Brain MR image denoising for Rician noise using pre-smooth non-local means filter
© Yang et al.; licensee BioMed Central. 2015
Received: 26 July 2014
Accepted: 19 November 2014
Published: 9 January 2015
Magnetic resonance imaging (MRI) is corrupted by Rician noise, which is image dependent and computed from both real and imaginary images. Rician noise makes image-based quantitative measurement difficult. The non-local means (NLM) filter has been proven to be effective against additive noise.
Considering the characteristics of both Rician noise and the NLM filter, this study proposes a frame for a pre-smoothing NLM (PSNLM) filter combined with image transformation. In the PSNLM frame, noisy MRI is first transformed into an image in which noise can be treated as additive noise. Second, the transformed MRI is pre-smoothed via a traditional denoising method. Third, the NLM filter is applied to the transformed MRI, with weights that are computed from the pre-smoothed image. Finally, inverse transformation is performed on the denoised MRI to obtain the denoising results.
To test the performance of the proposed method, both simulated and real patient data are used, and various pre-smoothing (Gaussian, median, and anisotropic filters) and image transformation [squared magnitude of the MRI, and forward and inverse variance-stabilizing trans-formations (VST)] methods are used to reduce noise. The performance of the proposed method is evaluated through visual inspection and quantitative comparison of the peak signal-to-noise ratio of the simulated data. The real data include Alzheimer’s disease patients and normal controls. For the real patient data, the performance of the proposed method is evaluated by detecting atrophy regions in the hippocampus and the parahippocampal gyrus.
The comparison of the experimental results demonstrates that using a Gaussian pre-smoothing filter and VST produce the best results for the peak signal-to-noise ratio (PSNR) and atrophy detection.
Magnetic resonance imaging (MRI) images of the brain have an important role in diagnosing many neurological diseases, such as Parkinson’s disease, Alzheimer’s disease (AD), brain tumors, and stroke. Analyzing MRI images can help surgeons make appropriate decisions. However, MRI noises degrade image quality, which negatively affects image processing and analysis works, such as registration, segmentation, classification, and visualization. To obtain reliable analysis results, removing MRI image noises is necessary before further image processing can be conducted.
The technique of removing noises from images is called “image denoising,” which is an important image pre-processing step. Although many image denoising methods have been developed, denoising remains a challenge because these methods produce artifacts and blurry images . Most denoising methods still cannot provide desirable results .
where noise n is independent and Gaussian distribution with zero-mean and known standard deviation, I 0 is the true signal, and I is the observed signal. Most denoising methods have been developed by using the additive noise model. These methods are classified into two major categories: spatial filtering and transform domain filtering methods.
Spatial domain techniques directly deal with image pixels. A spatial image filter is an image operation wherein each pixel or voxel value I(u) is transformed through a function of the intensities of the pixels or voxels within a neighborhood (u). Traditional spatial image filters include Gaussian , median , Wiener , diffusion , and bilateral filters. Gaussian and median filters remove noise in a small constant region and blur images. An anisotropic diffusion filter preserves the edges of images, but erases small features and generates a mask effect in uniform regions of the denoised images. These denoising methods significantly remove noise but cause blurred images and add artifacts to the images. A transform domain image filter transforms images from the space domain into another domain, such as the frequency and wavelet domains , and then processes images in the new domain. The wavelet thresholding method can significantly reduce noise, but introduces characteristic artifacts. In this study, we mainly focus on the spatial domain filter.
A non-local mean (NLM) algorithm has been proposed recently . This algorithm provides good edge-preserving results. Each pixel of the denoised image from the NLM algorithm can represent the weighted average of all pixels in the noisy image by using a Gaussian function as the smoothing function. The NLM filter has been proven to be an effective denoising method, particularly against additive noise.
Many MRI denoising methods have been reported. Gaussian filter smoothing is a key step in voxel-based morphometry (VBM) analysis . A Wiener filter uses its neighborhood to estimate its parameters . An anisotropic filter combines local linear minimum mean squared error (MSE) filters to remove MRI noise . A trilinear filter achieves edge-preserving results by integrating geometric, photometric, and local structural similarities . Noise estimation methods in the wavelet domain are also used in MRI denoising. MRI in the wavelet domain is decomposed into sub-bands at various scales. Coefficients are processed with soft or hard thresholding to estimate signal components . The NLM filter is also used in MRI denoising [14, 15].
The noise in the MRI images is modeled according to the coil number in the imaging system. In the single coil system, the noisy distribution in MRI images can be modeled as a Rician distribution , which assumes that the real part and imaginary part of the MRI image is an uncorrelated Gaussian distribution with a zero mean and equal variance. While in the multi-coil system (parallel MRI), the magnitude of noises follows a non-central Chi distribution  with a sum-of-squares (SoS) reconstruction. The noise in the MRI images acquired from the generalized auto calibrating partially parallel acquisition (GRAPPA) reconstruction  can be modeled by the non-central Chi distribution. Actually, the Rician distribution is a special case of the nonCentral Chi distribution. To apply the additive noise model, the nonCentral Chi distribution should be transformed into the Gaussian distribution in the transformed space. Therefore, the above-mentioned denoising methods can be applied to the transformed data. After inverse transformation of the denoising data, the noise is removed from the MRI images. Many mthods have been proposed to deal with the problem, such as the local moment distribution based method , nonlocal maximum likelihood (NLML) estimation , and the effective variance of noise from the composite magnitude signal of MR data .
