Wavelet-Domain Medical Image Denoising Using ... - Semantic Scholar

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IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 56, NO. 12, DECEMBER 2009

Wavelet-Domain Medical Image Denoising Using Bivariate Laplacian Mixture Model Hossein Rabbani, Member, IEEE, Reza Nezafat, and Saeed Gazor∗ , Senior Member, IEEE

Abstract—In this paper, we proposed novel noise reduction algorithms that can be used to enhance image quality in various medical imaging modalities such as magnetic resonance and multidetector computed tomography. The noisy captured 3-D data are first transformed by discrete complex wavelet transform. Using a nonlinear function, we model the data as the sum of the clean data plus additive Gaussian or Rayleigh noise. We use a mixture of bivariate Laplacian probability density functions for the clean data in the transformed domain. The MAP and minimum mean-squared error (MMSE) estimators allow us to efficiently reduce the noise. The employed prior distribution is mixture and bivariate, and thus accurately characterizes the heavy-tail distribution of clean images and exploits the interscale properties of wavelets coefficients. In addition, we estimate the parameters of the model using local information; as a result, the proposed denoising algorithms are spatially adaptive, i.e., the intrascale dependency of wavelets is also well exploited in the enhancement process. The proposed approach results in significant noise reduction while the introduced distortions are not noticeable as a result of accurate statistical modeling. The obtained shrinkage functions have closed form, are simple in implementation, and efficiently enhances data. Our experiments on CT images show that among our derived shrinkage functions usually BiLapGausMAP produces images with higher peak SNR. However, BiLapGausMMSE is preferred especially for CT images, which have high SNRs. Furthermore, BiLapRayMAP yields better noise reduction performance for low SNR MR datasets such as high-resolution whole heart imaging while BiLapGauMAP results in better performance in MR data with higher intrinsic SNR such as functional cine data. Index Terms—Image and multidimensional biosignal processing, image filtering and restoration, signal and image processing.

I. INTRODUCTION ONINVASIVE cardiac imaging using MRI or multidetector computed tomography (MDCT) has emerged as a rapidly progressing field, largely due to technical advances in imaging hardware and new imaging methodologies. Availability of MR systems with higher number of receiver channels has enabled acquisition of MR images with higher spatial or temporal resolution that could significantly increase the diagnostic value of MRI. Availability of MDCT with higher number of

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Manuscript received June 5, 2008; revised December 24, 2008. First published August 18, 2009; current version published November 20, 2009. Asterisk indicates corresponding author. H. Rabbani is with the Department of Biomedical Engineering, Isfahan University of Medical Sciences, Isfahan, Iran (e-mail: [email protected]). R. Nezafat is with the Department of Medicine, Harvard Medical School, Boston, MA 02115 USA (e-mail: [email protected]). ∗ S. Gazor is with the Department of Electrical and Computer Engineering, Queen’s University, Kingston, ON K7L 3N6, Canada (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TBME.2009.2028876

slices such as 256 has enabled whole heart coverage within a single heart beat with isotropic spatial resolution of 0.4 mm [1]. Although both imaging modalities could acquire images with submillimeter spatial resolution, the improved spatial resolution comes with a penalty of increased noise, especially in MRI. The MR noise is independent of the measured MR signal from the imaging object; however, it is a function of the resistance of the receiver coil and volume of the imaging object. The SNR measurement also depends on various imaging parameters involved in the image acquisition process. In general, the SNR in MRI can be expressed by (1) SNR ∝ ∆x ∆y ∆z Tacq where ∆x , ∆y , and ∆z are spatial resolution in x-, y-, and z-direction, respectively, and Tacq is the total acquisition time. The SNR is inversely proportional to spatial resolution, i.e., higher spatial resolution yields lower SNR. Thus, the spatial resolution of MR is limited by available SNR and the image acquisition time. Repeated acquisition and averaging could improve SNR in MRI; however, multiple averaging is limited by several factors such as limited breath-hold duration. An alternative approach to averaging is to use more SNR-efficient image acquisition techniques. For example, steady-state free precession MRI has superior SNR property compared to spoiled gradient echo sequences [2]. In this study, we will investigate a complimentary approach in which the image noise is reduced in an additional postprocessing step after data acquisition. The prior probabilistic knowledge of the noise-free image as well and noise characteristic of each imaging modality in wavelet domain will be exploited to enhance SNR [3], [4]. This proposed technique can be used for various imaging modalities including MR and MDCT. These algorithms are Bayesian estimators [5]–[20]. Using minimum mean-squared error (MMSE) estimator or averaged MAP, the solution requires a prior knowledge about the distribution of noise and clean wavelet coefficients. To derive this spatially adaptive wavelet-based denoising method, we employ a mixture of bivariate Laplace probability density function (pdfs) as a prior of clean data in the discrete complex wavelet transform (DCWT) domain. We estimate the parameters of this model using local image data [21]. 1) Prior distribution of MR and CT noise: The MR noise is usually characterized by a Rician distribution [22]–[29], [35]. For a low and high SNR image, the Rician noise pdf is well approximated by a Rayleigh pdf and a Gaussian pdf, respectively [24]. In addition, the noise involved in the complex MRI components is modeled by an additive white Gaussian noise process [29], [36]–[38]. Therefore, for the complex MR images, we suggest enhancing the

