Curvelet-based Bayesian Estimator for Speckle Suppression

Curvelet-based Bayesian Estimator for Speckle SuppressionCurvelet-basedBayesianEstimatorforSpeckleSuppressionin sonographyImagingAbstract.Ultrasound figures argon inherently affected by situation dissonance, and therefore the reduction of this tone is a crucial pre-processing step for their successful interpretation. Bayesian adhesion is a powerful signal estimation technique used for speckle noise removal in images. In the Bayesian-based despeckling schemes, the choice of suitable statistical models and the development of a shrinkage character for estimation of the noise- liberate signal argon the major concerns. In this paper, a novel curvelet-based Bayesian estimation scheme for despeckling of sonography images is developed. The curvelet co in force(p)s of the multiplicative adulteration model of the noisy ultrasound image are additively decomposed into noise-free and signal-dependent noise components. The Cauchy and two-sided exponential disseminations are assumed to b e statistical models for the noise-free and signal-dependent noise components of the observed curvelet coefficients, respectively, and an efficient low-complexity realization of the Bayesian estimator is proposed. The experimental results demonstrate the validity of the proposed despeckling scheme in providing a significant suppression of the speckle noise and simultaneously preserving the image details.KeywordsUltrasound imaging, curvelet convert, speckle noise, Bayesian estimation, statistical modeling.IntroductionUltrasound imaging is important for medical diagnosis and has the advantages of cost effectiveness, port-ability, acceptability and safety 1. However, ultrasound images are of relatively poor quality due to its contamination by the speckle noise, which considerably degrades the image quality and pass alongs to a negative impact on the diagnostic task. Thus, reducing speckle noise while preserving anatomic information is necessary to better delineate the regions of int erest in the ultrasound images.In the work of speckle suppression in ultrasound images, many spatial-based techniques that employ either single-scale or multi-scale filtering have been developed in the literature 2-4. Earlydeveloped single-scale spatial filtering 2 are limited in their capability for significantly reducing the speckle noise. More promising spatial single-scale techniques such as those using bilateral filtering 4 and nonlocal filtering 3 have been recently proposed.This work was back up in part by the Natural Sciences and Engineering Research Council (NSERC) of Canada and in part by the Regroupement Strategique en Microelectronique du Quebec (ReSMiQ).These techniques depend on the surface of the filter window, and hence, for a capable speckle suppression, they require large computational time. Alternatively, multi-scale spatial techniques 5, based on partial differential equations, have been investigated in the literature. These techniques are iterative and can c onstruct smooth images with preserved edges. However, important structural details are unfortunately degraded during the iteration process. As an appropriate alternative to spatial-based speckle suppression in ultrasound images, many other despeckling techniques based on different transform dobrinys, such as the ones of wavelet, contourlet, and curvelet, have been recently proposed in the literature 6-8. Wavelet transform has a good reputation as a tool for noise reduction but has the drawback of poor directionality, which makes its usage limited in many applications. utilise contourlet transform provides an improved noise reduction surgical procedure due to its property of fiexible directional decomposability. However, curvelet transform offers a higher directional sensitivity than that of contourlet transform and is to a greater extent efficient in representing the curve-like details in images.For the development of despeckling techniques based on transform domains, thresholding 7 has been presented as a technique to build linear estimators of the noise-free signal coefficients. However, the main drawback of this thresholding technique is in the difficulty of determining a suitable threshold value. To circumvent this problem, non-linear estimators 6 have been statistically developed based on Bayesian estimation formalism. For the development of an efficient Bayesian-based despeckling scheme, the choice of a suitable probability distribution to model the transform domain coefficients is a major concern. Also, while investigating a suitable statistical model, the complexity of the Bayesian estimation process should be taken into consideration. Consequently, special attention should be paid to the realization complexity of the Bayesian estimator that results from employing the selected probabilistic model in one of the Bayesian frameworks.In this paper, to achieve a satisfactory writ of execution for despeckling of ultrasound images at a displace computatio nal effort, a new curvelet-based Bayesian scheme is proposed. The multiplicative