In this study, we focus on the single coil system which produces noises with Rician distribution. To remove noise and apply the additive noise model, Rician noise should be transformed into an independent Gaussian noise. The squared magnitude correction is widely used for MRI denoising . The standard deviation of noise is estimated from a dark uniform background. Recently, a forward and inverse variance-stabilizing transformation (VST)  has been proposed for the Rician distribution. The forward VST removes the dependency of the noise variance on the observed image, and the inverse VST compensates the bias in the filtered image.
A frame for a pre-smoothing NLM (PSNLM) filter is proposed in the study. MRI noise is first transformed into additive noise and then smoothed by a traditional denoising filter (Gaussian, median, or anisotropic). Next, the weight of the NLM algorithm is calculated from the smooth image, and noise is removed by the NLM filter. Finally, inverse transformation is used to obtain accurate results. Squared magnitude transformations, as well as forward and inverse variance-stabilizing transformations (VST) , are adopted to transform noise in the proposed method.
where z is the complex MRI; z real and z imaginary are the real and imaginary components, respectively, which are independently corrupted by Gaussian white noises n1 and n2, respectively, with zero mean and standard deviation σ; and r and θ are the magnitude and phase of the original MRI, respectively.
where I 0 denotes the modified Bessel function of order zero.
Estimating noise from the measured magnitude |z| of MRI is difficult. The most commonly used method is to square the magnitude of MRI. Recently, VST is proposed for Rician noise processing.
The squared magnitude method
Background of the brain
The background of the brain can be estimated by using any segmentation method. In this study, the Otsu method , which is an automatic image segmentation technique, is adopted to extract the background region. This algorithm assumes that two classes of pixels or bi-modal histogram (e.g., foreground and background) are found in the image and exhaustively searches for the threshold that maximizes inter-class variance [22, 23]. The process includes median filtering, Otsu segmentation, morphological close operation, and filling the holes in the image.
The method of variance-stabilization
This method involves the optimal forward and inverse VSTs for Rician noise. Forward transformations (f) treat MRI noise as an additive noise with a unitary variance. Thus, noise can be removed by using the NLM filter. Denoising MRI can be achieved through inverse transformation (I).
According to the Eq. (10)-(12), to modify each noisy voxel |z|(i, j, k), its neighborhood |z|(N i,j,k ) is compared with other noisy voxel’s neighborhood |z|(N l,m,n ) in the entire image. It needs huge computation. Considering the computational efficiency, usually the search window is selected to be a much smaller than the size of the entire image and is s x × s y × s z .
The Pre-smooth Non-local Means Filter (PSNLM)
Summary of the PSENLM
(1) Extract background
(1) Transform noisy image by the forward variance-stabilization transformation
(2) Compute the noise variance σ 2 according Eq. (5)
(2) Smooth the transformed image
(3) Compute the squared magitude of noisy image
(3) Compute the weigh term from the smoothed image
(4) Smooth the squared magitude of noisy image
(4) Obtained final result by the inverse variance-stabilization transforming of the NLM denoising result
(5) Compute the weigh term from the smoothed image
(6) Correct bias by subtracting 2σ 2 from the NLM denoising result. The final result is obtained by squared root of the bias correction
where PSNLM1(i, j, k) and PSNLM2(i, j, k) are the voxels of PSNLM1 and PSNLM2, respectively, at position (i, j, k); and r(i, j, k) is the original MRI at position (i, j, k). A large PSNR indicates good results.
For real patient data, the performance of the proposed method in detecting brain atrophy among patients with AD is evaluated by using VBM [9, 26]. In VBM studies, all individual brain images are spatially normalized onto template space through a registration algorithm. Based on the estimated deformation field, each brain morphometry information can be measured by using a Jacobian map, which reflects regional volumetric changes that are compared with the template image. Subsequently, voxel-wise group analysis is performed by using SPM software to detect brain atrophy .
In this case, randn(size(r)) produces data with the same size as the ground truth image, and the mean and standard deviation of the data are 0 and 1, respectively. The real patient data that are used in this study are downloaded from the Alzheimer’s Disease Neuroimaging Initiative (ADNI) database , which includes many T1-weighted MRI images of AD patients and normal controls (NC). To test the performance of the proposed method, 20 NCs and 20 AD patients with T1-weighted MRI images of the baseline are randomly selected from the database.
Results and discussion
The experimental results are obtained from both simulated and real patient data.
