechies complex scalingfunction and this leads to complex Daubechies wavelettransform. The construction of complex Daubechieswavelet has been done as in [20,42].

The generating wavelet c(t) is given by

cðtÞ ¼ 2X

n

ð�1Þna1�nfð2t � nÞ (4)

and c(t) and f(t) share the same compact support [�M,M+1].

Any function f(t) can be decomposed into complexscaling function and mother wavelet as

f ðtÞ ¼X

k

cj0

k fj0 ;kðtÞ þ

Xjmax�1

j¼j0

djkcj;kðtÞ (5)

where j0 is a given resolution level, fcj0

k g and fdjkg are

known as approximation and detail coefficients.

2.2. Properties of Daubechies complex wavelet transform

The Daubechies complex wavelet function can be madesymmetric. The symmetry property of filter makes it easyto handle the boundary problems of the object [20]. Wehave used symmetric Daubechies complex wavelet (SDW)transform. SDW is also in linear phase and its linear phaseproperty allows it to retain the shape of the object duringreconstruction [19]. All the usual properties of realDaubechies wavelet bases are derived from the amplitude.Thus those properties are maintained in the complexsolution. However complex Daubechies wavelets exhibitsome other important properties.

2.1.1. Reduced shift sensitivity

Daubechies complex wavelet transform is approxi-mately shift invariant. A transform is shift sensitive if aninput signal shift causes an unpredictable change intransform coefficients. In discrete wavelet transform shiftsensitivity arises from downsamplers in the implementa-tion. Fig. 1 illustrates the reduced shift-sensitivity ofDaubechies complex wavelet transform. Fig. 1(a) shows aninput signal and shifted form of the input signal by onesample. Fig. 1(b) shows high-pass wavelet coefficients ofthe original and the shifted signals using DWT, while Fig.1(c) shows the corresponding high-pass complex waveletcoefficient magnitudes. From the figure it is quite clearthat the real wavelet transform is highly shift-sensitivewhereas the Daubechies complex wavelet transform isapproximate shift-invariant in nature.

2.1.2. Edge detection

Let fðtÞ ¼ kðtÞ þ ilðtÞ be a scaling function and cðtÞ ¼uðtÞ þ ivðtÞ be a wavelet function. Let l oð Þ and k oð Þ beFourier transforms of l(t) and k(t). Considering the ratio

aðoÞ ¼ � lðoÞkðoÞ

(6)

Clonda [20] observed that a(o) is strictly real-valued andbehaves as o2 for |o|op. This observation relates theimaginary and real components of scaling function: l(t)accurately approximates the second derivative of k(t), upto some constant factor.

From the above property, Eq. (6) indicates lðtÞ � aDkðtÞ.Here D represents second derivative. This gives multiscaleprojections as

hf ðtÞ;fj;kðtÞi ¼ hf ðtÞ; kj;kðtÞi þ ihf ðtÞ; lj;kðtÞi

� hf ðtÞ; kj;kðtÞi þ iahDf ðtÞ; kj;kðtÞi (7)

From Eq. (7), it can be concluded that the complex scalingfunction exhibit multiresolution averaging information (inthe real part) together with the corresponding Laplacian(in the imaginary part). Thus Daubechies complex wavelettransform can act as local edge detectors, since imaginarycomponent of complex scaling coefficients representstrong edges. We have developed a method for detectionof the strong edges, using only the imaginary componentof complex scaling coefficients at multiscale as discussedin Section 3.2.

3. The proposed model

3.1. The shrinkage function

The images obtained from the coherent imagingsystems like ultrasound images are corrupted by specklenoise. Within each resolution cell, a number of elementaryscatterers reflect the incident wave towards the sensor.The back scattered coherent waves with different phasesundergo the constructive or destructive interference in arandom manner. Thus the acquired image is corrupted bya random granular pattern that hinders the interpretationof the image content and reduces the detectability of thefeatures of interest [2]. A speckled image Y is commonlymodeled as

Y ¼ XN (8)

where X is a noise free image and N is a unit mean randomfield. Random variable X and N are considered to beindependent, when speckle is assumed to be fullydeveloped.

