Global denoising for 3D MRI
- Xi Wu^{1},
- Zhipeng Yang^{1},
- Jing Peng^{1} and
- Jiliu Zhou^{1}Email author
Received: 14 October 2015
Accepted: 28 April 2016
Published: 12 May 2016
Abstract
Background
Denoising is the primary preprocessing step for subsequent application of MRI. However, most commonly-used patch-based denoising methods are heavily dependent on the degree of patch matching. Due to the large number of voxels in the 3D MRI dataset, the procedure of searching sufficient similarity patches was limited by the empirical compromising between computational efficiency and estimation accuracy, and cannot fulfill the application in multimodal MRI dataset with different SNR and resolutions.
Methods
In this study, we propose a modified global filtering framework for 3D MRI. For each denoising voxel, the similarity weighting matrix is computed using the reference patch and other patches from the whole dataset. This large weighting matrix is then approximated using the k-means clustering Nyström method to achieve computational viability.
Results
Experiments on both synthetic and in vivo MRI datasets demonstrated that the proposed adaptive Nyström low-rank approximation could achieve competitive estimation compared with exact global filter while reducing the sampling rate by four orders of magnitude. In addition, the corresponding global filter improved patches-based method in both spatial and transform domain.
Conclusion
We propose a global denoising framework for 3D MRI which extracts information from the entire dataset to restore each voxel. This large weighting matrix of the global filter is approximated using Nyström low-rank approximation with an adaptive k-means clustering sampling scheme, which significantly reduce the sampling rate as well as the running time. The proposed method is capable of denoising in multimodal MRI dataset and can be used to improve currently used patch-based methods.
Keywords
Global denoising Nyström method 3D MRI k-means clusteringBackground
Magnetic resonance imaging is of ever increasing importance in clinical and research settings, owing to its accurate non-invasive 3D representation of the internal human structure. Nevertheless, MRI suffers significantly from noise arising during the acquisition procedure. Hence, denoising is used as the primary preprocessing step for subsequent clinical diagnosis and other applications, such as registration and segmentation.
A wide range of denoising methods have been proposed to estimate the latent image from noisy observations and many of them can be categorized as patch-based filters. In general, patch-based filters estimate each voxel in an MRI dataset using patches selected according to some predefined similarity criteria. For example, a non-local means filter [1] denoises images by minimizing the penalty term for the weighted distance between filtered image patches and other patches in the search window. This weighted distance was first defined as a decreasing function of the Euclidean distance between gray level intensities in the patches, and then adapted to various high-order operators to enhance the performance [2–5]. This strategy has also been implemented effectively in statistical methods, such as expectation–maximization-based Bayesian estimation [6] and maximum likelihood estimation [7] to promote more effective denoising of 3D MRI. On the other hand, based on the assumption that latent images have a sparse representation in some transform domain, numerous transforms have been proposed to filter noisy images. These include the discrete cosine transform (DCT) [8], principal component analysis [9], over-complete dictionaries [10], and high-order singular value decomposition [11]. The filtering process was implemented by applying a shrinkage function to the transform coefficients and recovering estimated patches with an inverse transform. For instance, the block matching in three dimensions (BM3D) method [8] collects a group of similar patches from the reference patches to construct a 3D array. After projecting the 3D array onto a 3D transform basis, the coefficients are truncated by a hard threshold and filtered patches are reconstructed by inversion of the transform.
Although promising results have been achieved, the performance of patch-based filters is still limited by the necessity of finding sufficiently similar patches. Specifically, if the similar patches cannot be effectively represented, the denoising process will be affected by mistaken truncating of the signal component or some other implementation due to a lack of redundancy provided by similar patches. Since the number of voxels of 3D MRI dataset is large, most of the currently used patch-based methods implemented a subset of the whole dataset as the searching window and compromise between computational efficiency and estimation accuracy empirically [2–4]. However, these predefined parameters cannot be used to fulfill various MRI dataset, especially when multimodal MRI has different signal to noise ratio (SNR) or resolutions.