degradation model representing an observed ultrasound image is decomposed into an additive model consisting of noise-free and signal-dependent noise components. double-faced exponential distribution is used as a prior statistical model for the curvelet coefficients of the signal-dependant noise. This model, along with the Cauchy distribution is used to develop a low-complexity Bayesian estimator. The performance of the proposed Bayesian despeckling scheme is evaluated on some(prenominal) syntheticallyspeckled and real ultrasound images, and the results are compared to that of some other existing despeckling schemes.Modeling of Curvelet CoefficientsThe multiplicative degradation model of a speckle-corrupted ultrasound image g(i,j) in the spatial domain is given byg(i,j) = v(i,j)s(i,j)(1)where v(i,j) and s(i,j) denote the noise-free image and the speckle noise, respectively. This model of the noisy obse rvation of v(i,j) can be additively decomposed as a noise-free signal component and a signal-dependant noiseg(i,j) = v(i,j) + (s(i,j) 1)v(i,j)= v(i,j) + u(i,j)(2)where (s(i,j) 1)v(i,j) represents the signal-dependant noise. Taking the curvelet transform of (2) at level l, we haveyl,d(i,j) = xl,d(i,j) + nl,d(i,j)(3)where yl,d(i,j), xl,d(i,j) and nl,d(i,j) denote, respectively, the (i,j)th curvelet coefficient of the observed image, the corresponding noise free image and the corresponding additive signal-dependant noise at direction d= 1,2,3,,D. In order to simplify the notation, we will henceforth drop both the superscripts drop off dand the index (i,j).In this work, in order to reduce the noise inherited in ultrasound images, we propose exploiting the statistical distinctives of the curvelet coefficients in (3) to derive an efficient Bayesian estimator. Thus, one needs to provide a prior probabilistic model for the curvelet coefficients of xandn. It has been shown that the distrib ution of the curvelet coefficients of noise-free images can be suitably modeled by the Cauchy distribution 9. The zero-mean Cauchy distribution is given bypx(x) = (/)(x2 + 2)(4)where is the dispersion parameter. The noisy observation is used to estimate the Cauchy distribution parameter by minimizing the function 2yyt(t) (t) edt(5)where y(t) is the empirical characteristic function corresponding to the curvelet coefficients yof22the noisy observation, y(t) = x(t)E(t), x(t) = et, and E(t) = e(/2)tdeviation Eobtained aswith the standardE=MAD(y(i,j))0.6745(6)In (6), MAD denotes the median value absolute deviation operation. Now, in order to mold theBayesian estimator, a prior statistical assumption for the curvelet coefficients of nof the signal dependant noise should also be assumed. From experimental observation, it is noticed that the tailpart of the empirical distribution of ndecays at a low rate. Hence, in this paper, we propose to usea two-sided exponential (TSE) distribution g iven by1pn(n) =en/2(7)where is a positive real constant referred to as the scale parameter. The method of log-cummulants(MoLC) is adoptive to estimate the parameter , and thus the estimated is obtained by using thefollowing expression = exp1N1 N2log(y(i,j))+ (8)N1N2i=1j=1where is the Euler-Mascheroni constant and N1 and N2 define the size N1 -N2 of the curveletsubband considered.Bayesian Estimator collectable to the fact that each of the Cauchy and TSE distributions has only one parameter, one could expect the process of Bayesian estimation to be of lower complexity. The value of the Bayes estimates x of the noise-free curvelet coefficients xof a subband under the quadratic loss function, which minimizes the mean square error (MSE), are given by the shrinkage functionx(y) =pxy(xy)xdxP pn(yx)px(x)xdx=P p(yx)p(x)(9)It is noted that a closed-form expression forx(y) given by the above equation does not exist.Thus, in order to obtain the Bayesian estimates for the noise-free curvelet co efficients, the two integrations associated with (9) are numerically performed for each curvelet coefficient. Since this procedure requires an excessive computational effort, the bayseian estimates are obtained by replacing the associated integrals in (9) with infinite series as suggested in 10. Accordingly, the Bayesian shrinkage function can be expressed asey/f (y) + ey/ f(y) + x(y) =(10)ey/f21(y) + 2 + ey/f22(y) + 2wheref11(y) = f12(y) = sin(/) Im E( y+ j)Si(/) + 12y+jcos(/) Re E1(+ Ci(/) ,(11)f(y) = f1y+ j(y) = sin(/) Re E()+ Ci(/)212211y+jcos(/) Im E1(Si(/) + 2,(12)1 = lim f12y(y) = sin(/) Si(/) + cos(/)Ci(/), and(13)= lim f11(y) =sin(/)Ci(/) +cos(/)Si(/) +(14)222yIn the equations above, j= 1, Imand Reare the imaginary and real parts, respectively, of a complex argument, and E1(), Si() and Ci() are, respectively, the exponential, sine and cosineintegral functions obtained as in 10. observational ResultsExtensive experimentations are carried out in order to study the performan ce of the proposed despeckling scheme. The results are compared with those of other existing despeckling schemes that use improved-Lee filtering 2, accommodative-wavelet shrinkage 6, and contourlet