Rician nonlocal means denoising for MR images using nonparametric principal component analysis
© Kim et al; licensee Springer. 2011
Received: 7 February 2011
Accepted: 14 October 2011
Published: 14 October 2011
Denoising is always a challenging problem in magnetic resonance imaging (MRI) and is important for clinical diagnosis and computerized analysis, such as tissue classification and segmentation. The noise in MRI has a Rician distribution. Unlike additive Gaussian noise, Rician noise is signal dependent, and separating the signal from the noise is a difficult task. In this paper, we propose a useful alternative of the nonlocal mean (NLM) filter that uses nonparametric principal component analysis (NPCA) for Rician noise reduction in MR images. This alternative is called the NPCA-NLM filter, and it results in improved accuracy and computational performance. We present an applicable method for estimating smoothing kernel width parameters for a much larger set of images and demonstrate that the number of principal components for NPCA is robust to variations in the noise as well as in images. Finally, we investigate the performance of the proposed filter with the standard NLM filter and the PCA-NLM filter on MR images corrupted with various levels of Rician noise. The experimental results indicate that the NPCA-NLM filter is the most robust to variations in images, and shows good performance at all noise levels tested.
Magnetic resonance (MR) images are affected by several types of artifact and noise sources, such as random fluctuations in the MR signal mainly due to the thermal vibrations of ions and electrons. Such noise markedly degrades the acquisition of quantitative measurements from the data. The noise in MR images obeys a Rician distribution [1–3]. Unlike additive Gaussian noise, Rician noise is signal dependent, and consequently separating the signal from the noise is difficult.
There is an extensive literature on Rician noise reduction in magnetic resonance imaging (MRI), varying from the use of traditional smoothing filters to more elegant methods. Most conventional mask-based denoising filters, such as Gaussian and Wiener filters , are conceptually simple. However, they will most likely fail to reduce Rician noise in MRI, as they usually assume that the noise is Gaussian. Restored images may often look blurred and may be corrupted by artifacts that are usually visible around the edges. One way to overcome the problems of simple smoothing is to use a nonlocal means (NLM) filter [5–10]. These methods make use of the self-similarity of images, in that many structures show up more than once in the image. The NLM filter takes advantage of the high degree of redundancy of any natural image and produces an optimal denoising result if the noise can be modeled as Gaussian. Unfortunately, the method requires computation of the weighting terms for all possible pairs of pixels, making it computationally expensive. A number of recent reports on NLM denoising focused on shortcuts to make the method computationally practical [11–14]. One of the most compelling strategies is to exclude many weight computations between the image neighborhood feature vectors. Azzabou, et al.  and Tasdizen [13, 14] proposed the so-called PCA-NLM filter, which uses the lower dimensional subspace of the space of image neighborhood vectors in conjunction with NLM using principal component analysis (PCA). More important, this approach was also shown to result in increased accuracy over those that use the full-dimensional ambient space.
There are, however, some applications for which the PCA-NLM filter is not recommended because PCA is sensitive to image features and the presence of noise in the data, and the PCA-NLM filter is, therefore, highly dependent on the settings of its parameters.
In this paper, we propose a nonparametric PCA-NLM filter that is a useful alternative to the PCA-NLM filter for Rician noise reduction in MR images. The proposed filter uses PCA with ranked data instead of the original pixel data. We refer to this as the NPCA- NLM filter. We estimate the subspace dimensionality from parallel analysis [15–17] based on the artificial rank correlation matrix. In contrast to the method reported by Tasdizen [13, 14], our estimation does not require the assumption of a Gaussian distribution and produces a more robust subspace dimensionality regardless of the images being denoised. We also propose a nonparametric method for optimal smoothing kernel width selection.
2.1 Rician noise in MRI
2.2 NLM filter
where Z i is a normalization constant and h acts as a smoothing parameter controlling the decay of the exponential function.
This method is too slow to be practical. The high computational complexity is due to the cost of weight calculation for all pixels in the image during the process of denoising. For every pixel being processed, the whole image is searched and the differences between corresponding neighborhoods computed.
