Bradley Sutton^{1}, Xi Peng^{1}, Fan Lam^{1}, and Zhi-Pei Liang^{1}

For dynamic imaging or parameter mapping applications, there is a significant amount of spatiotemporal correlations in the data, with many parts of the image sharing similar temporal signal profiles. This results in a data matrix that is low rank. Several methods have been developed to exploit this low-rank structure to achieve very high imaging speed. This talk will describe these low rank methods and demonstrate the imaging speed that can be achieved.

Magnetic resonance imaging data has a great deal of spatiotemporal correlations in the data, especially in data that collects multiple time points in dynamic imaging, multiple images for different diffusion orientation, and multiple contrasts for parameter mapping. Due to this large amount of spatiotemporal correlation, collecting the fully sampled spatiotemporal data is not necessary as there is significant shared information across space and time. Instead, we can formulate the sparse-data image reconstruction problem as an optimization problem to effectively take advantage of low rank structures in the data matrix or tensor.

One important approach to induce low rank structures is to represent
the high dimensional/multi-dimensional data using partially separable functions [1].
Specifically, we can represent the full spatiotemporal data, *s(k,t)*, as: $$ s(k,t) = \sum_{l=0}^L
\psi_l(k) \phi_l(t), $$where $$$\psi (k)$$$ are the spatial coefficients for the
temporal basis functions $$$ \phi(t) $$$. As a result, the following
spatiotemporal data matrix (or tensor) *C*, also called the Casorati matrix, will have a rank upper bounded by the order of the partial separability (PS) model, $$$L$$$ [1, 2], which is typically a small number due to the strong spatiotemporal correlation (i.e., $$$ \mathcal{R}(C)=L<\mathtt{min}(M,N) $$$). The Casorati matrix is as follows, where *s(k _{i},t_{j})* is the data from the

Another main approach to construct low rank matrices and tensors is to exploit linear predictability (LP) inherently embedded in the data, which can be derived from finite image support [6, 7], smooth image phase [6] and/or the signal behaviors along the temporal dimension [8, 9]. Accordingly, different Hankel matrices [6-10] can be constructed by special arrangements of the data, and these matrices will be low rank due to the LP property. A number of methods have been proposed to take advantage of these low rank structures for improving parallel imaging and accelerating high-dimensional imaging [6-13].

Many algorithms have been developed in the last few years to
solve the low rank matrix completion/recovery problems. One collection of
methods enforce the low rank nature implicitly through various kinds of penalties
on the norm of the data matrix, i.e. the nuclear norm or Schatten *p*-norm, as
was done in [3, 14]. Other methods focus on the low rank estimation problem by
explicitly enforcing the low rank nature of the data, representing the Casorati
matrix,* C*, as a multiplication of two
low rank matrices: $$$ C = UV$$$, where $$$U\in\mathcal{C}^{NxL}$$$
is a collection of basis functions spanning the k-space or image space
domain of size *N *in the data and $$$V\in\mathcal{C}^{LxM}$$$ is collection
of basis functions spanning the temporal domain in the data [1].
They can also be viewed as the matrix representations of the spatial coefficients
and temporal basis in the PS model. Through this framework, the matrix *C* is forced to be at most rank *L*, and
the associated optimization algorithm is usually more computationally
efficient due to the reduced numbers of unknowns.

Within the PS model formulation, there are two ways to go
about recovering the low rank data matrix [1].
First, we could simultaneously estimate the *U*
and *V* matrices, as was done in [2, 15].
A second approach is motivated by the
partial separability of space and time components. This approach estimates the
temporal basis functions, potentially using temporal navigators, using a
singular value decomposition. Simultaneously acquired sparse imaging data is used
to fit the spatial maps associated with the temporal components through
regularized least squares, potentially incorporating other penalties typical in
compressed sensing approaches [16].
With this structure, the navigator can be chosen to provide a good trade-off
between SNR and coverage of k-space while maintaining high temporal resolution
for the overall reconstruction [17, 18].
Using this approach, high quality dynamic imaging has been done in both cardiac
and speech applications, including full 3D imaging of the vocal tract at over
150 frames per second [19].