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RABBANI et al.: WAVELET-DOMAIN MEDICAL IMAGE DENOISING USING BIVARIATE LAPLACIAN MIXTURE MODEL

complex MR components assuming that noise is additive, Gaussian, and circularly symmetrical instead of denoising the magnitude components that have Rician distribution. This allows better image enhancement since the phase carries information about the noise process. In MDCT, the measurements are based on counting of the events in a discrete process; in such a case, a reasonable statistical model is that the measurements have independent Poisson distributions. Thus, the MDCT noise is fundamentally different from the MR noise and can be accurately characterized using a Poisson distribution [30], [31]. However, the Gaussian distribution is usually used (often implicit) for MDCT noise as an accurate continuous approximation for Poisson distribution [30]–[34]. Furthermore, the MDCT noise is nonstationary and is highly correlated between neighboring pixels (as each ray intersects many pixels). The algorithms proposed in this paper efficiently extract and exploit the nonstationary and correlation information of such a noise process. 2) Distribution of wavelet coefficients of natural images: Three important statistical properties of wavelets for natural images are compression, clustering, and persistence. The compression property is well reported in the literature (e.g., [9] and [10]) as image wavelet coefficients have heavy-tailed marginal pdf. Various pdfs (for example, see [7], [13], and [16]) are proposed to model this property. The clustering property is that if a particular wavelet coefficient is large/small, then spatial adjacent coefficients are also very likely large/small [7]. Usually spatially adaptive algorithms characterize this property by estimating the parameters of the pdfs using local neighboring data (see [8], [13], [16], and references therein). In order to take into account compression and clustering properties, several shrinkage functions based on local mixture distributions are proposed for image denoising [16], [18]. Although these methods perform well, quantitatively and qualitatively, they are unable to exploit the persistence property of wavelets [12], [13]. Indeed, large/small values of wavelet coefficients tend to propagate across scales and bivariate pdfs, which exploit the dependency between coefficients, have shown better statistical characterization of wavelets [12]–[15], [17], [19], [20]. In this paper, we use a mixture of bivariate Laplacian pdfs for noise-free data with parameters estimated locally that allow us to exploit all three properties simultaneously. In Section II, we find several shrinkage functions using averaged MAP (AMAP) and MMSE estimators in presence of Gaussian, Poisson, and Rayleigh noise. In [39], it has been shown that the joint histogram of wavelet coefficients fits the elliptically symmetric bivariate pdfs better than the circularly symmetric bivariate pdfs. Following [39], we also improve our proposed local bivariate mixture pdf by allowing different variances for each component of the bivariate pdf. In Section III, we apply the proposed shrinkage functions to the artificial 3-D noisy images and also to 3-D medical images including coronary MDCT data, cardiac MRI, and ultrasound data. Finally, we conclude the paper in Section IV.

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II. DENOISING USING A MIXTURE OF BIVARIATE LAPLACE PDFS We solve the noise reduction problem for Gaussian or Rayleigh noise. However, we use the solutions for removal of the Rician, speckle, and Poisson noise in the following sections as well. Let us denote the observed data by xk = sk + εk , where sk are the clean data and εk is the additive noise at the spatial position k. Applying DCWT to the observed data, we have yk = wk + nk , where yk , wk , and nk represent the noisy observation, clean data, and noise in DCWT domain, respectively. To include the interscale dependency (i.e., persistence property of wavelets), we denote ¯k + n ¯k y¯k = w

(2)

where β¯k = (βk , βp,k ) and βp,k is the parent of βk at the spatial position k. The MMSE estimate of clean data w ¯k is the posterior mean of clean data given noisy observation y¯k , and is given by +∞ +∞ ¯k |¯ yk ] = w k = E[w

¯k p(w ¯k )p(¯ yk |w ¯k )dwk dwp,k −∞ −∞ w +∞ +∞ ¯k )p(¯ yk |w ¯k )dwk dwp,k −∞ −∞ p(w

(3) ¯k ) represents the conditional (bivariate) pdf y¯k where p(¯ yk |w given w ¯k , and p(w ¯k ) is the pdf of the clean wavelet signal w ¯k . In yk |w ¯k ) results in (3), substituting various pdfs for p(w ¯k ) and p(¯ different estimators. In this paper, we use the following Gaussian and Rayleigh pdfs for the noise distribution:  2  1 2 exp − |¯y k −w¯2 k | , Gaussian 2π σ n 2σ n ¯k ) = |y −w p(¯ yk |w 2 y −w | |¯ y − w ¯ | k k p , k p , k  , Rayleigh. exp − k σ 2 k 2σ n4 n (4) We also assume the following local bivariate Laplacian mixture pdf for the clean image: p(w ¯k ) =

I

ai,k pi (w ¯k )

i=1

=

√ p I a exp − 2((|w |/σ w ) + (|w |/σ )) i,k k p,k i,k i,k i=1

w σp 2σi,k i,k

(5)

where ai,k ≥ 0, Ii=1 ai,k = 1, and I is the number of mixture components. In this case, we have E[wk wp,k ] = 0, p(wk ) = √ w √

I w i=1 (ai,k / 2σi,k )exp(− 2(|wk |/σi,k )), and p(wk , wp,k ) = p(wk )p(wp,k ). Thus, wk and wp,k are uncorrelated but not independent. As a result, this distribution can describe the interscale dependency of uncorrelated coefficients in adjacent scales. Since, the marginal pdfs are also mixture pdfs, this distribution can also describe the heavy-tailed nature of wavelets. For each p w , σi,k }Ii=1 coefficient w ¯k , we estimate the parameters {ai,k , σi,k in (5) from the noisy local data in square windows Nk centered at k, using an expectation–maximization (EM) algorithm. The resulting local pdf also describes the intrascale dependency between spatially adjacent coefficients.

