Sparse MRI reconstructs images from highly undersampled data and has great potential to accelerate dynamic magnetic resonance images (dMRI). Conventional sparse MRI methods usually impose sparsity constraint on 1D vectors stretched from 2D/3D images or patches [1-2], and thus cannot fully address the multidimentional nature of high dimensional MRI data. The purpose of this work is to investigate the feasibility of reconstruct images form highly undersampled dMRI data using a tensor based dictionary method which does not need reform multidimensional data into vectors and has the advantage of simultaneously exploiting sparseness along all direction of multidimensional data.

Given undersampled sequence k-t space data $$$\mathcal{Y}\in\mathbb{C}^{M_1\times{M_2}\times{T}}$$$, the proposed method reconstructs underlying image $$$\mathcal{X}\in\mathbb{C}^{N_1\times{N_2}\times{T}}$$$ using the following formulation:$$\begin{align*}&\mathop {\min }\limits_x\sum_{i=1}^L\left\| \mathcal{G}_i\right\|_0+\mu\left\|F_u\mathcal{X}-\mathcal{Y}\right\|_F^2\\&s.t.\ R_i \mathcal{X} = \mathcal{G}_i \times_1 D_1 \times_2 D_2 \times_3 D_3,\ i=1,2,\cdots,L \\&D_1^H \cdot D_1^H = I_{n},\ D_2^H \cdot D_2^H = I_{n},\ D_3^H \cdot D_3^H = I_{T}\ \end{align*}$$Therein, $$$R_i$$$ is an operator applied to $$$\mathcal{X}$$$ to generate a block $$$R_i\mathcal{X}\in\mathbb{C}^{n\times{n}\times{T}}$$$, $$$\mathcal{G_i}$$$ denotes coefficient tensor of the i-th block $$$R_i\mathcal{X}$$$ over the dictionaries $$$D_1$$$, $$$D_2 $$$ and $$$ D_3$$$, $$$F_u$$$ denotes the Fourier encoding operator and $$$\mathcal{G}=\{ \mathcal{G}_1,\ \mathcal{G}_2,\ \cdots,\ \mathcal{G}_L \}$$$.

The solution to the proposed problem is obtained by alternatively solving three unconstrained subproblems, i.e., sparse coding, dictionary updating and data consistency, with respect to one variable with others fixed. The closed-form solution is derived in each subproblem due to the unitary constraint on each elementary dictionary.

We perform quantitative and qualitative comparisons of the proposed method with that using overcomplete non-structured dictionary learning and temporal gradient sparsity (DLTG) [2]. A fully sampled short-axis cardiac cine data (courtesy of [2]) and an in vivo cardiac perfusion data set (courtesy of [3]) are used in implementations of the two methods. Quantitatively, the proposed method outperforms the DLTG method by approximately 2 dB improvement in terms of peak signal-to-noise ratio (PSNR) and more detail preservation measured by the structural similarity index (SSIM) [4] ($$$R<8$$$) (Fig. 1). It is also shown that the performance of the proposed method is comparable with DLTG in the case of high reduction factor ($$$R\geq8$$$). The DLTG slightly over smoothes the reconstruction along time due to the additional temporal gradient penalty, and the proposed method is able to better provide edge structures compared to the DLTG (Fig. 2). In the in vivo experiment, the DLTG reconstructions show slight motion blurring which is almost removed in the reconstructions by the proposed method (Fig. 3).We introduced a novel tensor dictionary learning based method, which takes advantage of the structure contained in all different dimensions simultaneously, for dynamic MRI reconstruction from under-sampled k-t space data. An orthonormal constraint is imposed on the elementary matrices of the tensor dictionary, which make the corresponding problem is analytically solved and thus significantly improve the computational efficiency. The reconstruction results clearly show the advantage of the tensor model for recovering the dynamic images from under-sampled measurements compared to the matrix model. In our model, the elementary matrices of the tensor dictionary are square and orthonormal. For future work, we plan to extend it to overcomplete case to further explore the sparsity of the 3D image patches. We also plan to combine a dictionary based model and a parallel MRI reconstruction to obtain higher acquisition acceleration.