- Open Access
Optshrink LR + S: accelerated fMRI reconstruction using non-convex optimal singular value shrinkage
- Priya Aggarwal^{1}Email author,
- Parth Shrivastava^{1},
- Tanay Kabra^{1} and
- Anubha Gupta^{1}
- Received: 9 February 2016
- Accepted: 23 December 2016
- Published: 10 January 2017
Abstract
This paper presents a new accelerated fMRI reconstruction method, namely, OptShrink LR + S method that reconstructs undersampled fMRI data using a linear combination of low-rank and sparse components. The low-rank component has been estimated using non-convex optimal singular value shrinkage algorithm, while the sparse component has been estimated using convex l ^{1} minimization. The performance of the proposed method is compared with the existing state-of-the-art algorithms on real fMRI dataset. The proposed OptShrink LR + S method yields good qualitative and quantitative results.
Keywords
- Accelerated functional MRI
- Low-rank recovery
- Sparse recovery
- Compressed sensing
- k–t acceleration
- Undersampling
1 Introduction
Functional magnetic resonance imaging (fMRI) is one of the most significant noninvasive and non-ionizing diagnostic imaging modality [1, 2]. It measures blood oxygenated level dependent (BOLD) signal for localizing brain activity [3]. However, despite the advancements in fMRI scanners, one of the biggest limitations of fMRI modality is slow imaging compared to the other medical imaging modalities [4].
Conventionally, parallel imaging techniques such as Sensitivity Encoding (SENSE) [5, 6], Generalized Autocalibrating Partially Parallel Acquisitions (GRAPPA) [7, 8], and Simultaneous Acquisition of Spatial Harmonics (SMASH) [9] are used to accelerate magnetic resonance imaging (MRI). Here, the basic principle involves use of multiple receiver coils with complementary sensitivity information. SENSE, GRAPPA, or SMASH reconstruct MRI images from multiple k-space undersampled images acquired on different coils. For the case of fMRI, only k–t GRAPPA is able to accurately reconstruct fMRI images [10]. However, this method introduces strong temporal autocorrelation in the data that limits the extent of undersampling of fMRI data [10].
Apart from parallel imaging, compressed sensing (CS)-based fMRI reconstruction is another attractive method of fMRI acceleration [11–17]. Similar to parallel imaging, less data are acquired in the k-space (spatial frequency domain) in CS resulting in accelerated fMRI data acquisition. However, unlike GRAPPA and SENSE that does not exploit information contained across time frames, CS exploits information across time frames leading to sparse representation and hence provides good reconstruction quality. Reconstruction of full fMRI data from this less or undersampled data requires efficient reconstruction algorithm. Researchers have proposed various methods for efficient reconstruction from undersampled k-space measurements [11–17]. These methods largely rely on compressive sensing and reconstruct data using an optimization framework under certain constraints. Often, fMRI data are assumed to be sparse in some transform domain. Theoretical studies have shown that it is possible to recover sparse signals by l ^{1} norm minimization [18]. For example, in [13], undersampled fMRI data are reconstructed using CS with sparsity of fMRI data in the wavelet domain, wherein orthogonal Daubechies wavelet is used as the sparsifying basis. This is to note that CS-based sparse recovery methods are being used extensively in many applications including other medical imaging modalities [19, 20] and in videos [21, 22].
In general, fMRI data matrix \({\mathbf{X }}\), i.e., one fMRI slice data stacked over time, is observed to be low rank. Hence, low-rank constraint can be imposed in the CS-based optimization framework to recover fMRI data slice by slice. Recently, k–t FASTER method has been proposed on similar lines that recovers fMRI signal via hard thresholding of singular values of low-rank data matrix X in the CS framework [11].
For quality accelerated fMRI reconstruction in noisy settings, improved low-rank matrix and sparse matrix estimation are necessary. There has been a great interest to recover low-rank matrix from noisy measurements in various fields such as statistical signal processing [26–28], machine learning [29], and estimation and classification problems [30]. This motivates us to explore an improved method of accelerated fMRI reconstruction that can recover denoised low-rank matrix and sparse component from the undersampled k-space data.