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TABLE I EXPRESSIONS OF gi (¯ y k ), w i , k , AND wi , k FOR GAUSSIAN AND RAYLEIGH NOISE

proposed formulas in [14] and [16] are for univariate (not bivariate) mixture pdfs. Parameters of the mixture model p w , σi,k }Ii=1 can be estimated using a modified EM algo{ai,k , σi,k rithm. For each coefficient w ¯k , a local squared window Nk with size |Nk | centered at pixel k is proposed and the parameters are iteratively estimated as follows. In the E-step, the responsibility factors ri,k are updated at each iteration by yk ) ai,k gi (¯ ri,k ← I , yk ) i=1 ai,k gi (¯

for i = 1, . . . , I.

(12)

p w In the M-step, the parameters {ai,k , σi,k , σi,k }Ii=1 are updated. The parameters {ai,k }Ii=1 are updated by 1 ai,k ← ri,j . (13) |Nk | j ∈N k

p w {σi,k , σi,k }Ii=1

Then, the parameters are approximately estimated as

2 j ∈N k ri,j yj w 2

σi,k ← max − σn , 0 (14) j ∈N k ri,j p σi,k

Substituting (5) into (3), we can write +∞ +∞

I ¯k )p(¯ yk |w ¯k )dwk dwp,k i=1 ai,k −∞ −∞ wk pi (w w k = I . +∞ +∞ ¯k )p(¯ yk |w ¯k )dwk dwp,k i=1 ai,k −∞ −∞ pi (w (6) To abbreviate (6), we denote +∞ wk pi (wk )p(yk |wk )dwk (7) w i,k = −∞ +∞ −∞ pi (wk )p(yk |wk )dwk +∞ +∞ gi (¯ yk ) = pi (w ¯k )p(¯ yk |w ¯k )dwk dwp,k (8) −∞

and get

−∞

(15)

p w One may assume σi,k = σi,k = σi,k (see [14]). In this case, the previous equation becomes

¯j2 j ∈N k ri,j y

− σn2 , 0 . (16) σi,k ← max 2 j ∈N k ri,j

Another parameter that must be estimated to implement our denoising algorithms is σn . This parameter is estimated using a robust median estimator for the finest scale of noisy wavelet coefficients [5] median(|yk |) , 0.6745 yk ∈ subband HH in first and second scales ς=C

I w k =

2 r y i,j p,j j ∈N

k ← max − σn2 , 0 . j ∈N k ri,j

i=1

I

ai,k w i,k gi (¯ yk )

i=1

ai,k gi (¯ yk )

.

(9)

Note that w i,k is the MMSE estimator of w ¯k given the ith component of (5). Alternatively, for each component, we can use the MAP estimator and obtain the following AMAP estimator:

I i,k gi (¯ yk ) i=1 ai,k w (10) w k =

I yk ) i=1 ai,k gi (¯ where w i,k is the MAP estimate of w ¯k for pi (w ¯k ) and is a feasible solution of ¯k ) ∂ log pi (w ¯k ) ∂ log p(¯ yk |w + = 0. (11) ∂wk ∂wk After some simplifications, we found the expressions for w i,k , yk ), as summarized in Table I. In this way, we w i,k , and gi (¯ find four new shrinkage functions namely BiLapGausMAP, BiLapGausMMSE, BiLapRayMAP, and BiLapRayMMSE. The

(17)

where C is a smoothing factor. A larger smoothing factor may cause blurring of some edges and a smaller smoothing factor may prevent the removal of noise. Thus, according to the preference of the clinician, the value of C allows a tradeoff; in this paper, it is set to 1.5. The proposed algorithms are summarized in Table II. III. EXPERIMENTAL RESULTS In this section, we evaluate our proposed algorithms for wavelet-based denoising. For this purpose, the six-tap filter proposed in [40] is employed for each dimension of DCWT (some useful codes and explanations can be found at http://taco.poly.edu/WaveletSoftware/). Our experiments show that using more than two mixture components is timeconsuming and does not improve the denoising results. So, in the following, we set I to 2 in the corresponding equations.


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TABLE II SUMMARY OF PROPOSED DENOISING ALGORITHMS

Fig. 1. Convergence of the EM algorithm for different initialization of parameters. Initialization I: a 01 , k = 0.5, [σ 1w, ,k0 , σ 2w, ,k0 , σ 1p ,,k0 , σ 2p ,,k0 ] =

w . Initialization II: as given by (18). Initialization III: [1, 0.8, 0.5, 0.3]σ 1,k

w . a 01 , k = 0.25, [σ 1w, ,k0 , σ 2w, ,k0 , σ 1p ,,k0 , σ 2p ,,k0 ] = [2, 1.8, 1.5, 1.3]σ 1,k