Wavelet coefficient of ultrasound images typicallyexhibit strong non-Gaussian statistics [23]. Waveletcoefficients of an image corrupted with speckle noisecan be modeled as generalized Gaussian distribution(GGD) [24]. However, it is not easy to work with GGD,due to its complex structure. Several alternative modelsare also proposed. Among those alternative methods amixture density of two zero-mean Gaussian distributionshas been proposed due to its relatively simple form and

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A. Khare et al. / Signal Processing 90 (2010) 428–439432

high accuracy in modeling the distribution of waveletcoefficients [25].

We have observed that the distribution of Daubechiescomplex wavelet coefficients follow the mixture of twoGaussian distribution. Fig. 2 shows the performance of themixture Gaussian model in matching the distributions ofreal and imaginary components of complex waveletcoefficients of an ultrasound image. Fig. 2 shows thatthe mixture Gaussian model follows the actual histogram

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Real Part of Wavelet Coefficient

Pro

babi

lity

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Imaginary Part of Wavelet Coefficient

Pro

babi

lity

blue - real PDFblack - Mix PDFgreen - Gauss 1 PDFred - Gauss 2 PDF

Fig. 2. PDF of complex wavelet coefficient matching with Gaussian PDF

model for an ultrasound image.

Fig. 3. (a) An original ultrasound image of hepatic vein

very well. A small discrepancy near zero does not reducethe effectiveness of the model. Based on the mixture-Gaussian model, a denoising method has been establishedas follows.

In wavelet domain, the decomposition operation canbe written as [11,26]

WY ¼W½Y � ¼W½XN�

¼W½X� þW½XðN � 1Þ�

¼WX þWS (9)

Since N is a unit mean random field, therefore

E½N� ¼ 1 and E½N � 1� ¼ 0 (10)

Therefore WX ¼W½X� and WS ¼W½XðN � 1Þ� ¼W½XNC �

are centered and uncorrelated random process [11,45] and

E½WX � ¼ 0; E½WS� ¼ 0; and E½WXWS� ¼ 0 (11)

The a posteriori probability density function conditionedto the observation can be written as [11]

PWX jWYðwX jwY Þ ¼

PWY jWXðwY jwXÞPWX

ðwXÞ

PWYðwXÞ

¼PWSjWX

ðwSjwXÞPWXðwXÞ

PWYðwY Þ

The estimate wX maximizing the a posteriori pdf, is

wX ¼ arg maxwX

ðPWX jWYðwX jwY ÞÞ

� �

d

dwXðlnðPWSjWX

ðwSjwXÞÞ þ lnðPWXðwXÞÞÞwX¼wX

¼ 0 (12)

We assume Gaussian model for PWSjWXðwSjwXÞ and

PWXðwXÞ [11]. The MAP criteria is then equivalent to

applying MMSE criteria.We assumed Gaussian distributions for both WX and

WS and in the Gaussian case, the uncorrelationðE½WXWS� ¼ 0Þ is sufficient to ensure the independencebetween WX and WS. Therefore, the likelihood termPWS jWX

ðwSjwXÞ in the Bayesian relation is equal to pdfPWSðwSÞ of WS.For removal of signal dependent noise, a shrinkage in

wavelet domain is most popular. Several shrinkageestimates are used for removal of signal dependent noise[14,27,43,44]. In the present work, we used a different

and (b) detected edges by the proposed method.

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A. Khare et al. / Signal Processing 90 (2010) 428–439 433

adaptive shrinkage estimate for speckle removal similar tothat of Foucher [11].