thresholding 7. Performance evaluation of the assorted despeckling schemes is conducted on synthetically-speckled and real ultrasound images. In the implementation of the proposed speckling scheme, the 5-level decomposition of the curvelet transform is applied. From the experimental observation, applying a higher level of decomposition of the curvelet transform does not lead to any improvement in the despeckling performance. Since the curvelet transform is a shift-variant transform, the cycle spinning 11 is performed on the observed noisy image to avoid any possible pseudo-Gibbs artifacts in the neighborhood of discontinuities. In the proposed despeckling scheme, only the detail curvelet coefficients are despeckled using the Bayesian shrinkage function in (10).The peak signal-to-noise r atio (PSNR) is used as a quantitative measure to assess the despeckling performance of the various schemes when applied on synthetically-speckled images. Table I gives the PSNR value obtained when applying the various schemes on two synthetically-speckled images of size 512-512, namely, Lenaand Boat. It is obviously seen from this table that, in all cases, the proposed despeckling scheme provides higher values of PSNR compared to that provided by the other schemes. To have a better insight on the despeckling performance of the various schemes, the results in Table 1 are visualized in aim 1. It is obvious from this figure that the superiority of the proposed scheme over the other schemes is more unvarnished when a higher level of speckle noise is introduced to the test images. In order to study the performances of the various despeckling schemes on real ultrasound images, two images obtained from 12 and shown in Figure 2 are used. Since the noise-free images cannot be made availabl e, one can only give a subjective evaluation of the performance of the various despeckling schemes. From Figure 2, it is clearly seen that the schemes in 2 and 6 provide despeckled images that suffer from the presence of visually noticeable speckle noise. On the other hand, the scheme in 7 severely over-smooth the noisy images thus providing despeckled images in which some of the texture details are lost. However, the proposed despeckling scheme results in images with not only a significant reduction in the speckle noise but also a good preservation of the textures of the original images.Table 1 The PSNR values obtained when applying the various despeckling schemes on LenaandBoatimages contaminated by speckle noise at different levels.342326307Proposed2826242220180.10.20.30.40.50.71Standard deviation of noise(a)322306287Proposed2624222018160.10.20.30.40.50.71Standard deviation of noise(b)Fig. 1 Quantitative comparison between the various despeckling schemes in terms of PSNR values(a ) Lenaimage (b) Boatimage.ConclusionIn this paper, a new curvelet-based scheme for suppressing the speckle noise in ultrasound images has been developed in the framework of Bayesian estimation. The observed ultrasound image is first additively decomposed into noise-free and signal-dependant noise components. The Cauchy and twosided exponential distributions have been used as probabilistic models for the curvelet coefficients of the noise-free and signal-dependant noise components, respectively, of the ultrasound image. The proposed probabilistic models of the curvelet coefficients of an observed ultrasound image has been employed to formulate a Bayesian shrinkage function in order to obtain the estimates of the noise-free curvelet coefficients. A low-complexity realization of this shrinkage function has been employed. Experiments have been carried out on both synthetically-speckled and real ultrasound images in order to demonstrate the performance of the proposed despeckling scheme. In comparison with some other existing despeckling schemes, the results have shown that the proposed scheme provides higher PSNR values and gives well-despeckled images with better diagnostic details.(b)(c)(d)(e)(f)(g)(h)(i)(j)Fig. 2 Qualitative comparison between the various despeckling schemes. (a)(b) Noisy ultrasound images. Despeckled images obtained by applying the schemes in (c)(g) 2 ,(d)(h) 6 ,(e)(i) 7 and (f)(j) the proposed scheme.ReferencesDhawan, A.P. Medical image analysis. Volume 31. hind end Wiley Sons (2011)Loupas, T., McDicken, W., Allan, P. An adaptive weighted median filter for speckle suppression in medical ultrasonic images. IEEE transactions on Circuits and Systems 36(1) (1989) 129-135Coupe, P., Hellier, P., Kervrann, C., Barillot, C. Nonlocal means-based speckle filtering for ultrasound images. IEEE transactions on image processing 18(10) (2009) 2221-2229Sridhar, B., Reddy, K., Prasad, A. An unsupervisory qualitative image enhancement using adaptive morphol ogical bilateral filter for medical images. International Journal of Computer Applications 10(2i) (2014) 1Abd-Elmoniem, K.Z., Youssef, A.B., Kadah, Y.M. real time speckle reduction and coherence enhancement in ultrasound imaging via nonlinear anisotropic diffusion. 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