only within less than five to ten iterations. We also observe that the initialization in (18) makes the EM algorithm considerably faster than random initialization. B. Denoising of Images Corrupted by Computer Generated Noise In Section III-A, the initialization of the algorithm is investigated. Then, in Section III-B, we apply our wavelet-domain methods and some recent algorithms to several noisy data where the noise is generated and added by software in order to quantitatively evaluate and compare these methods. This allows us to compare the denoised image with the clean image in terms of peak SNR (PSNR) defined by PSNR = 20 log(255/MSE), where MSE is the mean squared of the error between estimated data and clean data. In Section III-D and III-C, we compare these methods for denoising of medical 3-D cardiac MRI and coronary CT images, respectively. Finally, in Section III-E, we discuss more about the proposed method in this paper for enhancing other imaging modalities. The simulation results are summarized at the end of this section. A. Initialization To implement the local EM algorithm, we choose the window size in each dimension to 7 and initialize as follows: a01,k = 0.75 w ,0 σ1,k = σ 1,k p,0 σ1,k = 0.5σ 1,k

where

a02,k = 0.25

(18a)

w ,0 σ2,k = 0.8σ 1,k

(18b)

p,0 σ2,k = 0.3σ 1,k

(18c)

σ yj2 − σn2 , 0). 1,k = max((1/|Nk |) ∆

j ∈N k

The convergence of the EM algorithm have been studied and proven for a wide class of models (e.g., see [41] and references therein). However, the convergence analysis of the EM algorithm for many other distributions is an active research area. Fig. 1 shows three parameters for a typical subband and three different initializations. All other parameters exhibit similar learning curves. It is important to initialize the set of parameters of each component away from that of other components. In particular, the initialization in (18) allows the convergence of the variance of different components to take different values. Interestingly, our experiments reveal that all parameters converge

Here, we evaluate the proposed algorithms with some recent algorithms. In particular, we added a zero-mean white computergenerated Gaussian noise to the following 3-D images (referred to as datasets) and compare the denoised images with noisy and clean images: DS1: 8-bit, 160 × 224 × 48 “Salesman” video sequence (160 × 224 pixels/frame and 10 frames/s) (here, we treat the video as an example of a 3-D image); DS2: 8-bit, 512 × 512 × 40 abdominal MDCT image sequence from pelvis (this dataset is produced from a noisy dataset using BiLapGausMMSE); DS3: 8-bit, 512 × 512 × 20 cardiac MR image sequence (this dataset is produced from noisy data using BiLapRayMap). We applied the proposed algorithms to the noisy data in four different domains: 2-D and 3-D ordinary discrete wavelet transform (DWT) domains, and 2-D and 3-D DCWT domains. Our experiments show that the PSNR in 3-D DCWT is higher than other domains. For example, the PSNRs for a denoised 8-bit 128 × 128 × 24 MRI with σn = 30 shown in Fig. 2 using BiLapGausMAP in 2-D DWT, 3-D DWT, 2-D DCWT, and 3-D DCWT domains are 27.30, 28.74, 28.30, and 29.96, respectively. This result suggests that about 1.1 dB gain is obtained in average if 3-D transformation is employed instead of 2-D transformation. In addition, about 1.5 dB is obtained in average if an appropriate DCWT is employed instead of ordinary DWT. In this figure, the volume rendering of the proposed data is also illustrated for better qualitative evaluation. This figure also reveals that if we employ 3-D DCWT, some imaging details (which are important for medical image) are better preserved in the denoised image, while fewer visual artifacts are produced. Unfortunately, 3-D transforms require higher computational cost and simultaneous processing of multiple frames, which is not appreciated in some applications. Table III illustrates the PSNRs of 3-D images denoised employing hard and soft thresholding, BiLapGausMAP, and BiLapGausMMSE in DCWT domain for different noise levels σn = 10, 20, 30 and the aforementioned datasets. We understand from this table that usually BiLapGausMAP outperforms

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Fig. 3. Cardiac MRI (8-bit 288 × 288 × 30) denoising in 2-D DCWT domain. (a) One slice from an MRI data (used as clean reference). (b) Same slice from the denoised image using hard thresholding. (c) Same slice from the denoised image using BiLapGausMAP. (d) Same slice from noisy image σ n = 40. (e) Same slice from the denoised image using soft thresholding. f) Same slice from the denoised image using BiLapGausMMSE. TABLE IV PSNR COMPARISON OF AN 8-BIT 128 × 128 × 24 MRI SEQUENCE CORRUPTED WITH RICIAN NOISE IN VARIOUS LEVELS AND ENHANCED USING HARD AND SOFT THRESHOLDING, BILAPGAUSMAP, BILAPGAUSMMSE, AND BILAPRAYMAP IN 3-D DCWT DOMAIN

Fig. 2. Two first six images show a comparison between one slice of denoised 8-bit 128 × 128 × 24 MR data using BiLapGausMAP in different wavelet domains. (a) One slice of an MR. (b) Zero-mean white Gaussian noise is added (σ n = 30). (c) Denoised image in 2-D DWT. (d) Denoised image in 3-D DWT. (e) Denoised image in 2-D DCWT. (f) Denoised image in 3-D DCWT. The next six images are the corresponding volume rendering of the first six images. TABLE III PSNR COMPARISON OF 3-D IMAGES DENOISED EMPLOYING HARD AND SOFT THRESHOLDING, BILAPGAUSMAP, AND BILAPGAUSMMSE

