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Nonlinear filtering based on 3D wavelet transform for MRI denoising
EURASIP Journal on Advances in Signal Processing volume 2012, Article number: 40 (2012)
Abstract
Magnetic resonance (MR) images are normally corrupted by random noise which makes the automatic feature extraction and analysis of clinical data complicated. Therefore, denoising methods have traditionally been applied to improve MR image quality. In this study, we proposed a 3D extension of the wavelet transform (WT)based bilateral filtering for Rician noise removal. Due to delineating capability of wavelet, 3D WT was employed to provide effective representation of the noisy coefficients. Bilateral filtering of the approximation coefficients in a modified neighborhood improved the denoising efficiency and effectively preserved the relevant edge features. Meanwhile, the detailed subbands were processed with an enhanced NeighShrink thresholding algorithm. Validation was performed on both simulated and real clinical data. Using the peak signaltonoise ratio (PSNR) to quantify the amount of noise of the MR images, we have achieved an average PSNR enhancement of 1.32 times with simulated data. The quantitative and the qualitative measures used as the quality metrics demonstrated the ability of the proposed method for noise cancellation.
1. Introduction
Threedimensional magnetic resonance imaging (MRI) has, during the last several decades, benefited from a variety of technological developments resulting in increased resolution, signaltonoise ratio (SNR), and acquisition speed. However, fundamental tradeoffs among resolution, acquisition speed, and SNR combined with scientific, clinical, and financial pressures to obtain more data more quickly, can result in images that exhibit significant artifacts, e.g., noise, partial volume, and intensity nonuniformity. For instance, the need for shorter acquisition times for patients in certain clinical studies often undermines the ability to obtain images having both highresolution and high SNR. Another example concerns diffusiontensor (DT) MRI that has become quite popular over the last decade due to its ability to measure the anisotropic diffusion of water in structured biological tissue. DT MRI differentiates between the anatomical structures of cerebral white matter, which was previously impossible with MRI, in vivo, and noninvasively. The effects of Rician noise on DT MRI, however, are severe because of the inherent nature of the processhigher tissue anisotropy produces progressively lower intensities in diffusionweighted images that, in turn, are more susceptible to Rician noise. The efficacy of higherlevel post processing of MR and DTMR images, e.g., segmentation and registration, that assume specific models on regions of interests, e.g., homogeneous, is sometimes impaired by even moderate noise levels. Hence, it is necessary to remove the noise from MR image.
The removal of noise from noisy data to obtain the unknown signal is often referred to as denoising. Postprocessing filtering techniques with the advantage of not to increase the acquisition time have extensively been used in MRI denoising. Many image denoising methods have been proposed in previous research. The conventional approach [1, 2] was proposed to estimate the Rician noise level and perform signal reconstruction using a maximum likelihood method. Anisotropic diffusion [3–5] reduces image noise by considering a scale space, and it has been adapted to suppress the Rician noise in MR image [6]. Moreover, anisotropic diffusion filter combined with the Wiener filter [7] has been used for MRI denoising, which spatially averages pixels according to their correlation structure. The nonlocal means filter has been applied for feature preserved MRI denoising [8–12]. It builds an estimation of the restored pixel value by weighted averaging over a large portion of the pixels within the image. The weights are based on the similarity computed by comparing the patches instead of single point, and the edges and the details can both be well preserved.
Recently, wavelets have become a popular tool in various applications for data analysis and image processing. Application of wavelets for denoising of MR images has produced a large number of algorithms [13, 14]. Early study [15] was followed by multiscale products thresholding [16], which uses adjacent wavelet subbands to detach the edges from noise. Complex denoising of MR images using wavelets was proposed by Zaroubi and Goelman [17]. The method produces better SNR compared to the magnitude denoising scheme. Wu et al. [18] proposed a waveletbased background noise removal method in MRI. The proposed method can be used jointly with existing denoising methods to improve their effectiveness. Bilateral filtering in wavelet domain has been shown to preserve the edges efficiently [19]. Moreover, wavelet has been used for MRI denoising in combination with Radon transform, which estimates noise variance in different scales [20].