Shrinkage estimate of the noise-free wavelet coeffi-cient WX is WX ¼ ZWY , where Z is shrinkage factor. Basedon minimum mean square error (MMSE) criteria, theoptimal shrinkage factor Z0 is obtained by minimizingmean square error between WX and WX

Z0 ¼ arg minZ

E½ðWX �WXÞ2�

0 5 10 15 20 250

5

10

15

20

25

30

SMSE (in dB) of noisy image

SM

SE

(in

dB) o

f den

oise

d im

age Black - The Proposed Method

Red - The Weiner filterBlue - The Frost filterGreen - The Gamma filterMagenta - The Median filter

0 5 10 15 20 250

5

10

15

20

25

30

SMSE (in dB) of noisy image

SM

SE

(in

dB) o

f den

oise

d im

age Black - The proposed method

Red - The Kuan filterBlue - The Lee filterGreen - Log-transformed wavelet denoising

Fig. 4. Comparison of the proposed method with other methods at

different noise levels of speckle noise.

Table 1Signal to noise ratio (SNR) (in dB) measures obtained by different speckle rem

Method Image 1 Image 2

Noisy image 40.4365 37.0248

Weiner filter 43.1113 42.0356

Frost filter 42.2444 41.2529

Gamma filter 39.9324 39.3890

Median filter 40.3738 39.0865

Kuan filter 42.1492 41.3597

Lee filter 45.0725 42.7658

Log wavelet speckle filter 43.7598 41.2973

GenLik method 45.2470 43.4482

TLS-denoise method 45.3266 42.8197

The proposed method 46.0243 43.5243

Substituting WX ¼ ZWY in above equation, we get

Z0 ¼ arg minZ

E½ðZWY �WXÞ2�

Thus, the MMSE solution is in the form of

Z0 ¼ E½W2Y � � E½WY WS�

E½W2Y �

(13)

To calculate Z0, it is necessary to estimate all unknownparameters. In the above equation E½W2

Y � can be knowneasily, but it is difficult to estimate E½WY WS� due todependence of noise WS upon WY.

After replacing WY with WX+WS in E½WY WS�, we obtain

E½WY WS� ¼ E½WXWS� þ E½W2S � (14)

Because of the zero-mean mixture Gaussian distributionmodel of WY [11], its expectation E[WY] is also zero.Therefore

E½W2Y � ¼ s2

WYand E½W2

S � ¼ s2WS

and from Eq. (11), E½WXWS� ¼ E½WX �E½WS� ¼ 0Therefore, E½WY WS� becomes

E½WY WS� ¼ 0þ E½W2S � ¼ s2

WS(15)

Thus the new shrinkage factor is

Z0 ¼ E½W2Y � � E½WY WS�

E½W2Y �

¼s2

WY� s2

WS

s2WY

(16)

Here s2WY

is known and s2WS

have to be estimated.Foucher [11] estimated s2

WS, by assuming zero-mean

mixture Gaussian distribution model for WY as

s2WS¼ cjm2

XC2Nð1þ C2

XÞ (17)

where mX ¼ E½X� and C2X ¼ C2

WY� cjC

2N=cjð1þ C2

NÞ

CN ¼ffiffiffiffiffiffiffiffi1=L

p, for L-look image with L ¼ 1 for ultrasound

images.The parameter,

cj ¼X

k

ðhkÞ2

!2 Xl

ðglÞ2

!2ðj�1Þ

where h and g are the high-pass and the low-pass filters atdecomposition level j, respectively.

CWY¼ sWY

=mY is called normalized standard deviationof noisy wavelet coefficients.

oval methods.

