Recently nonlocal means (NLM) and its variants have been applied in the various scientific fields extensively due to its simplicity and desirable property to conserve the neighborhood information. The twostage MRI denoising algorithm proposed in this paper is based on 3D optimized blockwise version of NLM and multidimensional PCA (MPCA). The proposed algorithm takes full use of the block representation advantageous of NLM3D to restore the noisy slice from different neighboring slices and employs MPCA as a postprocessing step to remove noise further while preserving the structural information of 3D MRI. The experiments have demonstrated that the proposed method has achieved better visual results and evaluation criteria than 3DADF, NLM3D, and OMNLM_LAPCA.
As a significant imaging technique, magnetic resonance imaging (MRI) provides very important information to research the tissues and organs in the human body with noninvasive style. However, MRI is affected by several artifacts and noise sources. One of them is the random fluctuation of the MRI signal which is mainly due to thermal noise. Such noise seriously degrades the acquisition of any quantitative measurements from the data. Consequently, the denoising techniques are required to improve the quality of MRI.
Generally speaking, MRI denoising techniques can be classified as either filtering, transform, or statistical approach [
Due to its ability to perform decorrelation, PCA has also been used in image denoising. However, PCA requires that the number of images be bigger than the number of significant components of the image. The drawback has limited the application of PCA in the field of image denoising. The paper [
Actually, MRI is naturally a 3D image, which can be considered as tensor data on multidimensional space. From the aspect of superresolution reconstruction, the noise image can be considered as the degraded version of the original image. Therefore, this paper proposed a multidimensional structure preserving MRI denoising algorithm. The algorithm consists of two stages. On the first stage, the 3D variant of the nonlocal means technique is employed to reduce the noise, which takes full advantage of the neighbor information between different 3D MRI slices and has the capability of exploiting the underlying structure in the multidimensional image. On the second stage, for the result image obtained from the first stage, multidimensional principal component analysis is performed to suppress the remaining noise, which avoids the vectorization to preserve the neighborhood information for MRI image and is helpful to improve the computation cost. According to the experiments on 3D MRI image, the proposed algorithm is superior to restore the original image from noise compared with other stateoftheart methods.
For image denoising problem, the noisy data
The key issue of NLM is the computation of
However, the basic NLM has a great influence for computational efficiency, especially for 3D MR images. Consequently, [
Although PCA has been applied in image denoising widely, most denoising algorithms based on PCA assume that data lie on vector space and usually process the vectorization operation to make image into a vector. The vectorization destroys the structural information about the neighborhood.
Instead of data in vector space, any multidimensional data can be considered as tensor data in multidimensional space. Each tensor data will be treated by tensor decomposition [
Based on PCA on vector space, [
In tensor algebra, any tensor data
Due to the difficulty in the computation of
To the best of our knowledge, this is the first attempt to introduce MPCA to MRI image denoising. It should be noticed that image denoising based on MPCA is different from its application in machine learning.
For MRI denoising, 3D MRI is a 3rdorder tensor
NLM 3D filtering is applied to denoise and obtain the initial 3D images.
Then the initial 3D images are processed by MPCA to remove noise furtherly.
Several experiments were conducted to compare the proposed methods with related stateoftheart methods.
In order to illustrate the performance of the proposed method, several experiments were conducted to compare the proposed methods with related stateoftheart methods, including the 3D version of anisotropic diffusion filtering (3DADF) [
There are some free parameters that need to be set to obtain optimal performance. For 3D anisotropic diffusion filtering, the integration constant is the maximum value, the number of iterations is 4, and the gradient modulus threshold is 70. For NLM3D and OMNLM_LAPCA, the radius of the search area is 5 and the radius of similarity area is 2. For the proposed method, the number of preserved largest principal components is 140 (see below).
Three kinds of quality measurement are used to evaluate the denoising performance. The first one is the signaltonoise ratio (SNR), the second one is the peak signaltonoise ratio (PSNR), and the last one is the structural similarity index (SSIM) [
The SNR is computed as follows:
The PSNR is based on the root mean square error (RMSE) between the denoised image and original image:
In this part, the 3D T1weighted MRI image in the wellknown BrainWeb [
Example images of the BrainWeb database. (a) Noisyfree T1w image, (b) noisy image corrupted with a Rician noise at 3%, (c) noisy image corrupted with a Rician noise at 4%, and (d) noisy image corrupted with a Rician noise at 5%.
The denoising performance of different methods with different noise levels is compared based on SNR, PSNR, and SSIM, as shown in Table
The comparison of denoising methods with different noise levels on T1 weighted MRI Images.
Denoising method  3% Rician noise  4% Rician noise  5% Rician noise  