We use optimal singular value shrinkage denoising algorithm (OptShrink), a data-driven method, recently used for denoising of low-rank matrix [31]. We call the proposed method as Optshrink LR + S method. In [31], OptShrink has been shown to have improved performance over singular value thresholding (SVT) in the recovery of data with missing entries. The OptShrink method requires noisy low-rank matrix and its rank estimate as input and provides denoised low-rank matrix estimate.
The proposed Optshrink LR + S fMRI reconstruction method is compared with other offline fMRI reconstruction methods such as direct inverse Fourier transform (IFT), LR + S [15], and CS with wavelet sparsity [13] methods. We compare reconstruction results using different methods at both the subject- and group level at different acceleration factors. Our proposed OptShrink LR + S method reconstructs fMRI data with greater accuracy compared to other methods even at lower sampling ratios.
The rest of this paper is organized as follows. Section 2 discusses fMRI reconstruction problem and presents the proposed Optshrink LR + S reconstruction method. In Sect. 3, simulation results using Optshrink LR + S and some of the existing methods are presented on real fMRI data. Conclusions are presented in the last section.
2 Materials and methods
In this section, we present the mathematical formulation of fMRI reconstruction problem followed by details of the proposed Optshrink LR + S fMRI reconstruction method and description of the fMRI dataset used in simulations.
2.1 Problem formulation
The functional MRI imaging involves acquisition of contiguous brain slices over a number of time points. For each individual brain slice, Casorati matrix \({\mathbf{X }}\in {\mathbb {R}}^{n\times T}\) is formed by stacking one brain slice over all time points [32], i.e., \({\mathbf{X }}=\left\{ {\mathbf{x }}_{i},i=1,\ldots ,T \right\}\), where T is the number of time points and n is the number of voxels in one brain slice. Hence, each column \({\mathbf{x }}_{i}\) of \({\mathbf{X }}\) corresponds to data of a particular brain slice captured at one time point. Let us denote the undersampled k-space fMRI data of one brain slice captured over time by the matrix \({\mathbf{Y }}\).
2.2 Reconstruction using low-rank plus sparse decomposition
In this paper, we are interested in accelerated fMRI data reconstruction using low-rank plus sparse decomposition. Hence, in this subsection we first elucidate low-rank plus sparse reconstruction problem.
2.3 Proposed Optshrink LR + S method
In this subsection, we explain the proposed Optshrink LR + S reconstruction method wherein the problem in (3) is solved by breaking it into two subproblems of estimating \({\mathbf{L }}\) and \({\mathbf{S}}\) as described below.
2.3.1 S subproblem
2.3.2 L subproblem
Low-rank matrix recovery is ill-posed and NP hard [35]. One of the methods to solve this problem is via convex optimization using nuclear norm minimization [35]. Nuclear norm minimization implies \(l^{1}\) penalty on singular values of matrix L that supports matrix L to be low rank. Global minimum of convex nuclear norm minimization is obtained by soft thresholding on singular values, known as singular value thresholding (SVT) [36].
Recently, in [15], low-rank matrix is recovered using SVT, where noisy input low-rank matrix is initialized from the previous iteration. The key idea behind SVT is to shrink nonsignificant singular values toward zero while keeping the singular vectors unchanged. However, nuclear norm minimization is an over-relaxing recovery solution of low-rank matrix [37].
This algorithm is named as Optshrink [31]. OptShrink is a data-driven method, recently used for denoising of low-rank matrix in an application of signal recovery in missing data. It considers noisy low-rank matrix and its rank estimate [= m in (7)] as input, and provides denoised estimate of the low-rank matrix as the output. It is a non-convex solution that does weighing of singular vectors. It shrinks the corresponding singular values using truncated singular value decomposition (TSVD) and hence is called non-convex optimal SVT. This algorithm works better than SVT [31].