Recently, another group of methods stems from the emerging theory of low-rank matrix completion, which is derived from compressed sensing theory [12], were proposed to accomplish the accurate restoration from the whole 3D dataset effectively. Low-rank matrix completion technology relies on self-similarities across different slices or frames in an MRI dataset to construct a low-rank matrix and demonstrates significant benefits for various MRI applications, including reconstruction from highly under-sampled k-t space data [13], super-resolution [14], and denoising [15–17]. Lin exploited the self-similarity between multi-channel coil images and formulated denoising as a non-smooth convex optimization problem [15]. Lam incorporated low-rank modeling and an edge-preserving smoothing algorithm prior to denoising MRI data in a unified mathematical framework [16]. Recently, Talebi and Milafar [18] proposed a promising global approach based on low-rank approximation to denoise natural scene images. Rather than empirically predefining a part of the image as a search window, this global filter introduced the whole dataset as a search window and implemented a low-rank Nyström extension to approximate the large weighting matrix for practical applications [19].
In this study, we propose a modified global filtering framework for 3D MRI. The proposed framework estimates denoised voxels based on a similarity weighting matrix which is computed between the reference patches and other patches located throughout the entire dataset. Meanwhile, the large similarity weighting matrix is low-rank approximated using the Nyström method [19] to achieve computational viability. The contributions of this work are as follows: firstly, the global filtering framework is extended and applied to 3D MRI datasets. More importantly, since the computational burden is prohibitively high due to the sophisticated structure of 3D MRI data and other MRI modalities such as diffusion weighted imaging (DWI) [20], a simple and efficient sampling scheme, k-means clustering, is introduced to improve the Nyström method for applicable implementations. This adaptive sampling scheme enhances the matrix estimation accuracy and effectively improves the computational efficiency. This is significantly beneficial for global filtering.
In the remainder of this paper, we describe our global filtering framework in detail, together with the adaptive Nyström low-rank approximation. After quantitative and qualitative comparisons of the proposed global framework for multimodal MRI datasets, we consider the advantages and limitations of our proposed technique.
Methods
General model of iterative global denoising
It should be noted: firstly, since the global framework implemented the whole dataset to obtain the optimal similarity weight matrix, this scheme will be improvable for currently used patch-based denoising methods. Secondly, the computational burden of the proposed global framework is prohibitively high due to 3D structure of MRI dataset. As a result, we have implemented the Nyström method [19] in the following section in order to ameliorate this problem. We then introduce a k-means clustering which further improves the global framework to achieve applicable implementation for 3D MRI.
K-means Nyström approximation for global filtering
As mentioned above, the proposed global filter demands extremely high computational and storage costs. Fortunately, as mentioned previously, the proposed global framework only requires a portion of the ‘best’ eigenvectors and eigenvalues for efficient matrix approximation, rather than computing every element of W. This strategy can be effectively achieved by the Nyström method.
The Nyström method offers an efficient way to generate a low-rank approximation of the original matrix from a subset of its columns [19, 25, 26]. Given a matrix M, the eigenvectors can be estimated numerically:
Let the l × l matrix M _{1} denote the similarity weights of voxels in the subimage W _{1}, which samples l voxels from the entire image. The (n − l) × (n − l) matrix M _{2} denotes the similarity weights of voxels in subimage W _{2}, which contains (n − l) unsampled voxels. The l × (n − l) matrix M _{12} denotes kernel weights between voxels in M _{1} and M _{2}. The similarity matrix M can then be defined as.
In addition to fixed sampling methods, various adaptive sampling schemes have also been proposed to produce efficient low-rank approximations. The sparse greedy matrix approximation [30, 31] involves matching a pursuit algorithm to a new sample, randomly selected from a subset in successive rounds. The incomplete Cholesky decomposition [28] generates low-rank factorization through adaptive selection of columns based on potential pivots. The K-means clustering method stores centroids, which are used to generate informative columns [32].