among these methods. Fig. 3 shows one slice of an 8-bit 288 × 288 × 30 cardiac MRI (used as the clean reference), the denoised images using hard and soft thresholding, BiLapGausMAP, and BiLapGausMMSE, and the noisy image that is produced by adding noise (σn = 40) to the clean reference. From this figure, we observe that hard thresholding cannot remove the noise as well as the proposed methods. In addition, the soft thresholding results in blurred images. The proposed data in this paper belong to different modalities. In addition, where the noise is added to the data, it is generated according

to the noise characteristics of the corresponding modality. For example, we corrupted the central section of first 20 slices of DS2 with Poisson noise. In this case, the PSNR of hard thresholding, soft thresholding, Wiener filtering, BiLapGausMAP, and BiLapGausMMSE are, respectively, 31.16, 31.17, 31.15, 31.96, and 32.06. It is clear that our algorithms outperform the others. Furthermore, we corrupt the proposed ground truth data in Fig. 2 by Rician noise at various levels in order to provide a more realistic evaluation of the algorithms. The results are summarized in Table IV. As we expect, BiLapGausMAP and BiLapGausMMSE are superior for low-noise levels and BiLapRayMap outperforms the others for high-noise levels. The effect of C used in (17) is also observed. In this particular example, the value of C = 1.5 is more appropriate. C. Enhancement of CT Images In this section, we apply the proposed methods to a coronary artery MDCT dataset. As discussed previously, the Poisson distribution can characterize the signal count measurement


Fig. 4. Coronary MDCT image denoising. (a) One slice of a coronary MDCT. (b) Corresponding slice denoised using Wiener filter. (c) Corresponding slice denoised using BiLapGausMAP. (d) Corresponding slice denoised using BiLapGausMMSE.

on MDCT. Unlike Gaussian noise, separating signal from Poisson noise is not an easy task, since the parameter of the Poisson pdf is a function of the underlying signal intensity. To overcome this complication, the nonlinear-invertible function s = 2 I + (3/8) known as Anscombe’s transformation [34] is applied to the input image I. The distribution of the output image s is assumed to be Gaussian with the clean component as the mean. The input image I is processed successively as follows [21]: the image s is transformed into the wavelet domain, is enhanced and transformed back by the inverse wavelet transform, and finally, the inverse of the aforementioned transform I = (1/4) s2 − (3/8) is applied. Fig. 4 shows one slice from the input coronary MDCT and the corresponding slices from denoised images obtained by employing BiLapGausMMSE, BiLapGausMAP, and Wiener filtering and using C = 1.5 in (17) (the impact of C is studied in [18]). We observe that the proposed methods reduce the noise more efficiently than the Wiener filter, while the edges are preserved better. Fig. 5 also shows that BiLapGausMMSE outperforms BiLapGausMAP, as BiLapGausMMSE produces fewer visual artifacts especially for high SNRs.

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Fig. 5. Comparison between denoised abdominal MDCT using BiLapGausMAP and BiLapGausMMSE. (a) Comparison of a small section of one slice of denoised abdominal MDCT using BiLapGausMAP (left image) and BiLapGausMMSE (right image). (b) Comparison of volume rendering of a section of denoised abdominal MDCT using BiLapGausMAP (left image) and BiLapGausMMSE (right image).

D. Enhancement of MR Images Here, we use following two methods to reduce the noise in cardiac MRI. M1: We suggest this method if we only have the magnitude of complex MR data. In this case, we propose to apply BiLapGausMAP (or BiLapGausMMSE) for high-SNR segments of images and BiLapRayMAP for low-SNR image segments. This method is motivated by the fact that the Rician noise pdf involved in the magnitude information is well approximated by Rayleigh and Gaussian pdfs, respectively, for a low or high SNR [24]. Fig. 6 shows one slice of an input cardiac MRI data and the corresponding output slice of this method. Since this image has

Fig. 6. One slice of a cardiac MRI sequence (top image) and the corresponding denoised slice using method M1 (bottom image).

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Fig. 7. Cardiac MRI denoising employing BiLapGausMAP using method M2. (a) Real part of one slice of input cardiac MRI. (b) Real part of the corresponding slice of the denoised image. (c) Imaginary part of the corresponding slice of the input. (d) Imaginary part of the denoised slice. (e) Magnitude of the corresponding input slice. (f) Magnitude slice produced using (b) and (d).

low SNR, BiLapRayMAP outperforms BiLapGausMAP or BiLapGausMMSE. This method clearly preserves the important details of images and does not produce artifacts. M2: This method is applicable if the complex MR data are available. Since the complex noise of involved MR images is a Gaussian process, we suggest to apply BiLapGausMAP or BiLapGausMMSE to real and imaginary part of data separately. Then, the denoised data are obtained by calculating the magnitude of the enhanced real and imaginary parts. This method outperforms method M1 as the complex input image contains more information about the noise characteristic allowing more efficient enhancement. Fig. 7 show the real, imaginary, and magnitude of one slice of an input cardiac MR image and the corresponding slices employing BiLapGausMAP. Fig. 8 shows similar results employing BiLapGausMMSE. The mean SNR (MSNR) and contrast-to-noise ratio (CNR) are two quality measurements in MRI, which are defined as [29] MSNRROI =

µROI σ

CNR = |MSNRROI1 − MSNRROI2 | =

(19) |µR O I 1 − µR O I 2 | σ (20)

where µROI is the mean signal value computed for a small region of interest (ROI). The desired ROI can be a homogeneous

Fig. 8. Cardiac MRI denoising employing BiLapGausMMSE using method M2. (a) Real part of one slice of an input cardiac MRI sequence. (b) Real part of the corresponding slice of the denoised image. (c) Imaginary part of the corresponding slice of the input. (d) Imaginary part of the denoised slice. (e) Magnitude of the corresponding input slice. (f) Magnitude slice produced using (b) and (d).