In this study, we proposed a 3D extension of the wavelet transform (WT)based bilateral filtering ideas for Rician noise removal. Due to delineating capability of wavelet, 3D WT was employed to decompose the MR image into the approximation and the detailed subbands. Next, bilateral filtering of the approximate coefficients in a modified 3D neighborhood improved denoising efficiency and effectively preserved relevant edge features. Meanwhile, the detailed subbands were processed with a weighted NeighShrink (WNS) thresholding algorithm. At the end, inverse 3D WT was performed on the selected subbands to obtain final denoised image. In the proposed method, the combined property of 3D WT and the bilateral filter significantly reduces the blurring of image features.
The structure of this article is as follows. First we describe our proposed noise cancellation algorithm (Section 2). Then, we explain our experimental methodology and present the results with both synthetic and real images (Section 3). Finally, Section 4 is devoted to discussion and conclusion.
2. Materials and methods
2.1. Rician noise estimation
One main source of noise in MRI signal is the thermal noise. The signal component of the measurement is present in both real and imaginary channels; each of the two orthogonal channels is affected by white Gaussian noise. An MR image is usually reconstructed by computing the inverse discrete Fourier transform of the raw data. The magnitude image of the reconstructed MRI is used for visual inspection and for automatic computer analysis. Since the magnitude reconstruction is the square root of the sum of two independent Gaussian random variables, the magnitude image data are described by a Rician distribution [21].
The complex MRI data is given as
The noise in the complex raw data is zero mean Gaussian noise and the spatial MR image is the magnitude of the noisy raw data. Therefore, the magnitude of the noisy raw data z is given by
i.e., z is corrupted by Rician noise. Under these conditions, it is advantageous to take the square of the magnitude MR image. Its expectation reads
And we could obtain an unbiased estimator of y by taking
where the maximum function is applied to avoid physically meaningless complex values. Note that this unbiasing procedure relies on a proper estimate of the σ_{ n }parameter. To this end, many methods have previously been reported [22–25]. They are mainly based on the features of the Rayleigh background, for instance,
where μ is the mean value of the background of the squared magnitude of image, these methods are suitable for our method as long as the MR image contains background.
2.2. 3D WT
Wavelets are orthogonal basis functions that delve data into different spatiofrequency components. The delineating capability of wavelet leads to better discrimination between the noise and the signal. In terms of wavelet space decomposition, the separable 3D WT [26] can be expressed by a tensor product by
where ⊕ denotes space direct sum, L^{α} and H^{β}, respectively, represent the high and lowpass directional filters along directions of αaxis, where α ∈ {x,y,z}.
Figure 1 shows a separable 3D decomposition of a volume: after being applied on the rows and on the columns, the analysis filters followed by a 2 to 1 decimation are applied along the third dimension. After the decomposition, eight subvolumes of lower resolution were obtained: the approximation subvolume from resolution '1' named L^{x}L^{y}L^{z} and 7 subvolumes of details. The separable 3D wavelet provided an equal decorrelation of the original volume voxels in the three directions and the waveletbased denoising was achieved by modifying the contents in the octant subbands, followed by wavelet synthesis.
2.3. 3D bilateral filtering
For 2D image denoising, the original 2D bilateral filter [27] computes the similarity of two pixels by comparing the similarity of the two pixels' square neighborhood centered at the pixels. In a similar way, the similarity of two voxels in 3D image is determined by comparing the similarity of their cubic neighborhood centered at the voxels. For a given size, the cubic neighborhood contains more voxels than the square one. Thus, even for a small neighbor size, some voxels similar to the voxel being processed in a certain extent have low weight when using 3D neighbors. But, if 2D square neighbor is used to compare similarity in 3D images, many voxels not very similar to the voxel being processed will involve in the decision of the new value. To address this issue, we employed a new neighbor for computing the similarity of two voxels. The modified neighborhood consisted of one square in a plane and one line in another axis that was normal to the plane. The square neighbor and the line were both centered at the voxel being studied, as shown in Figure 2.