For an MRI dataset, the informative voxels are concentrated in only a portion of the columns and the number of background voxels is large. Motivated by this observation and the fact that k-means clustering can find a local minimum in the quantization error [33], we propose using centroid voxels obtained by k-means clustering as adaptive sampled voxels in the Nyström approximation. The reasons for choosing k-means clustering for adaptive voxel sampling are threefold and can be summarized as follows: firstly, the k-means clustering ensures most of the selected voxels are located in the informative parts of the dataset and excludes redundant sampling of voxels from the background area. Secondly, the centroids obtained by k-means clustering are distributed throughout the whole dynamic range, which is more representative of the whole dataset and thus improves the estimation accuracy. Thirdly, the k-means clustering itself has been proven to be more suitable for handling large datasets with minimal quantization errors [32]. Let k be the desired number of clusters: the larger k, the more accurate the approximation, together with higher computational burden. To explore the effectiveness of the proposed k-means clustering method, a quantitative comparison including uniform sampling and adaptive sampling, namely the sparse greedy matrix approximation and incomplete Cholesky decomposition, is proposed. The comparison was implemented using in-house MRI datasets, which included 35 different datasets from various parts of the human body. The performance of different sampling schemes using the Nyström approximation was evaluated using the relative accuracy defined in Ref. [19]:
Nyström reconstruction accuracy for various sampling methods for MRI datasets for three l/n percentages
l/n % | Uniform | ICL | SMGA | K-means |
---|---|---|---|---|
5 | 62.3 (0.6) | 67.7(0) | 73.8 (1.3) | 74.9 (0.8) |
10 | 73.6 (1.1) | 78.2(0) | 81.7 (1.5) | 86.2 (1.2) |
20 | 84.8 (1.7) | 83.4(0) | 89.6 (1.9) | 91.3 (1.1) |
Finally, W is orthogonalized to achieve the final denoising filter \({\tilde{\mathbf{W}}}\). Define:
This global filter \({\tilde{\mathbf{W}}}\) is approximated using the leading eigenvectors and eigenvalues and can be implemented for denoising using a small sampling rate which nearly matches the performance of the exact filter.
Experiments
The proposed framework was evaluated using state-of-the-art algorithms in both synthetic and in vivo MRI datasets. In our proposed global framework, both the non-local means [2] and BM3D [8] were implemented as baseline kernels in spatial and transform domain respectively. It should be noted that any patch-based denoising method could be used in our proposed global denoising framework. The k-means clustering scheme was implemented for the Nyström approximation and the sampling rate was set to 0.0005 %, which was defined and discussed in the following section. The non-local means filter and BM3D were implemented for comparison and the parameters were set as indicated in [2] and [8], respectively.
Two open-access datasets, namely BrainWeb (simulated images) [34] and the Internet Brain Segmentation Repository (IBSR—real images) [35] were selected for comparison. In addition, a real MRI dataset containing the spinal cord and a real diffusion-weighted MRI dataset were utilized to evaluate the adaptability and robustness of the proposed framework.
