Speckled Images Restoration Filter based onWeighted Multiplicative Regularization Approach

∗† Meriem HACINI, † Khalifa DJEMAL and ∗ Fella HACHOUF∗ Laboratoire d’ Automatique et de Robotique, Departement d’ electronique,

Universite Mentouri de ConstantineRoute d’Ain el Bey, 25000 Constantine, Algerie

e-mail: meriem.hacini@yahoo.fr, fhachouf@wissal.dz† IBISC Laboratory, University Evry val Essonnes, 40 Pelvoux Street,

91080 EVRY Courcouronnes Cedex, Francee-mail: djemal@iup.univ-evry.fr

Abstract—In this paper, a novel image denoising algorithmbased on a multiplicative regularization technique is proposed.The regularization employs a weighted total variation (TV )that is included as a multiplicative constraint. In this way, theappropriate regularization parameter will be controlled by theoptimization process itself. The new proposed method not onlyovercomes the disadvantage of generating artificial edges but alsohas the advantages of denoising and edges preservation of TVmodel. Experimental results show that the new method is effectivein removing speckle noise and image details are kept well.

Index Terms—Image Denoising, Multiplicative Regularization,Weighted Total Variation, Speckle Suppression.

I. INTRODUCTION

Image denoising is an important preprocessing task in imageprocessing domains like segmentation, feature extraction,texture analysis etc. Denoising refers to suppressing thenoise while retaining the edges and other important detailsstructures as much as possible.Mathematically, the restoration process is an ill-posedproblem. Therefore, regularization methods are usuallyemployed to obtain a solution satisfying some criteria. In thelast decade, total variation TV regularization functional hasbecome of increasing interest. Indeed, the TV model have theability of smoothing homogeneous regions of objects withoutdegrading the edges, which are important attributes of theimage. Edges location is unknown and they must be preservedwhen the object is reconstructed and regularized.

Some edge-preserving regularization methods have beenreported in the literature. However, most of them relyon information from a local neighborhood to determineedge existence: A penalty assigned to each pixel or cliqueof pixels depends only on pixel values within a smallfixed neighborhood [1-3]. The use of TV in a constrainedminimization problem was first proposed by Rudin et al[4]. Since then, the TV minimization functional has beensuccessfully used in image restoration [5-7], smoothing[8], and applied successfully for inverse problems [9],[10]. Recently, some new TV-based image restorationmethods performed through a recursive application have been

proposed. In [11 ] the authors suggest to tackle the difficultyof multiplicative noise by converting it into an additive one.Then they adopt the Split Bregman approach to solve theoptimization problem resulting from a total variation criterion.The ATV-filter [12] has recently addressed the restorationof speckled images using total variation minimization. Theauthor proposed a speckle reduction method based on Loupasspeckle noise model [13] and adopted them to the Rudin etal total variation method [4]. The minimization is obtainedusing the Euler-Lagrange equations. In this method, theregularization factor is automatically and globally computedin iterative way to avoid smoothing edges region.

Regularization based on additive total variation has a verypositive effect on the quality of the restoration for noisyimages.In this paper, we propose a new speckle denoising method,based on Loupas et al noise model [13] and a multiplicativeregularization technique. The new algorithm is inspired fromV an den Berg et al [14,15] works. The proposed methodworks in iterative way. It locally minimizes a total variationfactor in a multiplicative regularization problem through anoverlapping sliding speckled images and it preserves finedetails.

The performance of the filter has been tested on bothsynthetic and real ultrasound images. Results are comparedto those obtained by the additive total variation filter (ATV-filter)[12].

In the next section, the Weighted Total Variation Multiplica-tive regularization algorithm WMTVR-filter, is presented. Sec-tion III, covers discussion and presents comparative simulationresults. Conclusion and future work are drawn in Section IV.

II. PROPOSED SPECKLE REDUCTION ALGORITHM

In this section a new speckle reduction technique, -TheWeighted Total Variation based on Multiplicative Regulariza-tion filter ( WMTVR) - , is proposed to suppress the speckle

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from Ultrasound images. Several Multiplicative Regularizationtechniques have been developed in a variety of domains, butthere are no specific ones for speckle suppression. For thispurpose, in this section we propose a filter based on theminimization of a TV factor in a multiplicative regularizationproblem. The Loupas et al noise model [13] is used.

