Low-Rank Methods
Bradley Sutton1, Xi Peng1, Fan Lam1, and Zhi-Pei Liang1

1University of Illinois at Urbana-Champaign, United States

### Synopsis

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.

### TARGET AUDIENCE:

MRI Physicists that want to reduce data acquisition requirements for high spatial resolution and fast imaging.

### OUTCOME/OBJECTIVES:

At the end of this session, participants will be able to design MR acquisitions and image reconstruction approaches that utilize spatiotemporal correlations or low-rank structures in their data to reconstruct images from undersampled data with high effective acceleration factors.

### DESCRIPTION OF LOW RANK METHODS:

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(ki,tj) is the data from the ith kspace location at the jth time point in a dynamic acquisition, $$C = \begin{bmatrix}s(k_1, t_1) & s(k_1, t_2) & ... & s(k_1, t_M) \\ s(k_2, t_1) & s(k_2, t_2) & ... & s(k_2, t_M) \\ ... & ... & ... & ...\\ s(k_N, t_1) & s(k_N, t_2) & ...& s(k_N, t_M)\end{bmatrix}.$$ Note that, depending on the acquisition or other regularization penalties, the data can be in k-space or image space and the temporal dimension can be in time, frequency or any parameter dimension (e.g., for parameter mapping or diffusion imaging). We usually do not have time to sample the entire data matrix, so many of the entries are missing. For example, perhaps we only sampled the blue entries and the red entries are not sampled in the undersampled Casorati matrix shown below: $$C = \begin{bmatrix}\color{blue}{s(k_1, t_1)} & \color{red}{s(k_1, t_2)} & ... & \color{blue}{s(k_1, t_M)} \\ \color{red}{s(k_2, t_1)} & \color{blue}{s(k_2, t_2)} & ... & \color{red}{s(k_2, t_M)} \\ ... & ... & ... & ...\\ \color{red}{s(k_N, t_1)} & \color{red}{s(k_N, t_2)} & ...& \color{blue}{s(k_N, t_M)}\end{bmatrix}$$ The image reconstruction problem can then be formulated as completing the matrix C using the sparsely sampled data [3]. In addition to recovering sparsely sampled spatiotemporal data matrices, the low rank formulation can also be used to denoise high-dimensional data [4, 5].

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].

### CONCLUSION:

For many natural imaging applications, the desired high-dimensional data has significant correlations such that different parts of the image share similar signals profiles, e.g., the underlying movements or physical mechanisms in the image are shared between spatial locations. In these cases, the data matrix is inherently low rank which implies reduced numbers of degrees-of-freedom. Significant accelerations in imaging are possible through exploiting this low rank structure, along with other constraints, such as spatial-spectral sparsity.

### Acknowledgements

No acknowledgement found.

### References

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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.

Proc. Intl. Soc. Mag. Reson. Med. 26 (2018)