Image 3 Image 4 Image 5 Image 6

34.3720 31.2663 29.0466 27.1947

40.6713 37.2728 31.4130 27.2094

40.3678 38.4806 36.7043 34.8135

38.5900 38.6799 36.2117 33.3733

37.6680 35.2590 33.2556 30.2942

40.2710 38.6264 36.7439 34.7515

40.1807 36.5213 33.6769 31.3951

39.7478 36.6351 23.8631 23.5672

41.3571 37.7249 36.5907 35.7721

41.7626 38.2518 36.9642 34.3399

42.0984 38.7468 37.8801 35.9165

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Fig. 5. (a). Original kidney ultrasound image, (b) image degraded with simulated speckle noise. Results of various speckle suppression methods: (c) the

proposed method, (d) the Kuan filter, (e) the Frost filter, (f) the Weiner filter, (g) Log-transformed image wavelet based denoising[8], and (h) GenLik

method [9].

A. Khare et al. / Signal Processing 90 (2010) 428–439434

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A. Khare et al. / Signal Processing 90 (2010) 428–439 435

Substituting the value of C2X in Eq. (17), we obtain

s2WS¼ cjm2

XC2N 1þ

C2WY� cjC

2N

cjð1þ C2NÞ

!

¼ cjm2XC2

N

cjð1þ C2NÞ þ C2

WY� cjC

2N

cjð1þ C2NÞ

¼ m2XC2

N

cj þ C2WY

1þ C2N

(18)

Since E[Y] ¼ E[X] (or mY ¼ mX) with normalized specklenoise N, and CWY

¼ sWY=mY ,

so we obtain

s2WS¼ m2

Y C2N

cj þs2

WY

m2Y

ð1þ C2NÞ

(19)

s2WS¼

C2Nðcjm2

Y þ s2WYÞ

ð1þ C2NÞ

(20)

Now the shrinkage factor Z0 can be computed easily bysubstituting the value of s2

WSin Eq. (16).

3.2. Edge detection in complex wavelet domain

Based on the observation and Eq. (7), that sharp edgeshave large values of imaginary component of scalingcoefficients over many scales, we have used an edgedetector that uses the interscale product of the imaginaryvalues of scaling coefficients, similar to the noise filtrationtechnique developed by Xu et al. [21]. Large values ofdirect multiplication locate important edges. This ap-proach is straightforward, easier to implement and robust.

In order to calculate edge coefficients, correlation(actually products) among imaginary component of scal-ing coefficients at adjacent scales has been computed. Inthe 1-D case, the correlation is defined as

CLð2j; kÞ ¼

YL

r¼0

ImðWf ð2jþr ; kÞÞ (21)

where ImðWf ð2jþr ; kÞÞ denotes imaginary component ofcomplex scaling coefficient at kth point and j+rth level.The optimal value of L for computing correlation can bedetermined experimentally and by means of severalexperiments, we have found that the choice for L ¼ 3gives good results for edge detection.

Given the observations made above, we can offer thefollowing process of edge detection:

Table 2Average processing time required for different speckle removal methods.

(1)

Method Average processing time (in seconds)

Rescale the power of fCLð2j; kÞg1�k�n to that of

fWf ð2j; kÞg1�k�n.

(2) Identify an edge at position k if jCLð2

j; kÞj4jWf ð2j; kÞj.

Frost filter 5.195

Gamma filter 4.374

Median filter 1.186

Kuan filter 5.835

Lee filter 5.694

Log wavelet speckle filter 1.607

GenLik method 183.706

TLS-denoise method 1051.973

The proposed method 1.334

The above procedure can be easily extended fordetection of edges in 2-D images.

Fig. 3(a) and (b) shows an ultrasound image of hepaticvein and its edges computed by the above proposedalgorithm. Here L ¼ 3, where this value has been deter-mined experimentally considering the quality of theobtained results. From these figures, it is clear that

imaginary component of complex scaling coefficientshave good capability of edge detection at multiple levels.The edges detected by the proposed method are sharp.Multiple and false edge detection problem is not presentin the proposed method.

3.3. The algorithm

The proposed denoising algorithm is as follows:

1.

Transform the noisy image y into Daubechies complexwavelet coefficient w.

2.