SNR  PSNR  SSIM  SNR  PSNR  SSIM  SNR  PSNR  SSIM  
3DADF  13.7207  25.4512  0.7642  12.9584  25.4144  0.7624  12.1337  25.3515  0.7604 
OMNLM_LAPCA  18.5248  35.0833  0.9495  16.0899  33.2808  0.9234  14.1684  31.7119  0.8927 
NLM3D  25.9919  37.5705  0.976  24.0109  35.9133  0.965  20.8779  34.5518  0.9472 
Proposed 









The noisefree image, noisy image, denoised images, residuals, and detail denoised images of different methods with 5% Rician noise for different slices.
For the proposed method, it is required to determine the optimal number of the largest principal components. To study its influence on the denoising performance, the SNR, PSNR, and SSIM with different principal components with different noise levels are shown in Figures
Plots of SNR versus the number of principal components for different noise levels.
Plots of PSNR versus the number of principal components for different noise levels.
Plots of SSIM versus the number of principal components for different noise levels.
Based on the idea proposed in [
The denoised results with different order of MPCA and NLM3D (5% Rician noise). (a) First apply MPCA and then apply NLM3D but in (b) first apply NLM3D and then apply MPCA.
MPCA + NLM3D
NLM3D + MPCA
All denoising methods were performed in MATLAB 2015 on a Windows 7 computer equipped with an Intel Core i75600U, 2.6 GHz CPU and 8 GB RAM. To denoise typical 3D dataset with the size of
The comparison of computational time.
Denoising methods  The computational time(s) 

3DADF  17.6905 
OMNLM_LAPCA  32.8070 
NLM3D  110.7295 
Proposed 

It cannot be denied that, compared with other algorithms, the proposed method will spend more time to denoise 3D MRI since it makes use of the 3D structure information from neighboring slices. It is also believed that the implementation of the proposed methods using MATLAB/C MEX techniques and parallel computations on graphic processing units may significantly further accelerate the filtering.
To evaluate the proposed method on real clinical data, the experiments are conducted on real T1w MRI data. The data were acquired on a GE MR750 3.0T scanner. The anatomical images were scanned using a T1weighted axial sequence parallel to the anteriorcommissureposteriorcommissure line. Each anatomical scan has 156 axial slices (spatial resolution = 1 mm × 1 mm × 1 mm, field of view = 256 mm × 256 mm, time repetition (TR) = 8.124 ms). The noisyfree image and noisy image are shown in Figure
The example images of real clinical data and noisy data with 5% Rician noise.
For real clinical data, the denoised results are shown in Figure
The comparison of different methods with 5% Rician noise.
Denoising method  SNR  PSNR  SSIM 

3DADF  9.7543  24.8338  0.6717 
OMNLM_LAPCA  9.7943  30.42  0.8415 
NLM3D  18.7476  32.6647  0.9034 
Proposed  19.7629  33.6679  0.9236 
The denoised images, residuals, and detail denoised images of different methods with 5% Rician noise for different slices.
The paper has proposed a structure preserving MRI denoising algorithm. The method has integrated NLM3D and MPCA to restore noisy image from 3D neighborhood and has achieved a better result compared with some famous MRI denoising methods, such as 3DADF, OMNLM_LAPCA, and NLM3D. However, the confusing question of the proposed method is how to determine the optimal number of principal components, which will affect the denoising effect. So our next work will research the selection problem of principal components. We will consider the cumulative energy or the scoring of principal components in the future.
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work is supported by the National Natural Science Funds of China (Grant no. 61502059) and the Young Scientist Project of Chengdu University (no. 2013XJZ21). The authors are glad to to have learned a lot from the reviewers and the editor who made many excellent comments to improve the paper presentation.