Moreover, several filters are used to pre-smooth images. In this study, common filters, such as Gaussian, median, and anisotropic filters, are used to pre-smooth MRI images. The NLM algorithm has three parameters: patch size, search window size s x × s y × s z , and filtering degree h. In many NLM-based MRI denoising cases, patch size is set as 5 × 5 × 5, and search window size is set as 11 × 11 × 11 for MRI images with a voxel size of 1 × 1 × 1 mm 3 . Thus, the same parameters are used in this study. h is typically set as proportional to the noise level of the image. In our experiments, different h values are tested, and we selected the h value that produces the largest PSNR. For comparison, the PSNR of different methods are calculated. These methods include the original NLM algorithm, unbiased correction (UNLM) by the squared magnitude, as well as VST, PSNLM1, and PSNLM2 with Gaussian, median, and anisotrpic filters, respectively.
Comparison of PSNR
Comparison by visual inspection
Figure 12 shows the enlarged regions of the experimental results on the T1-weighted MRI image. Figure 12(a) shows an enlarged region of the original MRI, and Figure 12(b) shows noisy MRI with 17% Rician noise. Figure 12(c) presents the result of the original NLM. The edges are obviously blurred. Figure 12(d) and (h) present the results of the UNLM with the squared magnitude and VST, respectively. Figure 12(h) is clearer than Figure 12(d). Although both results are better than that of the original NLM algorithm, the images are still blurred. Figure 12(e) to (g) present the results of the PSNLM1 with median, Gaussian, and anisotropic filters, respectively, whereas Figure 12(i) to (k) present the results of the PSNLM2 with median, Gaussian, and anisotropic filters, respectively. All results are significantly improved compared with Figure 12(d) and (h). However, some artifacts are observed in Figure 12(e),(g),(i), and (k). Figure 12(f) and (j) show that the results are extremely close to and similar to those of the original image. Nevertheless, the result in Figure 12(j) is slightly better than those in the other figures. Figure 13 is another enlarged region of the experimental results on the T1-weighted MRI image. Comparably, there are some artifacts in Figure 13(e),(f),(g) and (i), Figure 13(c),(d) and (h) are slight blurred, while Figure 13(j) and (k) are clearer than the others. It is obvious that the results of PSNLM2 with Gaussian and median filters are better than the other methods.
Figure 14 presents the results on the PD-weighted MRI image with 17% Rician noise. Figure 14(a) shows an enlarged region of the original MRI image, and Figure 14(b) shows the noisy MRI image. Figure 14(c) presents the result of the original NLM. Figure 14(d) and (h) present the results of the UNLM with the squared magnitude and VST, respectively. Figure 14(e) to (g) present the results of the PSNLM1 with median, Gaussian, and anisotropic filters, respectively. Figure 14(i) to (k) present the results of the PSNLM2 with median, Gaussian, and anisotropic filters, respectively. It is obvious that Figure 14(c), (d) and (h) are blurred, while the pre-smooth technique significantly improves the result, as can be seen in Figure 14(j).
Figure 15 is another enlarged region of the experiment results on the PD-weighted MRI image. It is obvious that Figure 15(c),(d) and (h) are, to some extent, blurred. While the proposed pre-smooth technique greatly improves the result, as can be seen in Figure 15(j).
Real patient data
The ratios of the detecting atrophy region and the region of hippocampus and parahippocampal gyrus
The PSNR of noisy image, results of LMMSE, TV and PSNLM2
In this study, a PSNLM frame combined with image transformation is proposed for MRI denoising. After the Rician noise in MRI is transformed to the additive noise, the NLM filter is applied to the transformed image with weights computed from the pre-smooth image. The final denoising image is obtained from the inverse transformation of the NLM denoising result. Experiments are performed on the simulated data of the T1-, T2-, and PD-weighted MRI images that are downloaded from the BrainWeb database and real data (T1-weighted MRI images) downloaded from ADNI. Many traditional filters are tested as pre-smooth filters. In this paper, only the results using Gaussian, median, and anisotropic filters are presented because others such as trilateral filter and wavelet filter produce worse results. The proposed method is automatic and robust since it improves the denoising results greatly in all the experiments. One challenge problem for the proposed method is that it is comparably time consuming, which costs about 15 minutes on the image size 181×217×181 with 10 threads on the Dell server. The reason is that the image transformation of the NLM takes heavy computation. Recently, many works have been proposed with respect to decrease the computation time of the NLM, such as the FFT transformation , which is about fifty times faster than the original non-local algorithm and can reach the clinic application. The experiment results indicate that using the Gaussian pre-smoothing filter and VST produces the best results for the peak signal-to-noise ratio (PSNR) and atropy detection.
This research was supported by the National Basic Research Program of China (2013CB328806), the Key Projects in the National Science & Technology Pillar Program (2013BAI01B01), the National Hi-Tech Research and Development Program (2013AA013703), and the National Natural Science Foundation of China (61179020).
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