PSNR and SSIM of the compared methods for different MRI modalities and Rician noise levels
Noise level (%) | 1 | 3 | 5 | 7 | 9 | 11 | 13 | 15 | ||
---|---|---|---|---|---|---|---|---|---|---|
T1w | PSNR | NLM | 36.74 | 31.07 | 29.27 | 28.03 | 26.86 | 25.72 | 24.63 | 23.46 |
G-NLM | 36.23 | 32.11 | 29.16 | 28.93 | 27.34 | 27.01 | 25.85 | 24.37 | ||
BM3D | 43.40 | 36.65 | 33.48 | 31.26 | 29.26 | 27.47 | 25.97 | 24.40 | ||
G-BM3D | 42.56 | 36.78 | 33.32 | 32.23 | 30.67 | 29.26 | 26.78 | 26.12 | ||
SSIM | NLM | 0.9700 | 0.9042 | 0.8666 | 0.8319 | 0.7968 | 0.7617 | 0.7275 | 0.6907 | |
G-NLM | 0.9625 | 0.9041 | 0.8745 | 0.8437 | 0.8052 | 0.7861 | 0.7432 | 0.7231 | ||
BM3D | 0.9904 | 0.9632 | 0.9320 | 0.8993 | 0.8661 | 0.8326 | 0.8029 | 0.7697 | ||
G-BM3D | 0.9903 | 0.9638 | 0.9362 | 0.9016 | 0.8795 | 0.8563 | 0.8247 | 0.7968 | ||
T2w | PSNR | NLM | 33.37 | 27.52 | 24.43 | 22.99 | 22.16 | 21.55 | 20.99 | 20.42 |
G-NLM | 33.43 | 27.51 | 25.12 | 23.37 | 22.06 | 21.87 | 21,34 | 20.64 | ||
BM3D | 42.54 | 34.97 | 31.68 | 29.50 | 27.77 | 26.34 | 24.87 | 23.50 | ||
G-BM3D | 42.89 | 33.67 | 32.54 | 30.68 | 28.52 | 26.43 | 25.26 | 24.37 | ||
SSIM | NLM | 0.9677 | 0.9051 | 0.8637 | 0.8259 | 0.7850 | 0.7467 | 0.7078 | 0.6731 | |
G-NLM | 0.9658 | 0.9037 | 0.8764 | 0.8347 | 0.7938 | 0.7637 | 0.7286 | 0.7031 | ||
BM3D | 0.9898 | 0.9627 | 0.9334 | 0.9084 | 0.8768 | 0.8497 | 0.8201 | 0.7918 | ||
G-BM3D | 0.9884 | 0.9638 | 0.9467 | 0.9128 | 0.8856 | 0.8643 | 0.8432 | 0.8169 | ||
PDw | PSNR | NLM | 34.81 | 30.11 | 27.55 | 25.85 | 24.74 | 23.95 | 23.35 | 22.75 |
G-NLM | 36.78 | 32.22 | 29.53 | 26.84 | 24.97 | 24.65 | 24.36 | 23.77 | ||
BM3D | 43.43 | 36.40 | 33.28 | 31.27 | 29.62 | 28.25 | 27.10 | 25.86 | ||
G-BM3D | 43.56 | 37.37 | 34.56 | 32.78 | 30.62 | 30.17 | 28.96 | 26.77 | ||
SSIM | NLM | 0.9728 | 0.9075 | 0.8401 | 0.7856 | 0.7435 | 0.7088 | 0.6742 | 0.6463 | |
G-NLM | 0.9823 | 0.9234 | 0.8564 | 0.7958 | 0.7736 | 0.7297 | 0.7056 | 0.6573 | ||
BM3D | 0.9915 | 0.9665 | 0.9403 | 0.9144 | 0.8885 | 0.8678 | 0.8406 | 0.8163 | ||
G-BM3D | 0.9927 | 0.9711 | 0.9568 | 0.9347 | 0.8982 | 0.8768 | 0.8543 | 0.8256 |
Results
Figures 1, 2 and 3 demonstrate visual evaluation of the denoising results using different MRI modalities with a 15 % noise level. It can been observed that the proposed global denoising framework improves patch-based methods (whether non-local means or BM3D) in different MRI modalities. Specifically, denoising results for the global filter feature less intensity oscillation in homogenous areas compared with the original patch-based method. This finding can be more clearly observed in residual images. Moreover, the global filter can improve denoising of patch-based methods when there are high levels of noise. This finding is also in agreement with the quantitative comparison shown in Table 1, which shows that the peak signal-to-noise ratio and the structural similarity index achieved by the global framework always achieve the best results. Finally, both edge and feature preservation can also benefit from this global filter. It can clearly be observed that the residual images in the global filter contain less structural information compared with residuals from the original NLM or BM3D. This coincided with all modalities including T1w, T2w, and PDw.