Also, in [31], it has been shown that the solution of Optshrink is quite robust to input rank specification, and hence a rough estimate of rank [= m in (7)] at the input is sufficient. Another advantage of Optshrink is that there is no need to specify shrinkage parameter as is required in SVT [refer to \(\lambda _{2}\) in (6)]. In SVT, we need to tune \(\lambda _{2}\) for every dataset. It has been observed that Optshrink always outperforms SVT in the estimation of low-rank matrix.
In this paper, we propose to apply OptShrink for low-rank matrix estimation in fMRI reconstruction using low-rank plus sparse decomposition. fMRI data inherently have low signal-to-noise ratio (SNR). Hence, fMRI reconstruction with OptShrink for denoised low-rank matrix estimation should outperform existing low-rank plus sparse fMRI reconstruction method [15].
2.3.3 Overall solution of (3) using Optshrink LR + S
2.4 Dataset description
To assess the performance of reconstruction methods, we have used two fMRI datasets in this paper: (1) Task-based fMRI dataset with false belief task (OpenfMRI publicly available dataset)^{1} and (2) Resting-state Baltimore fMRI dataset (1000 functional Connectomes Project Data).^{2}
2.4.1 Task-based dataset
This dataset consists of acquisition of 36 axial interleaved brain slices with dimensions 72x72 at each time point with echo time (TE) equal to 35 ms and repetition time (TR) equal to 2 s [38]. These data are collected over 179 time points, resulting in the matrix \({\mathbf{X }}\) of size \(5184\times 179\) for one brain slice. During the false belief experiment, the subject had to answer questions about stories that referred either to a person’s false belief (mental trials) or to outdated physical representations such as an old photograph. For more details on this dataset, please refer to [38].
2.4.2 Resting-state dataset
These data are publicly available as part of the 1000 Functional Connectomes Project. This is a collection of resting-state fMRI dataset from a number of laboratories around the world. We use Baltimore resting fMRI data. This dataset consists of 23 subjects resting-state fMRI data, aged between 20 and 40 years of age, acquired while subjects’ eyes were open and fixated on a screen. The repetition time (TR) is 2.5 s, size of a brain volume at one time point is \(96\times 96\times 47\), and the total no. of time points over which data are captured is 123.
3 Simulation results
This is to note that we have illustrated undersampling of fMRI data by retrospective sampling on the Cartesian grid because it allows sampling patterns to maintain incoherency among the columns of matrix X [39–41]. However, radial-Cartesian sampling grid is more realistic from the point of view of actual data acquisition [10, 42]. Similarly in [12], variable density spiral sampling pattern has been used in MRI scanner and is shown to be robust against motion, off resonance, and gradients artifacts in compressed sensing fMRI application. However, our work is focused on development of robust reconstruction algorithm. This is to note that the proposed Optshrink LR + S method reconstructs fMRI data as a superposition of low-rank and sparse matrix, where the low-rank component represents background information that is highly correlated across data captured at different time points and sparse component represents the dynamic and uncorrelated part. Since these assumptions are characteristic of fMRI data, they will remain valid irrespective of the sampling strategy used. Hence, although the proposed work is general and can be used with any sampling pattern provided sampling incoherence is maintained, we project the use of realistic sampling patterns as the future work.
Data obtained from the database are called as original data in the manuscript. k-space data are acquired by considering 2-D Fourier transform of this original data. Since the original data are real and are provided without any phase information, we only considered the magnitude part of the reconstructed data. Thus, no assumption is made about the phase part of the data. This is to clarify that this is a standard method of testing newer MRI/fMRI reconstruction algorithms via simulation results.
3.1 Comparison with different methods
In this section, we provide results on fMRI reconstruction from undersampled k-space fMRI data using the proposed Optshrink LR + S method, existing LR + S method [15], direct inverse Fourier transform-based reconstruction, and reconstruction using CS with wavelet sparsity [13]. Below, we present a brief overview of each of these existing reconstruction methods used.