The neighborhood that consists of a square and a line contains part of the 3D structure of the voxel's neighborhood and contains fewer voxels than the cubic neighbor. Thus, more similar voxels could be identified. The 3D bilateral filter with improved neighborhood is as follows:
where g(p, q, m) represents the intensity value of voxel at position (p, q, m) of volume, O(i, j, k) represents the modified neighborhood of voxel at position (i, j, k), f(i, j, k) represents the filtered value, w_{ d } and w_{ r } are spatial and radiometric components of the bilateral filter, respectively, the parameters δ_{ d } and δ_{ r } control the behavior of the weights.
2.4. 3D WNS thresholding
Recent study on wavelet thresholding evolves as block processing, in which the coefficient is most likely to contain signal if its neighborhood also contains signal coefficient. This method is called NeighShrink [28]. Zhou and Cheng [29] have improved it by optimally choosing the NeighShrink parameters based on Stein's unbiased risk estimate (SURE). We extended the Neighshrink algorithm into 3D domain as following:
For the 3D wavelet w_{ ijk } coefficient to be shrunk, consider a cubic neighborhood B_{ ijk } centered at w_{ ijk }, as shown in Figure 3. The size of neighborhood is represented as 3 × 3 × 3. For ${S}_{ijk}^{2}=\sum _{\left(p,q,m\right)\in {B}_{ijk}}{w}_{pqm}^{2}$, the NeighShrink shrinkage formula is given by [29]
where ${\stackrel{\u2322}{\theta}}_{ijk}$ is the estimator of the unknown noiseless coefficient and λ is the threshold. The optimal λ for each of the highfrequency subbands is estimated using SURE.
In the Neighshrink method, all the wavelet coefficients are shrunk to achieve the purpose of denoising. The result of this may be the noise is reduced while the structural details important in medical image are blurred. As shown in a previous study [29], the oversmoothing could be compensated by exploiting the neighborhood statistics over a pixel to be denoised. By taking advantage of the essential feature of wavelet coefficients known as energy clustering within each subband, a 3D WNS method was proposed to preserve the structural information in MRI.
Let ${W}_{pqm}^{{D}_{i}}$ denotes the wavelet coefficient at location (p,q,m) in subband D^{i} which belongs to {L^{x}H^{y}H^{z}, H^{x}L^{y}H^{z}, H^{x}H^{y}L^{z}, H^{x}H^{y}H^{z}}, and K^{LHH}, K^{HLH}, K^{HHL}, K^{HHH} represent different weights of different wavelet subbands, respectively, as shown in Figure 4. Weighting factors were determined according to the 3D Directional Filter Banks (3D DFB) developed by Lu and Do [30]. Then, ${S}_{ijk}^{{D}_{i}}$ will be calculated as follows:
The estimator of the unknown noiseless coefficient ${\stackrel{\u2322}{\theta}}_{ijk}$ is determined according to Equation (12).
2.5. Algorithm summary
The proposed 3D wavelet domain improved bilateral filter with the WNS thresholding (3DWIBF) was summarized as follows (see Figure 5):

(1)
Compute the square of noisy MR image I to obtain its square magnitude I_{sq}.

(2)
Use 3D WT to get the approximation coefficients (L^{x}L^{y}L^{z}, L^{x}L^{y}H^{z}, L^{x}H^{y}L^{z}, L^{x}L^{y}H^{z}) and the detail coefficients (L^{x}H^{y}H^{z}, H^{x}L^{y}H^{z}, H^{x}H^{y}L^{z}, H^{x}H^{y}H^{z}) of I_{sq} (see Section 2.2).

(3)
The bias in approximation coefficients is removed by subtracting $4{\sigma}_{n}^{2}$ [15] (see Section 2.1).

(4)
These unbiased approximation coefficients are passed through the 3D improved bilateral filter (see Section 2.3).