Fig. 9. One slice from a sample input cardiac MR images and proposed ROIs for computation of MSNR and CNR reported in Tables V and VI.

area of tissue with high signal intensity (such as myocardium, ventricular, and arterial blood in cardiac MR images). The noise standard deviation (std) σ is computed from a large region outside the object (such as chest wall in cardiac MR images), which represents the background noise. The CNR represents the contrast of MSNR between two ROIs. Fig. 9 shows two slices of two input cardiac images from which we selected ROI samples. We enhanced the left image employing BiLapRayMAP and local soft thresholding. The resulting values for both MSNR and CNR for these ROIs indicate that BiLapRayMAP significantly outperforms the local soft thresholding. To show that this


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TABLE V STATISTICAL COMPARISON OF INPUT MRI USING SEVERAL ROIS SUCH AS SHOWN IN FIG. 9 WITH THE ENHANCED IMAGES OBTAINED FROM BILAPRAYMAP AND SOFT THRESHOLDING (METHOD M1) IN TERMS OF AVERAGED IMPROVEMENT OF MSNR (IMR-MSNR) AND IMPROVEMENT OF CNR (IMR-CNR) VALUES

TABLE VI STATISTICAL COMPARISON OF INPUT MRI USING SEVERAL ROIS SUCH AS SHOWN IN FIG. 9 WITH THE ENHANCED IMAGES OBTAINED FROM BILAPGAUSMAP AND BILAPGAUSMMSE (METHOD M2) IN TERMS OF IMPROVEMENT OF MSNR (IMR-MSNR) AND IMPROVEMENT OF CNR (IMR-CNR) VALUES

Fig. 10. (a) Section of one slice of input noisy MRI. (b) Enhanced corresponding section using method M2 and BiLapGausMAP.

result is statistically significant, we define the improvement of MSNR as follows for a given ROI: IMR-MSNR =

MSNR(enhanced output image) . MSNR(noisy input image)

(21)

The improvement of CNR (IMR-CNR) is defined similarly as before. We selected two set of ROIs such as those shown in Fig. 9 for left ventricle blood and myocardium; we calculated values of IMR-MSNR and IMR-CNR for these sets. The average values of IMR-MSNR and IMR-CNR over these two sets of ROIs (24 ROIs with average size 10 × 10) in Table V clearly indicate that BiLapRayMAP is significantly preferred. This table also gives the P-values obtained by applying the paired Student’s t-test on the IMR-MSNR (and IMR-CNR) data. These results indicate that it is highly likely (with probability of order of 1−0.0021) that BiLapRayMAP outperforms soft thresholding. We also applied BiLapGausMAP and BiLapGausMMSE (method M2) on multiple images, selected several random ROIs from each image, and calculated IMR-MSNR and IMR-CNR values for each of these ROIs. Table VI summarizes the results of these experiments providing the mean and the std of the data. Since the std is smaller than the difference between the means, it could be concluded that the enhancement using BiLapGausMMSE is preferred. For evaluation of the proposed algorithms in more details, we consider a whole heart coronary exam with low SNR with spatial resolution of 1.4 × 1.4 × 1.8. A section of one slice of this image is shown in Fig. 10. We applied method M2 using BiLapGausMAP to this 8-bit 448 × 448 × 150 input sequence. Comparing the input and the enhanced data, we note the improvement visually. In addition, we noted the following. 1) We computed the CNR between left ventricular blood to myocardium for the input as well as for the enhanced data. The CNR is improved by a ratio of 2.39 =

Fig. 11. (a) Comparison of profile values for the input and denoised data using method M2 and BiLapGausMAP. (Left image) A zoomed section of input image from Fig. 10(a). (Right image) Corresponding section of enhanced image from Fig. 10(b). (b) Comparison between profile values of indicated arrow in Fig. 11(a).

CNR(enhanced image)/CNR(input) as a result of the image enhancement. 2) The MSNR of myocardium is improved by a ratio of 2.4379 = MSNR(enhanced image)/MSNR(input). 3) The MSNR of arterial blood was improved by a ratio of 2.4393 = MSNR(enhanced image)/MSNR(input). The profile of indicated arrow shown in Fig. 11(a) is compared for the input image and the enhanced image in Fig. 11(b). From the profile of data, we observe that the accuracy along the line between the local maxima and the local minima is increased. Therefore, we conclude that the proposed method improves the contrast of the image while attenuates the noise.

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Fig. 13. (a) Noise-free image. (b) Noisy image (σ = 0.3). (c) Denoised image using the method proposed in [43]. (d) Denoised image with Pizurica’s method [11]. Denoised image with BiLapGausMAP using (e) 15 × 15, (f) 3 × 3, (g) 7 × 7, and (h) 11 × 11 window sizes.