Detection of strong edges:(a) Separate real and imaginary components of wave-

let coefficient.(b) Detect strong edges in noisy image using the

procedure described in Section 3.2.

3. Wavelet shrinkage:

(a) Shrink the complex wavelet coefficients (except atedge points) with shrinkage factor described in Eq.(16).

(b) Apply the inverse wavelet transform to getestimated image.

4. Experiments and results

Here we present performance of the proposed methodand compare our results with other conventional despecklingmethods like the median, the Weiner, the Frost, the Lee, theGamma MAP and the Kuan filters. The proposed method isalso compared with recent wavelet based speckle removalmethods, viz. wavelet based statistical log-transformed imagedenoising [8], generalized likelihood ratio based waveletdomain noise removal technique (GenLik) [9], and total leastsquare denoising algorithms (TLS-denoise) [41].

Speckles occur when a coherent source and a non-coherent detector are used to interrogate the medium, whichis rough on the scale of the wavelength. Ultrasound B-scanmedical images represent the back-scattering of an ultra-sound beam from structures inside the body. Diffusescattering arises when there are a large number of scattererswith random phase within the resolution cell of theultrasound beam and this diffuse scattering causes specklein medical images. Speckle noise is in the form of granulartexture. Speckles degrade the ability to resolve details anddetect objects of size comparable to the speckle size.

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For the experimentation purpose, the medical ultra-sound images were acquired using the LOGIC 9 (GEmedical system, Milwaukee, WI, USA). A C5-1 convex-array transducer with a frequency range of 1–5 MHz wereused. The ultrasound exams were performed by employ-ing adaptive gain compensation, extreme resolution,tissue harmonic and compound imaging through trans-verse, longitudinal scans to examine the abdominalorgans including liver, gall bladder and kidney. The imageswere recorded with a resolution 476� 400 pixels with 256gray levels. Experiments have been performed for many

Fig. 6. One representative set of images (hepatic vein) for visual evaluation. De

GenLik method [9], (d) the median filter, (e) the Gamma filter, (f) the Weiner fi

images, but for the results reported in the present paper,we used 25 different diagnostic ultrasound images ofhepatic vein, portal vein, gall bladder, kidney and liver.

As a quantitative performance measure, we have usedsignal to mean square error (SMSE) [22] and signal tonoise ratio (SNR) [1]. The SMSE is defined as

SMSE ¼

PKi¼1X2

i

ðPK

i¼1ðXi � XiÞ2Þ

(22)

noised image by (a) the proposed method, (b) log wavelet denoising, (c)

lter, (g) the Lee filter, and (h) the Kuan filter.

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Fig. 6. (Continued)

A. Khare et al. / Signal Processing 90 (2010) 428–439 437

where X is original image, X is denoised image, and K isnumber of pixels in image

For denoising in real wavelet domain we used db6wavelet, as its performance is better, as reported in [7] andfor complex wavelet domain shrinkage, we apply thesymmetric Daubechies wavelet 14 (SDW14) as it isoptimal for denoising in complex domain [14]. For thesimulation experiments on 2D ultrasonic medical images,we selected a clean 256�256 liver grayscale ultrasoundimage as a reference image. For making noisy images, wehave artificially added speckle noise of different amountby using our MATLAB program. These noisy images havebeen denoised by different methods. Fig. 4 showsperformance of the proposed method over others in termsof SMSE values. Fig. 4(a) shows the performance of theproposed method in comparison with the weiner filter,the frost filter, the gamma filter and the median filter,while Fig. 4(b) shows the comparison of the proposedmethod with the Kuan filter, the Lee filter and the log-transformed wavelet based denoising method. From thesefigures it is clear that the performance of the proposedtechnique is far better in all noise regions, or in otherwords, the proposed shrinkage in complex waveletdomain, with edge preservation, results in removing morenoise compared to the conventional spatial domainmethod as well as real wavelet based speckle removalmethod in logarithmic transform domain.