3 Proposed NPCA-NLM denoising
3.1 NPCA approach
where (R i ◦ e p ) denotes the inner product of the two vectors.
where is the new normalizing term. Note that the proposed approach with d = Q is equivalent to the useful NLM filter when applied to ranks rather than the original observations, i.e., Equation 4 with d = Q becomes Equation 2 when calculated on the ranks.
3.2 Optimal smoothing parameter selection
Slope and intercepts used in determining h for various subspace dimensionality of 7 × 7 neighborhoods
d = 6
d = 10
Intercept ( α)
Unlike Tasdizen , the linear fit parameters in Table 1 do not require the assumption of a Gaussian distribution of the noise with which the images are corrupted. Therefore, we expect that h produced by these parameters will yield results for a much larger set of images than that from which they were learned.
3.3 Automatic dimensionality selection
4 Experiments and results
where μ x and μ y are means of x and y, respectively; and are variances of x and y, respectively; and cov xy is covariance of x and y. The constants were set as follows: c1= 0.01L and c2 = 0.03L, and L was 255 for 8-bits/pixel gray scale images.
4.1 Visual quality comparison
As shown in Figure 6, the three filters based on the NLM filter performed well on images with low noise variance (σ = 10). The differences in performance of these filters are difficult to distinguish in the restored images for low noise, but inspection of images with high noise in Figure 7 showed that the denoising effects of the NPCA-NLM filter and PCA-NLM filter were almost identical, except for slight blurring of the output from the PCA-NLM filter. However, the impact of noise on the standard NLM filter was clearly visible, and the restored images contained considerable noise spots.
4.2 Quantitative comparison
PSNR values for various Rician noise levels
σ = 10
σ = 40
MAE values for various Rician noise levels
σ = 20
σ = 30
σ = 40
DSSIM values for various Rician noise levels
σ = 20
σ = 30
σ = 40
Of the three filters investigated, the NPCA-NLM filter appeared to be the most robust to variations in images, performing well at all noise distributions tested.
We proposed an NPCA-NLM filter, which is a useful alternative to the PCA-NLM filter for Rician noise reduction in MR images. The filter uses PCA with ranked data instead of the original pixel data. The image neighborhood vectors used in the NLM filter are projected onto a lower dimensional subspace using NPCA. Therefore, the lower dimensional projections are not only used as search criteria, but also for computing similarity weights resulting in better accuracy in addition to reduced computational cost.
We estimated the subspace dimensionality from parallel analysis based on the artificial rank correlation matrix. We demonstrated that the numbers of components varied more significantly with noise level for PCA than for NPCA. Therefore, the number of principal components was more robust to variations in the noise as well as in the images for NPCA than for PCA. We also proposed a nonparametric method for optimal smoothing kernel width selection that produces results for a much larger set of images than that from which they were learned.
We investigated the performance of the proposed filter in comparison with the standard NLM filter and the recently proposed the PCA-NLM filter for various levels of Rician noise corruption. The experimental results showed that the NPCA-NLM filter was the most robust to variations in images, performing well at all noise levels tested.
This work was supported by the RACS 2010-2014, Production of fine-scale scenario of future climate change using regional climate models and analysis of uncertainties.