Besides leveraging the correlations in the entire data set, low rank can be applied to local domains, such as local image space or local k-space [6, 7, 10]. Additionally, the correlations across multiple channels can be placed in a low rank framework to enable calibration free parallel imaging [11].

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5. Lam F, Babacan SD, Haldar JP, Weiner MW, Schuff N, Liang ZP. Denoising diffusion-weighted magnitude MR images using rank and edge constraints. Magn Reson Med. 2014;71(3):1272-84. PMCID: PMC3796128.

6. Haldar JP. Low-rank modeling of local k-space neighborhoods (LORAKS) for constrained MRI. IEEE Trans Med Imaging. 2014;33(3):668-81. PMCID: PMC4122573.

7. Lee D, Jin KH, Kim EY, Park SH, Ye JC. Acceleration of MR parameter mapping using annihilating filter-based low rank hankel matrix (ALOHA). Magn Reson Med. 2016;76(6):1848-64.

8. Peng X, Ying L, Liu Y, Yuan J, Liu X, Liang D. Accelerated exponential parameterization of T2 relaxation with model-driven low rank and sparsity priors (MORASA). Magn Reson Med. 2016;76(6):1865-78.

9. Nguyen HM, Peng X, Do MN, Liang ZP. Denoising MR spectroscopic imaging data with low-rank approximations. IEEE transactions on bio-medical engineering. 2013;60(1):78-89. PMCID: PMC3800688.

10. Zhang T, Pauly JM, Levesque IR. Accelerating parameter mapping with a locally low rank constraint. Magn Reson Med. 2015;73(2):655-61. PMCID: PMC4122652.

11. Shin PJ, Larson PE, Ohliger MA, Elad M, Pauly JM, Vigneron DB, Lustig M. Calibrationless parallel imaging reconstruction based on structured low-rank matrix completion. Magn Reson Med. 2014;72(4):959-70. PMCID: PMC4025999.

12. Ongie G, Jacob M. Off-the-Grid Recovery of Piecewise Constant Images from Few Fourier Samples. Siam J Imaging Sci. 2016;9(3):1004-41.

13. He JF, Liu QG, Christodoulou AG, Ma C, Lam F, Liang ZP. Accelerated High-Dimensional MR Imaging With Sparse Sampling Using Low-Rank Tensors. Ieee Transactions on Medical Imaging. 2016;35(9):2119-29.

14. Lingala SG, Hu Y, DiBella E, Jacob M. Accelerated dynamic MRI exploiting sparsity and low-rank structure: k-t SLR. IEEE Trans Med Imaging. 2011;30(5):1042-54. PMCID: PMC3707502.

15. Bhave S, Lingala SG, Johnson CP, Magnotta VA, Jacob M. Accelerated whole-brain multi-parameter mapping using blind compressed sensing. Magn Reson Med. 2016;75(3):1175-86. PMCID: PMC4598248.

16. Zhao B, Haldar JP, Christodoulou AG, Liang ZP. Image reconstruction from highly undersampled (k, t)-space data with joint partial separability and sparsity constraints. IEEE Trans Med Imaging. 2012;31(9):1809-20. PMCID: PMC3434301.

17. Christodoulou AG, Hitchens TK, Wu YL, Ho C, Liang ZP. Improved subspace estimation for low-rank model-based accelerated cardiac imaging. IEEE transactions on bio-medical engineering. 2014;61(9):2451-7. PMCID: 4190021.

18. Fu M, Zhao B, Carignan C, Shosted RK, Perry JL, Kuehn DP, Liang ZP, Sutton BP. High-resolution dynamic speech imaging with joint low-rank and sparsity constraints. Magn Reson Med. 2015;73(5):1820-32. PMCID: 4261062.

19. Fu M, Barlaz MS, Holtrop JL, Perry JL, Kuehn DP, Shosted RK, Liang ZP, Sutton BP. High-frame-rate full-vocal-tract 3D dynamic speech imaging. Magn Reson Med. 2017;77(4):1619-29.