3.1.1 Low-rank plus sparse (LR + S) method [15]
We empirically selected \(\lambda _{1}=2\) and \(\lambda _{2}=200\) in the above Eq. (14) using the L-curve method [43]. Minimum normalized mean square error (NMSE) is obtained in the L-curve at the above chosen \(\lambda\)s for the existing LR + S method. This is to note that we used same values of \(\lambda\)s in the proposed Optshrink LR + S method. Thus, the values of \(\lambda\)s are optimally selected for the existing LR + S method and not for the proposed Optshrink method for presenting the comparative results.
3.1.2 Direct IFT
3.1.3 CS with wavelet sparsity (CSWD) [13]
- 1.
Slices reconstructed using LR + S method show a decline in quality with decrease in the number of radial sampling lines. On the other hand, reconstruction results with the proposed Optshrink LR + S method are quite consistent and the reconstruction quality does not fall by a great deal with the reduction in number of sampling lines.
- 2.
Slices reconstructed using different radial sampling patterns consistently show that LR + S method output is blurred and the slices have artifacts at the center and the boundary compared to the proposed Optshrink LR + S. This observation indicates that there is SNR loss with LR + S method that may lead to incorrect brain activation detection. On the other hand, slices reconstructed using the proposed Optshrink LR + S method are very clear and free of artifacts.
- 3.
Reconstruction using Optshrink LR + S method is robust to input rank specification. Hence, rank definition is not a bottleneck in the proposed Optshrink LR + S.
All the above observations indicate that we can reconstruct fMRI data with greater quality by sampling much lesser measurements in k–t space with the proposed Optshrink LR + S method compared to the existing LR + S method. Hence, higher acceleration rate is possible with Optshrink LR + S method.
Moreover, Optshrink LR + S results are similar with different input rank specification. These results are in line with the qualitative results observed with the reconstructed slice quality in Figs. 3 and 4. This is to note that the proposed Optshrink LR + S method estimates a denoised version of low-rank matrix and hence yields better results.
3.2 Group-level analysis
Reconstruction results with different methods on two datasets
Dataset | Method | NMSE | PSNR | ||||
---|---|---|---|---|---|---|---|
6 lines | 12 lines | 24 lines | 6 lines | 12 lines | 24 lines | ||
Task-based dataset | Direct IFT | 0.3088 | 0.2177 | 0.1595 | 3.54 | 6.58 | 9.63 |
CS with wavelet sparsity [13] | 0.2435 | 0.219 | 0.138 | 4.828 | 7.79 | 12.07 | |
LR + S [15] | 0.1992 | 0.1215 | 0.0699 | 6.583 | 10.87 | 15.67 | |
Proposed Optshrink LR + S (r = 1) | 0.0497 | 0.0442 | 0.0401 | 18.62 | 19.65 | 20.49 | |
Proposed Optshrink LR + S (r = 2) | 0.0501 | 0.0437 | 0.0401 | 18.571 | 19.76 | 20.49 | |
Proposed Optshrink LR + S (r = 3) | 0.0496 | 0.0435 | 0.0401 | 18.649 | 19.78 | 20.49 | |
Resting-state dataset | Direct IFT | 0.4067 | 0.286 | 0.1979 | 2.93 | 6.09 | 9.415 |
CS with wavelet sparsity [13] | 0.2917 | 0.2054 | 0.1193 | 5.42 | 8.465 | 13.19 | |
LR + S [15] | 0.2351 | 0.1198 | 0.0576 | 7.321 | 13.16 | 19.52 | |
Proposed Optshrink LR + S (r = 1) | 0.0469 | 0.0359 | 0.031 | 21.32 | 23.64 | 24.91 | |
Proposed Optshrink LR + S (r = 2) | 0.0462 | 0.036 | 0.0311 | 21.44 | 23.62 | 24.88 | |
Proposed Optshrink LR + S (r = 3) | 0.0473 | 0.0364 | 0.0312 | 21.24 | 23.52 | 24.85 |
3.3 Subject-level statistical analysis on activation maps
In this section, we would like to study the effectiveness of Optshrink LR + S method with reference to brain activation detection. To this end, reconstruction is performed on the task-based dataset (false belief task) using LR + S method and Optshrink LR + S (rank = 1) method. Preprocessing of the original and the reconstructed fMRI dataset are performed using SPM12.^{3} We performed motion correction that is used to suppress motion-related artifacts. In general, motion correction is followed by smoothing as a preprocessing step so that the noise is Gaussian-distributed (by Central Limit Theorem). This establishes the validity of statistical tests using general linear model (GLM)-based analysis, a univariate approach, used for detecting brain activation in task-based fMRI data [45]. Since Optshrink LR + S method is supposed to provide denoised low-rank matrix, we tested the robustness of the proposed method on brain activation detection both with and without smoothing in the preprocessing pipeline.