Figure 4 shows denoising results using the proposed global filter with a real MRI dataset from IBSR. Since the original images already contain noise, neither ideal residual images nor quantitative results can be obtained. As demonstrated with the synthetic dataset, the proposed global filter attains good results in terms of both noise removal and edge preservation. Similar to the results with the synthetic dataset, the denoising results from the global filter demonstrate a stable intensity in the central region of the target tissue and sharper edges. This can be better observed from line profiles which have smaller amplitude differences near the edges and attenuate the visible artifacts manifested in the image.
Figure 5 demonstrates the consistency of the proposed global framework for a real spinal cord MRI dataset. Upon close inspection of the central area, it can be seen that the denoised results using the proposed method include a more explicit boundary, which matches the results obtained with the synthetic datasets. The segmentation results in the denoised image, which produce smoother contours in the spinal cord image, may be beneficial to further applications.
Figures 6 and 7 demonstrate the effect of shape and orientation information on the diffusion tensor computed using the denoised results in diffusion-weighted imaging. The original image of the PD orientation map and the fractional anisotropy contain obvious artifacts. Compared with the results obtained using the non-local means and BM3D methods, the proposed global framework exhibits visually significant improvements in both PD and fractional anisotropy. For the two methods using the proposed global framework, the results appear smoother and preserve more fine structural detail.
Discussion
We proposed a global filtering framework for 3D MRI. Compared with commonly used patch-based methods, the proposed global filter uses all of the voxels in an input image to denoise every single voxel. Since the involvement of all voxels in the 3D MRI dataset results in a prohibitively high computational burden, we implemented a k-means clustering Nyström method to produce a low-rank approximation in order to achieve a viable algorithm.
The primary concern for efficiently generating a low-rank matrix approximation using the Nyström method is the sampling scheme. In this study, a k-means clustering algorithm was proposed as an adaptive sampling scheme, which effectively improved both estimation accuracy and computational efficiency. Compared with commonly-used fixed sampling schemes, which acquire voxels from the whole dataset, the k-means clustering method excludes redundant sampling of background voxels which form a large portion of the MRI dataset and concentrates the sampling in the informative area. Moreover, since the k-means clustering obtained centroids covering the whole dynamic range of gray levels, it ensures a fully representative selection of voxels for the Nyström approximation. From this point, a more elegant clustering algorithm, together with more effective clustering criteria, retains the potential for further improvements.
Running time (in seconds) of the compared methods
Volume size | NLM | G-NLM | BM3D | G-BM3D |
---|---|---|---|---|
181 × 217 × 181 | 2412.5 | 1091.3 | 845.9 | 487.6 |
Conclusions
We have proposed a global filtering framework for a 3D MRI dataset, which used the whole dataset to denoise each voxel. Due to its size, the large global denoising weight matrix was low-rank approximated using the Nyström method. In addition, k-means clustering was implemented as an adaptive sampling scheme to further improve the approximation accuracy and computational efficiency. The proposed global filtering framework is not restricted to any specific patch-based algorithm and our results show that it can be used to improve most patch-based methods.
Declarations
Authors’ contributions
XW conceived and designed this study and is responsible for the manuscript, ZY and JP participated in the analysis of the DWI dataset and real dataset respectively, JZ participated in the conception and design of this work and helped to draft the manuscript. All authors read and approved the final manuscript.
Acknowledgements
This work was supported by the National Basic Research Program of China (973 Program) 2014CB360506, National Science Foundation of China (Grant No. 61303126).
Competing interests
The authors declare that they have no competing interests.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The Creative Commons Public Domain Dedication waiver (http://creativecommons.org/publicdomain/zero/1.0/) applies to the data made available in this article, unless otherwise stated.