(5)
Denoise the detail coefficients using 3D WNS thresholding technique (see Section 2.4).

(6)
Compute inverse 3D WT of the filtered approximation and the denoised detail coefficients to obtain the estimate of I_{sq} (see Section 2.2).

(7)
The square root of the resultant gives the denoised magnitude MR image.
3. Experiments and results
3.1. Experimental data description
We have carried out experiments with both simulated and real data. To conduct the experiments over synthetic data, three simulated MR images (T1, T2 and PD) with 1 mm^{3} voxel resolution (8bit quantization) from the Brainweb phantom [31] were used. Each image contained 181 × 217 × 181 voxels. To simulate Rician noise, we added zero mean Gaussian noise to the real and imaginary parts of the simulated MR data and afterwards the magnitude image was computed.
To evaluate the proposed approach on real clinical data, three datasets were used. Informed consent was obtained from all volunteers in accordance with our institution's policies regarding human subjects. The first dataset consisted of an MPRAGE T1w volumetric sequence (256 × 240 × 176 voxels with a voxels resolution of 1 mm^{3}) acquired on a Siemens 1.5T Vision scanner. The acquisition parameters were TR = 9 ms, TE = 4 ms, flip angle = 10°, TI = 2 ms, TD = 200 ms.
The second dataset was obtained with a TSEFLAIR volumetric sequence (256 × 256 × 160 voxels with a voxels resolution of 0.94 × 0.94 × 1 mm^{3}) acquired on a Philips Gyroscan 3 Tesla scanner (Best, Netherlands) using a sensitivity encoding (SENSE) acceleration factor of 2, TR = 14 ms, and TE = 140 ms. Although parallel acquisition techniques such as SENSE or generalized autocalibrating partially parallel acquisitions introduce a spatially varying noise variance across the image, we used this dataset here to show the capability of the proposed approach on MR images with spatially varying noise.
Finally, we have obtained an image with a particularly large amount of noise by courtesy of Huiping Shi (Qiqihaer Medical College, Qiqihaer, China), as to assess the performance of the methods under extreme conditions. It was obtained on a 0.5 Tesla NeusoftPhilips, with parameters TR = 20 ms, TE = 5 ms, flip angle = 90°, field of view = 26 cm, matrix = 256 × 160, slice thickness = 2 mm. The resulting 3D image has size 512 × 512 × 40 voxels. Its original values are in the range [0, 255].
3.2. Quantitative and qualitative metrics
The efficiency of the denoising methods was compared quantitatively and qualitatively. For quantitative assessment, the mean squared error (MSE) and the Structural Similarity index (SSIM) [32] were evaluated. These measures were computed with the noisefree MR images as the ground truth. MSE is an objective measure that quantifies the deviation of estimated values from the true value
where I(i) and I_{0}(i) are the pixel (voxel) values at position i of the original image and the denoised image, respectively. M denotes the number of the pixels in each image. MSE yields the same relative ordering among methods as the power signaltonoise ratio (PSNR) and the root mean squared error (RMSE):
Though, MSE is most commonly employed similarity metric, it is not optimal with respect to the perceived quality. Our second performance measure was the SSIM, which was employed to study the structural and perceptual similitude between the original and denoised images. SSIM is an effective alternative that improvizes the error measures and is also consistent with the visual perception. As mentioned in previous section, MRI consists of delicate structural details, for which MSE is not sufficient to quantify the restored information. Therefore, SSIM is also used to study the structural and perceptual closeness between the denoised and the original images. The SSIM index is estimated locally over a 11 × 11 window, which moves pixelbypixel over the entire image. The final value of SSIM is the mean of SSIM index calculated over the N local regions. The SSIM between the image X and Y is calculated from
where μ is the mean intensity, δ denotes the standard deviation, and the constants c_{1} = 0.01, c_{2} = 0.03 were chosen as recommended in Wang et al. [32].
The value of SSIM lies between [1, 1]. Alternatively, the SSIM can also be given in percentage (%). Larger value of SSIM means high similarity between the compared images.