- Gudbjartsson H, Patz S: The Rician distribution of noisy MRI data. Magn Reson Med 1995, 34(6):910-914.View ArticleGoogle Scholar
- Macovski A: Noise in MRI. Magn Resonance Med 1996, 36(3):494-497.View ArticleGoogle Scholar
- Nowak RD: Wavelet-based Rician noise removal for magnetic resonance imaging. IEEE Trans Image Process 1999, 8(10):1408-1419.View ArticleGoogle Scholar
- Khireddine A, Benmahammed K, Puech W: Digital image restoration by Wiener filter in 2D case. Adv Eng Softw 2007, 38: 515-516.View ArticleGoogle Scholar
- Buades A, Coll B, Morel JM: A review of image denoising algorithms, with a new one. Multiscale Model Simul 2005, 4(2):490-530.View ArticleMathSciNetMATHGoogle Scholar
- Buades A, Coll B, Morel JM: A non-local algorithm for image denoising. Proc IEEE Trans IEEE Conf Computer Vision and Pattern Recognition 2005, 60-65.Google Scholar
- Orchard J, Ebrahim M, Wang A: Efficient nonlocal-means denoising using the SVD. Proc Int Conf Image Processing 2008, 1732-1735.Google Scholar
- Coupe P, Hellier P, Kervann C, Barillot C: NonLocal means-based speckle filtering for ultrasound images. IEEE Trans Image Process 2009, 18(10):2221-2229.View ArticleMathSciNetGoogle Scholar
- He L, Greenshields IR: A nonlocal maximum likelihood estimation method for Rician noise reduction in MR images. IEEE Trans Med Imaging 2009, 28(2):165-172.View ArticleGoogle Scholar
- Manjon JV, Coupe P, Mari-Bonmati L, Collins DL, Robles M: Adaptive non-local means denoising of MR images with spatially varying noise levels. J Magn Reson Imaging 2010, 31: 192-203.View ArticleGoogle Scholar
- Azzabou N, Paragios N, Guichard F: Image denoising based on adapted dictionary computation. Proc Int Conf Image Processing 2007, III: 109-112.Google Scholar
- Coupe P, Yger P, Prima S, Hellier P, Kervrann C, Barillot C: An Optimized blockwise nonlocal means denoising filter for 3-D magnetic resonance images. IEEE Trans Med Imaging 2008, 27(4):425-441.View ArticleGoogle Scholar
- Tasdizen T: Principal components for non-local means image denoising. Proc Int Conf Image Processing 2008, 1728-1731.Google Scholar
- Tasdizen T: Principal neighborhood dictionaries for nonlocal means image denoising. IEEE Trans Image Process 2009, 18(12):2649-2660.View ArticleMathSciNetGoogle Scholar
- Horn JL: A rationale and test for the number of factors in factor analysis. Psycho-metrica 1965, 30(2):179-185.View ArticleGoogle Scholar
- Glorfeld LW: An improvement on horn's parallel analysis methodology for selecting the correct number of factor's to retain. Educ Psychol Meas 1995, 55(3):377-393.View ArticleGoogle Scholar
- Ledesma RD, Valero-Mora P: Determining the number of factors to retain in EFA:an easy-to-use computer program for carrying out parallel analysis. Practical Ass Res Eval 2007, 12(2):1-11.Google Scholar
- Sijbers J, den Dekker AJ, Scheunders P, Dyck DV: Maximum-likelihood estimation of Rician distribution parameters. IEEE Trans Med Imaging 1998, 17(3):357-361.View ArticleGoogle Scholar
- Sen PK: Estimates of regression coefficient based on Kendall's tau. J Am Stat Assoc 1968, 63: 1379-1389.View ArticleMATHGoogle Scholar
- Kaiser HF: The application of electronic computers to factor analysis. Educ Psychol Meas 1960, 20: 141-151.View ArticleGoogle Scholar
- Fabrigar LR, Wegener DT, MacCallum RC, Strahan EJ: Evaluating the use of exploratory factor analysis in psychological research. Psychol Methods 1999, 3: 272-299.View ArticleGoogle Scholar
- Zwick WR, Velicer WF: Comparison of five rules for determining the number of components to retain. Psychol Bull 1986, 99: 432-442.View ArticleGoogle Scholar
- Cattell RB: The scree test for the number of factors. Multivar Behav Res 1966, 1: 245-276.View ArticleGoogle Scholar
- Wang Z, Bovik AC, Sheikh HR, Simoncelli EP: Image quality assessment: From error visibility to structural similarity. IEEE Trans Image Process 2004, 13(4):600-612.View ArticleGoogle Scholar
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