Statistical analysis results for uncorrected
Method | 6 lines | 12 lines | 24 lines | |||||||
---|---|---|---|---|---|---|---|---|---|---|
Cluster size | Z score | MNI position | Cluster size | Z score | MNI position | Cluster size | Z score | MNI position | ||
1 | LR + S [15] (without smoothing) | 12 | 3.7 | 23 −31 62 | 14 | 3 | −6 −13 29 | 36 | 4.06 | 45 −24 34 |
2 | LR + S [15] (with smoothing) | 93 | 2.41 | 58 14 15 | 50 | 3.74 | 54 −24 35 | 122 | 4.56 | −57 −30 28 |
3 | Proposed Optshrink LR + S (r = 1) (without smoothing) | 35 | 4.94 | 39 −20 34 | 57 | 5.64 | −9 −16 33 | 42 | 4.52 | 42 −23 34 |
4 | Proposed Optshrink LR + S (r = 1) [with smoothing (FWHM = 6 mm)] | 218 | 4.93 | −54 −24 24 | 204 | 4.34 | −54 −27 28 | 162 | 4.6 | −54 −27 28 |
3.4 Reproducibility of resting-state networks
In this section, we test the efficacy of the proposed Optshrink LR + S method on resting-state fMRI dataset. We evaluate performance in terms of reproducibility of resting-state networks (RSNs). We compare RSNs of Optshrink LR + S-based reconstructed data with the RSNs obtained from the fully sampled original fMRI data that is considered as the ground truth. RSNs are identified using the spatial independent component analysis (ICA) of GIFT toolbox.^{4}
Before ICA is applied, data are preprocessed. The first five fMRI brain volumes are discarded followed by slice-time correction. Next, realignment is done for motion correction followed by spatial normalization onto the Montreal Neurological Institute (MNI) space (3-mm isotropic voxels). In the end, brain volumes are spatially smoothed with a Gaussian kernel [full width half maximum (FWHM) = 6 mm].
After preprocessing, we utilize InfomaxICA algorithm in GIFT to obtain 100 independent spatial components. We identified 54 RSNs from mean maps of all five fully sampled ground truth fMRI data (corresponding to five subjects) after removing artifact components. These RSNs can be broadly categorized into 10 RSNs: (1) Visual network (VN), (2) Somatomotor network (SMN), (3) Limbic network (LN), (4) Dorsal attention network (DAN), (5) Ventral attention network (VAN), (6) Default mode network (DMN), (7) Frontoparietal network (FPN), (8) Temporal + Frontal network (TFN), (9) Subcortical network (SCN), and (10) Cerebellar network (CN). We also ran ICA on the reconstructed data. These dataset are reconstructed using 16.49% (12 radial lines) acquired samples in k-space using Optshrink LR + S with rank one. We identified 56 RSNs from mean spatial components. These RSNs can be further classified into various categories as mentioned above.