Table 1 presents the performance of the proposedmethod in term of signal to noise ratio (SNR) for sixdifferent simulated noisy images. High value of SNRindicates better noise removal. It is evident from the tablethat the proposed method is more successful in specklenoise suppression than others. The visualization of theperformance of the proposed method on one representa-tive simulated image is shown in Fig. 5.

Processing time required by the proposed method iscomparatively less than other methods. It is presented inTable 2. We have taken one clean ultrasound image andrun the MATLAB programs of different methods, for 100random simulated speckle noisy images, on Pentium IVmachine with 2.8 GHz processing speed and 2 GB RAM.Finally the average run time of each method is noted.From Table 2, it can be observed that the computationtime required by the proposed method is less than thetime required by conventional methods—the Frost, theGamma, the Kuan and the Lee filters. Computation time ofthe median filter is slightly lower than the proposedmethod, but its performance is poorer than the proposedmethod. The proposed method is faster than the real-wavelet based despeckling method (log wavelet specklefilter) due to use of complex wavelet transform. Themethods like GenLik and TLS-denoise are model basedmethods. Although the denoising performance of TLSdenoise and GenLik are close to the proposed method, as

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Fig. 7. Application of the proposed method on one real diagnostic ultrasound images: (a) original image and (b) denoised image.

A. Khare et al. / Signal Processing 90 (2010) 428–439438

can be observed from Table 1, but the processing timerequired by the GenLik and TLS denoise is much higherthan the processing time required for the proposedmethod, as presented in Table 2.

The visual evaluation varies from expert to expert andis subject to observer visibility. For the visual evaluation ofthe proposed method a total of 25 medical ultrasoundimages captured by LOGIC 9 (GE medical system,Milwaukee, WI, USA) have been taken. These images weredenoised by the proposed method as well as othermethods. These images were evaluated by three differentexperts—before and after denoising. The experts wereasked to assign a score in the one-to-five scale corre-sponding to low and high subjective visual perceptioncriteria. The experts evaluated the images according to theamount of clinical information and the amount of specklepresent in the images.

One representative set of images for visual evaluationis shown in Fig. 6. The experts suggested that the resultsproduced by the proposed method, the log-waveletdenoising [8], GenLik method [9], the median filtering,the Gamma MAP filter and the Weiner filter are approxi-mately same from clinical point of view. The performanceof local statistics filters, like Lee and Kuan filters, are poorthan the proposed method from clinical point of view. Butfrom the speckle removal capability point of view, theproposed method performs better than other methods. Itcan also be observed from the numeric values of SMSE andSNR given in Fig. 4 and Table 1, respectively. Theapplication of the proposed method on another represen-tative diagnostic ultrasound image is given in Fig. 7.

5. Conclusions

In this paper, we proposed a novel Daubechies complexwavelet transform based despeckling method for ultra-sound images. The use of complex wavelet transform indenoising adds the advantage of approximate shift-invariance over discrete wavelet transform. The artifacts,such as the ringing effect present in discrete wavelettransform are absent here. The shrinkage factor in theproposed method is in its simplest form, which makes the

method simple to apply. The adaptability of the proposedmethod is due to parameters involved in the computationof shrinkage factor, which is directly computed for givenultrasound image. The shrinkage was applied only on themagnitude of wavelet coefficients, thus preserving thephase and in turn the shape of the object in image. One ofthe problems in denoising is the selection of trade-offbetween the smoothness and the edge-preservation. Inthe real wavelet transform, the edge information requiresto be computed separately while in complex waveletdomain it is present inherently in the imaginary compo-nent of complex wavelet coefficients. Incorporating thisproperty in our technique we have designed the edge-sensitive despeckling method.

Acknowledgments

This work was supported in part by the center fordistributed sensor network, and the Systems biologyinfrastructure establishment grant provided by GIST,South Korea. The authors would like to thank anonymousreviewers for their constructive comments, which im-proved quality of the paper.

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