TABLE VII IMPACT OF WINDOW SIZE AND SCALES ON THE PERFORMANCE BILAPGAUSMAP COMPARED WITH THE PROPOSED METHOD IN [43] USING A WINDOW SIZE OF 7 × 7

Fig. 12. Denoising of 8-bit 128 × 256 × 64 3-D ultrasound data from wrist using BiLapGausMMSE and BiLapGausMAP for window sizes 7 and 15. (a) Comparison of a small section of one slice of data (top-left image) and denoised data using BiLapGausMMSE with window size 7 (bottom-right image) and BiLapGausMAP with window size 7 (bottom-right image) and window size 15 (top-right image). (b) Comparison of volume rendering of data (top-left image) and denoised data using BiLapGausMMSE with window size 7 (bottomright image) and BiLapGausMAP with window size 7 (bottom-right image) and window size 15 (top-right image).

E. Other Modalities and More Discussions The proposed denoising method in this paper can be modified for enhancement of produced data (3-D as well as 2-D) from other modalities. For example, we here consider ultrasound images that are mainly contaminated by the speckle noise. Applying a logarithm operator, this multiplicative noise can be approximately converted to the additive Gaussian noise model [15], [42]. Thus, we can apply BiLapGausMAP and BiLapGausMMSE to the logarithm of data in the sparse domain, and apply the exponential function to the enhanced data. Fig. 12 shows a comparison between an 8-bit 128 × 256 × 64 3-D ultrasound data from wrist and denoised data using BiLapGausMAP and BiLapGausMMSE for three scales of 2-D DCWT. In this figure, we can also see the effect of two window sizes 7 × 7 and 15 × 15. It is clear that using a bigger window results in a smoother output image. The effect of window size depends on the type of input image and the noise level. To understand this effect, we evaluate the algorithms in the presence of computer-generated noise. Similar to the proposed method in [43], we evaluate our method in four window sizes 3, 7, 11, and 15. To this end, we add a white Gaussian noise to the image in Fig. 13 in the log domain and employ our denoising algorithms BiLapGausMAP and BiLapGausMMSE and enhance the noisy image for various window sizes. The results over five runs are concluded in Table VII. As we can observe in Fig. 13, small window size produces more visual artifacts and in most cases using window size

7 × 7 (that is used in the most simulations in this paper) leads to better denoising results. However, it is clear from images with high level of noise that bigger windows are more appropriate. In this table, we can also see the effect of number of scales for wavelets. We can conclude that usually using less than three scales does not result in acceptable performance in terms of PSNR and produces high level of artifact in the final image. On the other hand, using more than three scales for low level of noise slightly improves the results by considerable incensement in computational cost. So, in most cases in this paper, we have set the number of scales to 3 except for high-level noise, which has been set to 5. Note that the proposed method in [43] is based on using univariate Laplacian mixture models for image denoising. In Table VII, we also compare the proposed approach in this paper that is based on bivariate mixture distribution (to further exploit the persistence property of wavelets) with the previous approach [43] to show the actual advantage of using bivariate Laplacian mixture prior (over univariate Laplacian mixture


Fig. 14. Comparison between error images of a section of denoised data using the method proposed in [43] and BiLapGausMAP. (From left to right) Noisefree section, noisy section, error section obtained using the method proposed in [43], and BiLapGausMAP.

TABLE VIII SNR IMPROVEMENT IN DECIBELS OBTAINED BY INCORPORATING COMPRESSION, CLUSTERING, AND PERSISTENT PROPERTIES FOR WAVELET IMAGE MODELING TO EVALUATE SEPARATE IMPACT OF PROPERTIES

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various (additive white Gaussian) noise levels 10, 25, 40, and 60 for various test images, including 512 × 512 “Lena” image that contains smooth regions, textures, and edges, 512 × 512 “Boat” image consisting smooth regions and edges, and 256 × 256 “House” image that mainly contains homogeneous textures. In this table, we observe that the improvements obtained from different properties depend on the image type and the noise level. Simulations show that using the persistent and the clustering properties together improves the results especially for crowded and noisy images. For instance, for 512 × 512 “Crowd” image corrupted with additive white Gaussian noise with stds 30, 40, 60, and 100, BiLapGaussMAP, respectively, results in 0.4, 0.6, 0.7, and 0.9 dB higher PSNRs than the method proposed in [43]. This means a model with more details, which employs both the persistent and the clustering properties, can better match to the images with more details. In contrast, a simpler model, which only employs the clustering property, is often sufficient for simpler images (e.g., uncrowded images).

F. Summary of Experimental Results

prior). To highlight the advantages of our method with respect to the method proposed in [43], we compare the error image of a section of denoised images (in Fig. 13) with these methods in Fig. 14. It is evident that the proposed method in this paper removes the noise with less visual artifacts. Although in Table VII, the results of the denoising using BiLapGaussMAP are comparable with those of the method in [43] (which is obtained using local univariate Laplacian mixture model), however, Fig. 14 reveals that BiLapGaussMAP results in significantly less artifacts. In particular, for medical image processing, even marginal reduction in artifacts is important. This result indirectly implies that in statistical modeling of wavelets, the persistent property plays a more important role than the other two properties, i.e., compression and clustering properties. Clearly, the intrascale dependency (clustering property) is the most important dependency among the main statistical dependencies of wavelets. To confirm this claim, in Table VIII, the SNR improvement results for each property separately are reported, where: 1) for the compression property, we used “univariate nonlocal Laplacian mixture” model instead of Laplacian pdf; 2) for the clustering property, we used “local Laplacian pdf”; and 3) for the persistent property, we used “bivariate circular symmetric Laplacian pdf.” In this table, we use