Authors’ Affiliations
References
- Buades A, Coll B, Morel J-M. A review of image denoising algorithms, with a new one. Multiscale Model Simul. 2005;4(2):490–530.MathSciNetView ArticleMATHGoogle Scholar
- Manjón JV, Carbonell-Caballero J, Lull JJ, et al. MRI denoising using non-local means. Med Image Anal. 2008;12:514–23.View ArticleGoogle Scholar
- Manjón JV, Coupé P, Buades A, et al. New methods for MRI denoising based on sparseness and self-similarity. Med Image Anal. 2012;16(1):18–27.View ArticleGoogle Scholar
- Coupé P, Yger P, Prima S, et al. An optimized blockwise nonlocal means denoising filter for 3-D magnetic resonance images. IEEE Trans Med Imaging. 2008;27(4):425–41.View ArticleGoogle Scholar
- Xi W, Shujuan L, Wu M, et al. Non-local denoising using anisotropic structure tensor for 3D MRI. Med Phys. 2013;40:101904.View ArticleGoogle Scholar
- Awate S, Whitaker R. Feature-preserving MRI denoising: a nonparametric empirical Bayes approach. IEEE Trans Med Imag. 2007;26(9):1242–55.View ArticleGoogle Scholar
- He L, Greenshields IR. A nonlocal maximum likelihood estimation method for Rician noise reduction in MR images. IEEE Trans Med Imag. 2009;28:165–72.View ArticleGoogle Scholar
- Dabov K, Foi A, Katkovnik V, et al. Image denoising by sparse 3-D transform-domain collaborative filtering. IEEE Trans Image Process. 2007;16(8):2080–95.MathSciNetView ArticleGoogle Scholar
- Manjon J, Coupe P, Concha L, et al. Diffusion weighted image denoising using overcomplete local PCA. PLoS ONE. 2013;8(9):e73021.View ArticleGoogle Scholar
- Elad M, Aharon M. Image denoising via sparse and redundant representations over learned dictionaries. IEEE Trans Image Process. 2006;15(12):3736–45.MathSciNetView ArticleGoogle Scholar
- Zhang X, Xu Z, Jia N, Yang W, Feng Q, Chen W, Feng Y. Denoising of 3D magnetic resonance images by using higher-order singular value decomposition. Med Imag Anal. 2015;19:75–86.View ArticleGoogle Scholar
- Donoho DL. Compressed sensing. IEEE Trans Inf Theory. 2006;52(4):1289–306.MathSciNetView ArticleMATHGoogle Scholar
- Lingala SG, Hu Y, DiBella E, Jacob M. Accelerated dynamic MRI exploiting sparsity and low-rank structure: k-t SLR. IEEE Trans Med Imag. 2011;30(5):1042–54.View ArticleGoogle Scholar
- Shi F, Cheng J, Wang L, et al. LRTV: MR image super-resolution with low-rank and total variation regularizations. IEEE Tran Med Imag, 2015: 1.Google Scholar
- Fan L, Babacan SD, Haldar JP, et al. Denoising diffusion-weighted magnitude mr images using rank and edge constraints. Magn Reson Med. 2014;71:1272–84.View ArticleGoogle Scholar
- Xu L, Wang C, Chen W, et al. Denoising multi-channel images in parallel MRI by low rank matrix decomposition. IEEE Trans Appl Supercond. 2014;24(5):1–5.MathSciNetGoogle Scholar
- Nguyen HM, Xi P, Do MN, et al. Denoising MR spectroscopic imaging data with low-rank approximations. IEEE Trans Bio-Med Eng. 2013;60(1):78–89.View ArticleGoogle Scholar
- Talebi H, Milanfar P. Global image denoising. IEEE Trans Image Process. 2014;23(2):755–68.MathSciNetView ArticleGoogle Scholar
- Kumar S, Mohri M, Talwalkar A. Sampling methods for the Nyström method. J Mach Learn Res. 2012;13:981–1006.MathSciNetMATHGoogle Scholar
