Nonlinear filtering based on 3D wavelet transform for MRI denoising
- Yang Wang^{1}Email author,
- Xiaoqian Che^{2} and
- Siliang Ma^{1}
https://doi.org/10.1186/1687-6180-2012-40
© Wang et al; licensee Springer. 2012
Received: 20 July 2011
Accepted: 21 February 2012
Published: 21 February 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 signal-to-noise 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.
Keywords
1. Introduction
Three-dimensional magnetic resonance imaging (MRI) has, during the last several decades, benefited from a variety of technological developments resulting in increased resolution, signal-to-noise ratio (SNR), and acquisition speed. However, fundamental trade-offs 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 high-resolution and high SNR. Another example concerns diffusion-tensor (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 process--higher tissue anisotropy produces progressively lower intensities in diffusion-weighted images that, in turn, are more susceptible to Rician noise. The efficacy of higher-level post processing of MR and DT-MR 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. Post-processing 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 multi-scale 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 wavelet-based 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].
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
where ⊕ denotes space direct sum, L^{ α } and H^{ β }, respectively, represent the high- and low-pass directional filters along directions of α-axis, where α ∈ {x,y,z}.
2.3. 3D bilateral filtering
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:
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 high-frequency 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.
The estimator of the unknown noiseless coefficient ${\stackrel{\u2322}{\theta}}_{ijk}$ is determined according to Equation (12).
2.5. Algorithm summary
- (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 (8-bit 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 MP-RAGE 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 TSE-FLAIR 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 Neusoft-Philips, 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 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 state-of-the-art filtering algorithms: the unbiased nonlocal means filter (UNLM) [12], the adaptive blockwise non-local means filter (ABONLM) [9], and the 2D wavelet domain bilateral filter (2DW-BF) [19].
Comparisons of experimental results in MSE
Algorithm | Test image and noise level | ||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
T1-weighted MR image | T2-weighted MR image | PD-weighted MR image | |||||||||||||
1% | 3% | 5% | 7% | 9% | 1% | 3% | 5% | 7% | 9% | 1% | 3% | 5% | 7% | 9% | |
UNML | 2.50 | 11.42 | 24.71 | 43.69 | 68.12 | 3.31 | 18.62 | 38.68 | 64.97 | 97.78 | 2.68 | 13.85 | 29.52 | 48.78 | 71.20 |
ABONLM | 3.28 | 19.78 | 51.07 | 97.98 | 161.39 | 6.55 | 24.76 | 56.19 | 101.03 | 159.83 | 4.01 | 18.82 | 46.08 | 87.25 | 141.02 |
2DW-BF | 3.08 | 11.56 | 26.06 | 51.56 | 62.26 | 7.23 | 20.33 | 57.30 | 68.27 | 90.03 | 3.65 | 15.19 | 30.63 | 43.67 | 62.77 |
3DW-IBF | 2.07 | 8.20 | 15.11 | 22.31 | 29.58 | 4.62 | 15.54 | 29.21 | 50.10 | 67.38 | 3.52 | 11.18 | 20.65 | 26.12 | 41.58 |
Comparisons of experimental results in SSIM
Algorithm | Test image and noise level | ||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
T1-weighted MR image | T2-weighted MR image | PD-weighted MR image | |||||||||||||
1% | 3% | 5% | 7% | 9% | 1% | 3% | 5% | 7% | 9% | 1% | 3% | 5% | 7% | 9% | |
UNML | 0.984 | 0.926 | 0.869 | 0.814 | 0.763 | 0.987 | 0.941 | 0.895 | 0.852 | 0.814 | 0.985 | 0.934 | 0.880 | 0.829 | 0.783 |
ABONLM | 0.943 | 0.862 | 0.827 | 0.798 | 0.770 | 0.967 | 0.902 | 0.865 | 0.839 | 0.816 | 0.968 | 0.897 | 0.855 | 0.823 | 0.796 |
2DW-BF | 0.967 | 0.929 | 0.931 | 0.896 | 0.873 | 0.991 | 0.948 | 0.899 | 0.867 | 0.856 | 0.997 | 0.930 | 0.881 | 0.836 | 0.830 |
3DW-IBF | 0.989 | 0.934 | 0.921 | 0.913 | 0.908 | 0.989 | 0.942 | 0.916 | 0.901 | 0.882 | 0.988 | 0.946 | 0.903 | 0.890 | 0.855 |
3.5. Validation on clinical dataset
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 "through-plane" structure. 2D denoising ignores through-plane 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 state-of-the-art 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 noise-generated gradients in high-noise 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.
Declarations
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/).
Authors’ Affiliations
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