Our experimental results reveal the following. 1) The 3-D DCWT (compared to 2-D DWT, 2-D DCWT, and 3-D DWT) is preferred for enhancement of 3-D images in terms of PSNR. In some applications, the 3-D DCWT may be computationally expensive or may not be implemented in real time. In such a case, our results suggest 2-D DCWT as the second alternative option for the implementation of the enhancement algorithms for better visual performance and higher PSNR. 2) For coronary MDCT noise reduction, our experiments suggest to apply either BiLapGausMAP or BiLapGausMMSE after Anscombe’s transformation. In most cases, BiLapGausMAP produces slightly better PSNR; however, better visual quality especially for high-SNR CT images is obtained using BiLapGausMMSE. 3) If complex MR data are available for enhancement, we can apply either BiLapGausMMSE or BiLapGausMAP to the real and imaginary parts of data separately. However, similar to CT data, BiLapGausMAP is preferred for lowSNR data, and usually, BiLapGausMMSE is preferred for high-SNR data. 4) If we only have the magnitude of complex MR data, BiLapRay is preferred for low-SNR data whereas for highSNR MR images, BiLapGausMAP is preferred. 5) For cardiac MRI denoising, our methods improve both MSNR and CNR. This means that by reducing the noise, the contrast is also improved, which has impact on the medical image interpretation. 6) By comparing our methods with hard and soft thresholding in the wavelet domain, we conclude that hard thresholding is not able to remove the noise as well as our proposed methods. Note that in contrast to the proposed methods, the soft thresholding produces a blurred image. 7) The proposed method could also be modified and applied to data from other imaging modalities.

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IV. CONCLUSION In this paper, new denoising methods are proposed for 3-D medical images. We modeled the noise-free data in the DCWT domain with a local bivariate Laplacian mixture distribution. This mixture distribution is able to simultaneously characterize the most important statistical properties of wavelets including heavy-tailed nature, and interscale and intrascale dependencies. We use both MAP and MMSE estimators in the presence of Gaussian, Rayleigh, and Poisson noise and obtain several shrinkage functions namely BiLapGausMAP, BiLapGausMMSE, and BiLapRayMAP. In our extensive simulations, the proposed methods illustrate impressive noise reduction ability for images such as coronary MDCT and cardiac MRI in terms of the MSNR and CNR. Although BiLapGausMAP produces images with higher PSNR, however, the visual quality using BiLapGausMMSE is often preferred especially for highSNR CT images. For low-SNR MR images, BiLapRayMAP has better visual performance and BiLapGausMAP is preferred for high-SNR MR images.

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Hossein Rabbani (M’09) was born in Iran in 1978. He received the B.Sc. degree (with the highest honors) in electrical engineering (communications) from Isfahan University of Technology, Isfahan, Iran, in 2000, and the M.Sc. and Ph.D. degrees in bioelectrical engineering from Amirkabir University of Technology (Tehran Polytechnic), Tehran, Iran, in 2002 and 2008, respectively. From January 2007 to July 2007, he was with the Department of Electrical and Computer Engineering, Queen’s University, Kingston, ON, Canada, as a Visiting Researcher. He is currently with the Department of Biomedical Engineering, Isfahan University of Medical Sciences, Isfahan. His current research interests include multidimensional signal processing, multiresolution transforms, probability models of sparse domain’s coefficients, and image restoration.

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Reza Nezafat received B.Sc. degree in electrical engineering in 1998 and the Ph.D. degree in biomedical engineering from Johns Hopkins School of Medicine, Baltimore, MD, in 2006. He currently serves as the Director of Translational Imaging Program in the Cardiovascular Division at the Beth Israel Deaconess Medical Center (BIDMC), Boston, MA, and an Assistant Professor of Medicine at Harvard Medical School, Boston. His research interests include cardiovascular magnetic resonance imaging and image-guided therapy.

Saeed Gazor (S’94–M’95–SM’98) received the B.Sc. degree (summa cum laude, with highest honors) in electronics engineering and the M.Sc. degree (summa cum laude, with highest honors) in communication systems engineering from Isfahan University of Technology, Isfahan, Iran, in 1987 and 1989, respectively, and the Ph.D. degree in signal and ´ image processing from Département Signal, Ecole Nationale Supérieure des Télécommunications, Telecom Paris, Paris, France, in 1994. From 1995 to 1998, he was an Assistant Professor in the Department of Electrical and Computer Engineering, Isfahan University of Technology. From January 1999 to July 1999, he was a Research Associate at the University of Toronto. Since 1999, he has been on the faculty at Queen’s University, Kingston, ON, Canada, where he is currently a Professor in the Department of Electrical and Computer Engineering and is also cross-appointed in the Department of Mathematics and Statistics. His current research interests include statistical and adaptive signal processing, image processing, cognitive radio and signal processing, array signal processing, speech processing, detection and estimation theory, multiple-input–multiple-output communication systems, collaborative networks, channel modeling, and information theory. Prof. Gazor received a number of awards including the Provincial Premier’s Research Excellence Award, the Canadian Foundation of Innovation Award, and the Ontario Innovation Trust Award. He is a member of the Professional Engineers Ontario. He is currently an Associate Editor for the IEEE SIGNAL PROCESSING LETTERS.