4 Conclusion
In this paper, we have proposed a new accelerated fMRI method, named Optshrink LR + S method, for fMRI reconstruction from undersampled k–t space data. The proposed method exploits sparsity and low-rank decomposition with denoising to improve fMRI reconstruction accuracy. Comparison results demonstrate that the reconstruction performance of the proposed Optshrink LR + S method is superior to existing methods at various acceleration factors. While the performance of the existing methods falls rapidly at faster acceleration rates, Optshrink LR + S method performs consistently. Quantitative and qualitative results, group-level and subject-level analyses, show the superior performance of the proposed method. In addition, Optshrink LR + S method provides enhanced brain activation maps that is an added but most useful advantage of the proposed method. MATLAB implementation of proposed algorithm is available online.^{5}
Declarations
Acknowledgements
The first author would like to thank Visvesvaraya research fellowship, Department of Electronics and Information Technology, Ministry of Communication and IT, Government of India, for providing financial support for this work.
Compliance with ethical standards
Conflicts of interest
On behalf of all authors, the corresponding author states that there is no conflict of interest.
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.
Authors’ Affiliations
References
- Feinberg DA, Yacoub E (2012) The rapid development of high speed, resolution and precision in fMRI. NeuroImage 62(2):720–725. doi:10.1016/j.neuroimage.2012.01.049 20 years of fMRI20 years of fMRIView ArticleGoogle Scholar
- Ogawa S, Lee TM, Kay AR, Tank DW (1990) Brain magnetic resonance imaging with contrast dependent on blood oxygenation. Proc Natl Acad Sci USA 87(24):9868–9872 2124706[pmid]View ArticleGoogle Scholar
- Worsley K, Friston K (1995) Analysis of fMRI time-series revisited—again. NeuroImage 2(3):173–181. doi:10.1006/nimg.1995.1023 View ArticleGoogle Scholar
- Frank LR, Buxton RB, Wong EC (2001) Estimation of respiration-induced noise fluctuations from undersampled multislice fMRI data. Magn Reson Med 45(4):635–644. doi:10.1002/mrm.1086 View ArticleGoogle Scholar
- Pruessmann KP, Weiger M, Scheidegger MB, Boesiger P et al (1999) Sense: sensitivity encoding for fast MRI. Magn Reson Med 42(5):952–962View ArticleGoogle Scholar
- Tsao J, Boesiger P, Pruessmann KP (2003) k–t blast and k–t sense: dynamic MRI with high frame rate exploiting spatiotemporal correlations. Magn Reson Med 50(5):1031–1042View ArticleGoogle Scholar
- Griswold MA, Jakob PM, Heidemann RM, Nittka M, Jellus V, Wang J, Kiefer B, Haase A (2002) Generalized autocalibrating partially parallel acquisitions (grappa). Magn Reson Med 47(6):1202–1210View ArticleGoogle Scholar
- Huang F, Akao J, Vijayakumar S, Duensing GR, Limkeman M (2005) k–t grappa: a k-space implementation for dynamic MRI with high reduction factor. Magn Reson Med 54(5):1172–1184View ArticleGoogle Scholar
- Sodickson DK, Manning WJ (1997) Simultaneous acquisition of spatial harmonics (smash): fast imaging with radiofrequency coil arrays. Magn Reson Med 38(4):591–603View ArticleGoogle Scholar
- Chiew M, Graedel NN, McNab JA, Smith SM, Miller KL (2016) Accelerating functional MRI using fixed-rank approximations and radial-cartesian sampling. Magn Reson Med 76:1825–1836View ArticleGoogle Scholar