Visual assessment of the residual image was employed for qualitative evaluation. The residual image was obtained by subtracting the denoised image from the noisy image [12]. The residual image was required to verify the traces of anatomical information in clinical image removed during denoising. So, this could reveal the excessive smoothing and blurring of small structural details contained in the image.
3.3. Validation on simulated dataset
We have compared, qualitative and quantitatively the performance of our proposed algorithm with optimal estimated parameters (R_{neighbor} = 7, δ_{ d } = 5, δ_{ r } = 1.5σ_{ n }) with other three stateoftheart filtering algorithms: the unbiased nonlocal means filter (UNLM) [12], the adaptive blockwise nonlocal means filter (ABONLM) [9], and the 2D wavelet domain bilateral filter (2DWBF) [19].
Our algorithm was quantitatively compared, using the synthetic data referred in Section 3.1. The values of MSE and SSIM obtained for the synthetic data using the aforementioned denoising techniques were tabulated in Tables 1 and 2, respectively. The performance of 3DWIBF with weighted Neighshrink is better than UNLM, ABONLM, and 2DWBF with respect to the quantitative metrics. Lower value of MSE and higher value of SSIM showed that our method significantly outperforms others.
The results on T1weighted images were shown in Figure 6. The original MR image (Figure 6a) and the noisy image with 5% Rician noise added (Figure 6b) were presented. Some structural details were smoothed in the results obtained by the UNLM filter (Figure 6c). While the UNLM filter could preserve the distinct edge features, it failed to preserve small structural details. The results obtained by the ABONLM filter (Figure 6d) were better than those by the UNLM filter. However, some regions were blurred and some useful information was lost. We found that the 2DWBF filter has the coarse edge effect (Figure 6e) compared to our proposed filter (Figure 6f).
The results on T2weighted images with 7% Rician noise added were shown in Figure 7. The result obtained by the 2DWBF filter (Figure 7e) reserved more information than the result obtained by the UNLM filter (Figure 7c) and that obtained by the ABONLM filter (Figure 7d). However, the 2DWBF filter smoothed the homogeneous regions. We observed that the result obtained by our proposed filter (Figure 7f) could recover as much information as that by the 2DWBF filter.
The results on PDweighted image were shown in Figure 8. The noisy image with 9% Rician noise added was presented (Figure 8b). We found that the interface between gray and white matter in the result by the UNLM filter (Figure 8c) was away from that of the original image (Figure 8a), and the result by the ABONLM filter (Figure 8d) was oversmoothed. There was still some obvious noise in the result by the 2DWBF filter (Figure 8e). Both the small structures and the boundaries were preserved well by the 3DWIBF filter (Figure 8f).
3.5. Validation on clinical dataset
Since the original images already have noise, neither ideal residual image nor quantitative results can be obtained. The denoising results obtained for the T1weighted brain image were shown in Figure 9. The residual images of the UNLM filter (Figure 9c), the ABONLM filter (Figure 9e), and the 3DWIBF method (Figure 9i) did not reveal significant anatomical information. The residual image of the 2DWBF filter showed the extent of smoothing along the edges of the image (Figure 9g). While the 2DWBF filter could preserve the distinct edge features, it blurs the heterogeneous regions and hence, reduces the contrast between the gray and the white matter regions. The results of filtering SENSE images reconstructed from patient brain data were shown in Figure 10. The ULM filter and the 2DWBF method left unfiltered noise in the image (Figure 10b, f). With the ABONLM method, the edges were blurred, reducing the image sharpness (Figure 10d). Alternatively, with the 3DWIBF method, the edges were preserved well and the contrast between the gray and the white tissues were also preserved well (Figure 10h). The results obtained for the T1weighted brain image with heavy noise were shown in Figure 11. With the 2DWBF method, the sharpness along the edges was smoothed (Figure 11f). Comparing the results of the ULM filter (Figure 11b) and the ABONLM method (Figure 11d), it was evident that the 3DWIBF method preserves well the details (Figure 11h).