- Johansen-Berg H, Behrens T. Diffusion MRI, 2ed. Cambridge: Academic Press; 2013.Google Scholar
- Milanfar P. A tour of modern image filtering: new insights and methods, both practical and theoretical. IEEE Signal Process Mag. 2013;30(1):106–28.MathSciNetView ArticleGoogle Scholar
- Foi A. “Noise estimation and removal in MR imaging: the variance stabilization approach,” In: 2011 IEEE International symposium on biomedical imaging: from nano to macro, p. 1809–14, 2011.Google Scholar
- Aja-Fernández S, Alberola-López C, Westin C-F. Noise and signal estimation in magnitude MRI and Rician distributed images: a LMMSE approach. IEEE Trans. Image Process. 2008;17(8):1383–98.MathSciNetView ArticleGoogle Scholar
- Maggioni M, Katkovnik V, Egiazarian K, Foi A. Nonlocal transform-domain filter for volumetric data denoising and reconstruction. IEEE Trans Image Process. 2013;22:119–33.MathSciNetView ArticleGoogle Scholar
- Williams C, Seeger M, “The effect of the input density distribution on kernel-based classifiers,” Proceedings of the 17th international conference on machine learning, pp. 1159–66, 2000.Google Scholar
- Kumar S, Mohri M, Talwalkar A. “Sampling techniques for the Nyström method”. In Conference on artificial intelligence and statistics, 2009.Google Scholar
- Drineas P, Ravi K, Mahoney MW. Fast Monte Carlo algorithms for matrices ii: computing a low-rank approximation to a matrix. SIAM J Comput. 2006;36(1):158–83.MathSciNetView ArticleMATHGoogle Scholar
- Drineas P, Mahoney MW. On the Nyström method for approximating a Gram matrix for improved kernel-based learning. J Mach Learn Res. 2005;6:2153–75.MathSciNetMATHGoogle Scholar
- Belabbas MA, Wolfe PJ. On landmark selection and sampling in highdimensional data analysis. Philos Trans: Math, Phys Eng Sci. 2009;367(1906):4295–312.MathSciNetView ArticleMATHGoogle Scholar
- Smola AJ, Bernhard S. “Sparse greedy matrix approximation for machine learning,” In International conference on machine learning, 2000.Google Scholar
- Shai F, Katya S. Efficient svm training using low-rank kernel representations. J Mach Learn Res. 2002;2:243–64.MATHGoogle Scholar
- Zhang K, Tsang I, Kwok J, “Improved Nystrom low-rank approximation and error analysis,” ICML; 2008. p. 1232–39.Google Scholar
- Gersho A, Gray R. Vector quantization and signal compression. Boston: Kluwer Academic Press; 1992.View ArticleMATHGoogle Scholar
- Kwan RK, Evans AC, Pike GB. MRI simulation-based evaluation of image-processing and classification methods. IEEE Trans Med Imaging. 1999;18:1085–97.View ArticleGoogle Scholar
- Worth AJ. MGH CMA internet brain segmentation repository (IBSR). 2010. http://www.cma.mgh.harvard.edu/ibsr/
- Zhou W, Bovik AC, Sheikh HR, et al. Image quality assessment: from error visibility to structural similarity. IEEE Trans Image Process. 2004;13:600–12.View ArticleGoogle Scholar
- Aja-Fernández S, Alberola-López C, Westin C-F. Noise and signal estimation in magnitude MRI and RICIAN distributed images: a LMMSE approach. IEEE Trans Image Process. 2008;17:1383–98.MathSciNetView ArticleGoogle Scholar
- Gonzalez RC, Woods RE. Digital Image Processing. 2nd ed. NJ: Prentice Hall; 2002.Google Scholar