- Chiew M, Smith SM, Koopmans PJ, Graedel NN, Blumensath T, Miller KL (2015) k–t faster: acceleration of functional MRI data acquisition using low rank constraints. Magn Reson Med 74(2):353–364. doi:10.1002/mrm.25395 View ArticleGoogle Scholar
- Fang Z, Van Le N, Choy M, Lee JH (2015) High spatial resolution compressed sensing (hsparse) functional MRI. Magn Reson Med 76:440–455View ArticleGoogle Scholar
- Holland DJ, Liu C, Song X, Mazerolle EL, Stevens MT, Sederman AJ, Gladden LF, D’Arcy RCN, Bowen CV, Beyea SD (2013) Compressed sensing reconstruction improves sensitivity of variable density spiral fMRI. Magn Reson Med 70(6):1634–1643. doi:10.1002/mrm.24621 View ArticleGoogle Scholar
- Lu W, Li T, Atkinson I, Vaswani N (2011) Modified-cs-residual for recursive reconstruction of highly undersampled functional MRI sequences. In: Image Processing (ICIP), 2011 18th IEEE International Conference on, pp 2689–2692. doi:10.1109/ICIP.2011.6116222
- Singh V, Tewfik A, Ress D (2015) Under-sampled functional MRI using low-rank plus sparse matrix decomposition. In: Acoustics, Speech and Signal Processing (ICASSP), 2015 IEEE International Conference on, pp 897–901. doi:10.1109/ICASSP.2015.7178099
- Yan S, Nie L, Wu C, Guo Y (2014) Linear dynamic sparse modelling for functional MR imaging. Brain Inform 1(1–4):11–18. doi:10.1007/s40708-014-0002-y View ArticleGoogle Scholar
- Zong X, Lee J, Poplawsky AJ, Kim SG, Ye JC (2014) Compressed sensing fMRI using gradient-recalled echo and EPI sequences. NeuroImage 92:312–321View ArticleGoogle Scholar
- Candes E, Romberg J, Tao T (2006) Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information. IEEE Trans Inform Theory 52(2):489–509. doi:10.1109/TIT.2005.862083 MathSciNetView ArticleMATHGoogle Scholar
- Afonso M, Sanches JM (2015) Image reconstruction under multiplicative speckle noise using total variation. Neurocomputing 150(Part A):200–213View ArticleGoogle Scholar
- Shen Y, Li J, Zhu Z, Cao W, Song Y (2015) Image reconstruction algorithm from compressed sensing measurements by dictionary learning. Neurocomputing 151(Part 3):1153–1162View ArticleGoogle Scholar
- Eslahi N, Aghagolzadeh A, Andargoli SMH (2016) Image/video compressive sensing recovery using joint adaptive sparsity measure. Neurocomputing 200:88–109View ArticleGoogle Scholar
- Gao X, Jiang F, Liu S, Che W, Fan X, Zhao D (2016) Hierarchical frame based spatial temporal recovery for video compressive sensing coding. Neurocomputing 174(Part A):404–412View ArticleGoogle Scholar
- Candès EJ, Li X, Ma Y, Wright J (2011) Robust principal component analysis? J ACM 58(3):11:1–11:37. doi:10.1145/1970392.1970395 MathSciNetView ArticleMATHGoogle Scholar
- Chandrasekaran V, Sanghavi S, Parrilo PA, Willsky AS (2010) Rank-sparsity incoherence for matrix decomposition. Technical reportGoogle Scholar
- Moore B, Nadakuditi R, Fessler J (2014) Improved robust PCA using low-rank denoising with optimal singular value shrinkage. In: Statistical Signal Processing (SSP), 2014 IEEE Workshop on, pp 13–16. doi:10.1109/SSP.2014.6884563
- Jolliffe I (2005) Principal component analysis. Wiley, Hoboken. doi:10.1002/0470013192.bsa501 View ArticleMATHGoogle Scholar
- Scharf LL (1991) The SVD and reduced rank signal processing. Signal Process 25(2):113–133. doi:10.1016/0165-1684(91)90058-Q MathSciNetView ArticleMATHGoogle Scholar
- Tufts D, Shah A (1993) Estimation of a signal waveform from noisy data using low-rank approximation to a data matrix. IEEE Trans Signal Process 41(4):1716–1721. doi:10.1109/78.212753 View ArticleMATHGoogle Scholar