A. Statements and Problem Formulation

Throughout this paper, Ω will denote the image domain in<2 on which the image intensity function f is defined; p =(x, y) denotes the location in Ω | p |= (x2 + y2)

12 denotes

the Euclidean norm, and ‖ . ‖Ω denotes the L2 norm in thedomain Ω.Let the observed intensity function d(x, y) denotes the pixelvalue of a noisy image Ω for x, y ∈ Ω. Let f(x, y) denotesthe desired clean image. Loupas et al in [12] have modeled anUltrasound image with speckle noise as:

d(x, y) = f(x, y) +√f(x, y).b(x, y) (1)

Where b(x, y) is a multiplicative noise considered as Gaussiancentered with standard deviation σb, and

√f(x, y).b(x, y)

term models the ultrasound noisy images.

The image restoration problem is then to obtain a reasonableapproximation of the original data f(x, y) from the given noisyimage d(x, y).Problem (1) is ill-posed. Consequently, matrix systems aris-ing upon the discretization of this equation are highly ill-conditioned. To stabilize the problem in (1) an iterative regu-larization problem through an overlapping sliding window iscarried out.In this work, we propose to minimize a weighted total variationfunction:

JTV (f,Ω) =1

NΩ.

∫Ω

W 2(f)‖ ∇f ‖2dxdy (2)

Where, the Weighted function is defined respectively as fol-lows:

W (f) =1√

| ∇f |2 +δ2(3)

Considering the model of Loupas et al in (1), the total variationfunction JTV is minimized taking into account a multiplicativeregularization functional in the following way:

JR(d, f,Ω) = ηR

∫Ω

(d− f√f

)2dxdy (4)

withηR =

1∫Ωd2dxdy

(5)

Assuming that d represents the observed data and f anoptimal solution of our problem, the cost functional to beminimized is:

J(d, f,Ω) = JTV (f,Ω).JR(d, f,Ω) (6)

Let us assume the image fapprox represents an approximativesolution of the problem. A variation of the value for the imageintensity can be written as a linear combination of the fapproxand a generic ζ.

f = fapprox + α.ζ (7)

Where, α is a real parameter.In this way, the minimization problem takes the form:

f = argminα

[J(d, (f + αζ),Ω)] (8)

B. Numerical Resolutions

In numerical implementation, the total variation is initial-ized considering JTV = 1. In equation (3), the term δ is a realparameter. It was introduced to decrease the total variation as afunction of the number of iterations by increasing δ. To achievethis behavior, δ is taken as:

δ =1

| d− f |(9)

1) Algorithm: The WMTVR-filter starts the updating of thesystem solution as follows.Define d to be the observed image and fn the approximatesolution of the minimization problem. Then, we suppose thatfn−1 is known, the sequence fn is updated as:

f0 = dfn = fn−1 + αn.ζn n = 1, ...

(10)

where αn is a real parameter, and ζn is the conjugate gradientupdate image, i.e. In a first instance, as the update direction forthe image we take the gradient of J with respect to changesin the image f at (n-1)th iteration i.e:

ζ0 = g0 = −ηM .d2−f2

0

f20

ζn = gn + γnPR.ζn−1 n = 1, ...

(11)

where γnPR is the Polak − Ribiere conjugate gradient direc-tions , i.e,

γnPR =Real〈gn, gn − gn−1〉Ω

‖gn−1‖2Ω(12)

and gn is the gradient of the cost functional Jn with respectto f evaluated at the (n− 1)th iteration,

III. EXPERIMENTS AND DISCUSSION

In this section, experiments are conducted on simulated andreal ultrasound images with the proposed WMTVR-filter. Theresults are discussed and compared to the output of the ATV-filter [12].