4. Discussion and conclusion
The sources that introduce uncertainty in voxel intensity are many and are generally derived from one of two categories: thermal noise and physiological noise. Other sources may also exist in the electronics of the acquisition system, such as digitization, but these can be minimized in an ideal condition. Thermal noise is usually considered as "white noise" because it is expected that its power should be equal for all frequencies within the readout bandwidth. Because MR images are reconstructed using the Fourier transform, the variance that characterizes the uncertainty due to thermal noise is constant throughout the imaging volume. But the physiological noise differs. In our study, we could only estimate the variance that characterizes the uncertainty of the MR measurement due to thermal noise. But we expected to estimate the variance component due to physiological noise such as flow, MR spin history effect. To this end, it requires many repeated acquisitions. It would also be difficult to isolate the variance due to patient motion in such repeated measurements (which is something the registration step is actually trying to diminish). Therefore, we did not estimate the noise introduced by physiological effects.
In this study, we proposed a 3D extension of the wavelet domain bilateral filtering ideas for Rician noise removal. Due to the delineating capability of wavelet, 3D WT was employed to decompose the MR image into the approximation and the detailed subbands. Compared to 2D WT, the inherent advantages of 3D WT is apparent due to improved ability to model "throughplane" structure. 2D denoising ignores throughplane signal correlations; each slice is treated independently. Consider a slender fiber tract. If the imaging plane is orthogonal to the fiber axis, it would be much more difficult to distinguish the "structure" from spurious, bright, noise pixel. 3D denoising uses structural correlations from all three principle planes, and is robust to fiber axis orientation.
Considering the similarities in the wavelet domain where the data and noise can be efficiently discriminated, 3D bilateral filtering of the approximation coefficients in a modified neighborhood eliminates the higher magnitude noise components carried into the approximation subbands. Utilizing a group of a square and a line as neighbor to replace the original cubic neighbor for weight estimation improves the computation accuracy. Noticing the fact that the wavelet subband coefficients of different orientations have different properties of energy clustering, a WNS thresholding has been proposed to threshold the noisy coefficients in the detailed subbands. Exploiting the interscale dependencies among the detailed coefficients tends to improve the performance of wavelet thresholding, and thus, it also enhances the denoising efficiency for MR images with spatially varying noise. In summary, the utilization of neighborhood similarities using wavelet domain bilateral filter and Neighshrink improves the noise cancellation efficiency and preserves the structural information effectively.
Experiments were carried out on both simulated and real datasets. Quantitative results using two different quality measures show a better behavior of the proposed scheme when compared to other stateoftheart filters for different noise levels.
An ideal filter for MR images with spatially varying noise levels must be able to improve the image PSNR while preserving image important structures and avoiding the generation of artifacts. The results indicated that parameter σ_{ n }chosen as a constant in all image areas may lead to the enhancement of noisegenerated gradients in highnoise areas that could be incorrectly identified as anatomical structures of different tissue type characteristics for the imaging modality such as vessels. Therefore, it is necessary to derive a strategy that optimizes the choice of σ_{ n }with respect to the local characteristics of the considered neighborhoods. Hence, this study can be extended to make the choice of σ_{ n }locally adaptive, optimizing the denoising procedure for all the noise levels.
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Acknowledgements
We would like to thank McConnell Brain Imaging Center (BIC) of the Montreal Neurological Institute, McGill University, for providing access to the MR data in the BrainWeb Database (http://www.bic.mni.mcgill.ca/brainweb). We would also like to thank VisAGeS of the IRISA Institute, University of Rennes I, for providing online applications (http://www.irisa.fr/visages/benchmarks/).
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Wang, Y., Che, X. & Ma, S. Nonlinear filtering based on 3D wavelet transform for MRI denoising. EURASIP J. Adv. Signal Process. 2012, 40 (2012). https://doi.org/10.1186/16876180201240
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Keywords
 magnetic resonance imaging
 3D image denoising
 3D wavelet transform
 bilateral filtering
 enhanced NeighShrink thresholding