- Drineas P, Kannan R, Mahoney MW (2006) Fast monte carlo algorithms for matrices II: computing a low-rank approximation to a matrix. SIAM J Comput 36(1):158–183. doi:10.1137/S0097539704442696 MathSciNetView ArticleMATHGoogle Scholar
- Klema V, Laub A (1980) The singular value decomposition: its computation and some applications. IEEE Trans Autom Control 25(2):164–176. doi:10.1109/TAC.1980.1102314 MathSciNetView ArticleMATHGoogle Scholar
- Nadakuditi R (2014) Optshrink: an algorithm for improved low-rank signal matrix denoising by optimal, data-driven singular value shrinkage. IEEE Trans Inf Theory 60(5):3002–3018. doi:10.1109/TIT.2014.2311661 MathSciNetView ArticleGoogle Scholar
- Liang ZP (2007) Spatiotemporal imaging with partially separable functions. In: Biomedical Imaging: From Nano to Macro, 2007. ISBI 2007. 4th IEEE International Symposium on, pp 988–991. doi:10.1109/ISBI.2007.357020
- Amaldi E, Kann V (1998) On the approximability of minimizing nonzero variables or unsatisfied relations in linear systems. Theoret Comput Sci 209(12):237–260. doi:10.1016/S0304-3975(97)00115-1 MathSciNetView ArticleMATHGoogle Scholar
- Donoho D (2006) Compressed sensing. IEEE Trans Inform Theory 52(4):1289–1306. doi:10.1109/TIT.2006.871582 MathSciNetView ArticleMATHGoogle Scholar
- Candes EJ, Recht B (2009) Exact matrix completion via convex optimization. Found Comput Math 9(6):717–772. doi:10.1007/s10208-009-9045-5 MathSciNetView ArticleMATHGoogle Scholar
- Cai JF, Candès EJ, Shen Z (2010) A singular value thresholding algorithm for matrix completion. SIAM J Optim 20(4):1956–1982. doi:10.1137/080738970 MathSciNetView ArticleMATHGoogle Scholar
- Gu S, Zhang L, Zuo W, Feng X (2014) Weighted nuclear norm minimization with application to image denoising. In: Computer Vision and Pattern Recognition (CVPR), 2014 IEEE Conference on, pp 2862–2869. doi:10.1109/CVPR.2014.366
- Moran JM, Jolly E, Mitchell JP (2012) Social-cognitive deficits in normal aging. J Neurosci 32(16):5553–5561. doi:10.1523/JNEUROSCI.5511-11.2012 View ArticleGoogle Scholar
- Zhang S, Block KT, Frahm J (2010) Magnetic resonance imaging in real time: advances using radial flash. J Magn Reson Imaging 31(1):101–109. doi:10.1002/jmri.21987 View ArticleGoogle Scholar
- Geethanath S, Reddy R, Konar AS, Imam S, Sundaresan R, Venkatesan R (2013) Compressed sensing MRI: a review. Crit Rev Biomed Eng 41(3):183–204View ArticleGoogle Scholar
- Lingala SG, Hu Y, DiBella E, Jacob M (2011) Accelerated dynamic MRI exploiting sparsity and low-rank structure: kt SLR. IEEE Trans Med Imaging 30(5):1042–1054View ArticleGoogle Scholar
- Graedel NN, McNab JA, Chiew M, Miller KL (2016) Motion correction for functional MRI with three-dimensional hybrid radial-cartesian EPI. Magn Reson Med. doi:10.1002/mrm.26390 Google Scholar
- Hansen PC (1992) Analysis of discrete ill-posed problems by means of the L-curve. SIAM Rev 34(4):561–580. doi:10.1137/1034115 MathSciNetView ArticleMATHGoogle Scholar
- Lin T, Herrmann F (2007) Compressed extrapolation. Geophysics 72:77–93View ArticleGoogle Scholar
- Lindquist MA (2008) The statistical analysis of fMRI data. Stat Sci 23(4):439–464. doi:10.1214/09-STS282 MathSciNetView ArticleMATHGoogle Scholar
- Friston K, Ashburner J, Kiebel S, Nichols T, Penny WE (2006) Statistical parametric mapping: the analysis of functional brain images. Academic Press, LondonGoogle Scholar