A. Implementation

In terms of processing time, the implementation of theWMTVR-filter depends on the image size. Running tests arecarried out on a i3 CPU @2.13GHZ.The best filters are usuallyobtained for small windows (typical values of a=3, 5 or 7),which ensure a reasonable processing time.

B. Results and evaluations

Tests have been carried on several noisy images (see Fig. 1and Fig. 3). The performance of the proposed WMTVR-filterand ATV-filter [12] are comparatively evaluated through thesubjective appearance of the filtered images. The ATV-filterwas applied using global processing while the proposedWMTVR-filter is based on a local one. Quantitative resultsare based on the estimation of the profile’ s image along agiven line, (see Fig. 2 and Fig. 4)

1) Test on Simulated Speckle images: In the first test, asynthetic image is obtained using spatially correlated specklenoise. The speckle can be simulated by low-pass filtering anda complex Gaussian random field, taking the magnitude ofthe filtered output [16]. The low- pass filtering is performedby averaging the complex values in a 3x3 sliding window.A short correlation term was found sufficient [17] to modelrealistic images.

Fig. 1(c) and Fig. 1(e) show respectively the resultingspeckled synthetic images processed with the ATV and theproposed WMTVR-filters.Both filter results satisfy well filtering; smoothinghomogeneous areas while preserving important structuressuch as edges (see Fig. 1(d) and (f)).Considering the edge map obtained after running our filters(see Fig. 1(e) and (f)), it appears that the main edges are notaltered while the number of noisy edges are reduced.

For this purpose, in order to examine the edges and pointpreserving capability of the processed filters, Fig. 2 plotsthe profile of the image along a given line. The image(seeFig. 2(a)) has three geometrical structures. So, it shouldhave three gray levels. It is noticed that the data processedby the two filters yield continuous edges and keep the highcontrast in (see Fig. 2(c) and Fig. 2(d)). We conclude thatthe multiplicative regularization method undertaken by theproposed WMTVR-filter could reduces speckles effectively inhomogeneous regions but preserves the edges well.

2) Tests on Real Ultrasound Images: The performance ofthe processed filters are compared using the real obstetricalultrasound image Fig. 3 (a). This image allows performanceevaluations under various conditions since it contains edgeswith high contrast and uniform areas. The results of theprocessed filters are given in (Fig. 3(c)-(e)). The two filtersprovide good noise attenuation and smoothing but it is alsoclear that the edges are not well conserved in the ATV filteroutput. Fig. 3 (d).

(a) The input image (b) Edge map of (a) computed as themagnitude of the Sobel gradient

(c) The ATV-filter (d) Edge map of (c) computed as themagnitude of the Sobel gradient

(e) The proposed WMTVR-filter (f) Edge map of (e) computed as themagnitude of the Sobel gradient

Fig. 1. Test on Synthetic images

To more closely evaluate the output from each filter, con-sider a single scan line running through the image. Fig 4 showsthe profile of one given line of the original and processedultrasound images.Once more, The edge maps and the scan line show thealgorithms efficiency. The ATV-filter performs well in noiseattenuation but produces sharpened contours (see Fig. 3d).The WMTVR-filter lights the best balance between noiseattenuation and detail preserving.

IV. CONCLUSION

In this paper we have proposed a new algorithm for denois-ing process. The proposed approach is based on a weightedtotal variation factor which is included as a multiplicativeconstraint. The algorithm is an iterative scheme and uses the

(a) Image with the white line for which theprofile was calculated

(b) The Profile of the input image

(c) The Profile of the filtered image using the ATV-filter

(d) The Profile of the filtered image using the proposed WMTVR-filter

Fig. 2. Profile of the Synthetic images

(a) The input image (b) Edge map of (a) computed as themagnitude of the Sobel gradient

(c) The ATV-filter (d) Edge map of (c) computed as themagnitude of the Sobel gradient

(e) The proposed WMTVR-filter (f) Edge map of (e) computed as themagnitude of the Sobel gradient

Fig. 3. Test on Real Ultrasound Images

Conjugate Gradient technique. Results show that this algorithmachieve image quality superior to the existing methods withlittle increase in computational time.

REFERENCES

[1] Ja. I.Koo and Song. B. Park , Speckle Reduction with edge preservationin medical ultrasonic images using a homogeneous region growing meanfilter (HRGMF), Rasonic Imaging. 13, pp. 211-237, 1991.

[2] S. Alenius, U. Ruotstainen and J. Astola , Using Local Median as theLocation of the Prior Distribution in Iterative Emission TomographyImage Reconstruction, IEEE Transactions on Nuclear Science 45 (6),pp. 3097-3104, 1998.

(a) Image with the white line for which theprofile was calculated

(b) The Profile of the input image

(c) The Profile of the filtered image using the ATV-filter

(d) The Profile of the filtered image using the proposed WMTVR-filter

Fig. 4. Profile of the Synthetic images

[3] A. Thakur, R. S. Anand, Speckle Reduction in Ultrasound Medical Imagesusing Adaptive Filter based on Second Order Statistics, Journal ofMedical Engineering and Technology 31 (4), pp. 263-279,2007.

[4] L. Rudin, S. Osher and C. Fatemi , Nonlinear Total Variation based onNoise Removal Algorithm, Physica 30, pp.259-268, 1992.

[5] C. R.Vogel, M. E. Oman, Fast, Robust Total Variation- based Reconstruc-tion of Noisy, Blurred Images, IEEE Transaction on Image Processing.7(6), pp. 813-824, 1998.

[6] Da. Hong. Xu, Run. Xu and Sheng. Wang, Accurate Numerical Methodfor Total Variation-Based Image Deblurring, International Conferenceon Machine Learning and Cybernetics , 5, pp. 2765-2769, 2008.

[7] A. Abubaker, P. M. Van Den Berg, A Multiplicative Regularization Ap-proach for Deblurring Problems, Image Processing, IEEE Transactionson,13(11), pp 1524 - 1532. November 2004.

[8] F. Catte, P. L. Lions, J. M. Morel and T. Coll, Image Selective Smoothingand Edge Detection by Nonlinear Diffusion, SIAM J. Numer. Anal, 29, pp. 182-193, 1992.

[9] A. Abubaker, P. M. Van Den Berg, Total Variation as a MultiplicativeConstraint for Solving Inverse Problems, IEEE Transaction ImageProcessing, 10(9), pp. 1384- 1392, 2001.

[10] P. Rodriguez, and B. Wohlberg , Efficient Minimization method fora Generalized Total Variation Functional, Image Processing, IEEETransactions on, 18(2), pp. 322- 332, 2009.

[11] J. M. Bioucas-Das, and M. A. T. Figueiredos,Total variation restorationof speckle images using a split-Bregman algorithm, Image Processing(ICIP), 2009 16th IEEE International Conference on, pp. 3717-3720,2009.

[12] K. Djemal, Speckle Reduction in Ultrasound images by Minimization ofTotal Variation,, ICIP 2005. IEEE International Conference on, 3, pp.357- 360, 2005.

[13] T. Loupas, W. N. McDicken, P. L.Allan, An Adaptive Weighted MedianFilter for Speckle Suppression in Medical Ultrasonic Images, IEEETransactions on Circuits and Systems, 36(1), pp. 129-135, 1989.

[14] Van Den Berg, P. M. and A. Abubakar, Contrast source inversionmethod: State of art, Progress In Electromagnetics Research, Vol. 34,189-218, 2001.

[15] P. M. Van Den Berg,A. Abubaker, and J. T. Fokkema , MultiplicativeRegularization for Contrast Profile Inversion, RADIO SCIENCE, 38(2),2003.

[16] A. Pizurica, W. Philips, , I. Lemahieu, and M. Acheroy,A versatilewavelet domain noise filtering technique for medical imaging, Medicalimaging, IEEE Transactions on, 22(3), 323- 331, 2003.

[17] A. Achim, A. Bezerianos and P. Tsakalides,, Novel Bayesian Multi scaleMethod for Speckle Removal in Medical Ultrasound Images, IEEETrans.Med. Imag., 20(8), pp. 772-7783, 2001.