Dynamic magnetic resonance imaging (dynamic MRI) is used to visualize living tissues and their changes over time. In this paper, we propose a new tensor-based dynamic MRI approach for reconstruction from highly undersampled (k, t)-space data, which combines low tensor train rankness and temporal sparsity constraints. Considering tensor train (TT) decomposition has superior performance in dealing with high-dimensional tensors, we introduce TT decomposition and utilize the low rankness of TT matrices to exploit the inner structural prior information of dynamic MRI data. First, ket augmentation (KA) scheme is used to permute the 3-order (k, t)-space data to a high order tensor and low rankness of each TT matrix is enforced with different weights. To reduce the computational complexity, we replace the nuclear norm of TT matrices with the minimum Frobenius norm of two factorization matrices to avoid singular value decomposition. Secondly, the l 1 norm of the Fourier coefficients along the temporal dimension is added as a sparsity constraint to further improve the reconstruction. Lastly, an effective algorithm based on the alternating direction method of multipliers (ADMM) is developed to solve the proposed optimization problem. Numerous experiments have been conducted on three dynamic MRI data sets to estimate the performance of our proposed method. The experimental results and comparisons with several state-of-the-art imaging methods demonstrate the superior performance of the proposed method. INDEX TERMS Dynamic MR imaging, tensor train, sparsity, the alternating direction multiplier method.
Tensor singular value decomposition (t‐SVD) provides a novel way to decompose a tensor. It has been employed mostly in recovering missing tensor entries from the observed tensor entries. The problem of applying t‐SVD to recover tensors from limited coefficients in any given ortho‐normal basis is addressed. We prove that an n × n × n3 tensor with tubal‐rank r can be efficiently reconstructed by minimising its tubal nuclear norm from its O(rn3n log2(n3n)) randomly sampled coefficients w.r.t any given ortho‐normal basis. In our proof, we extend the matrix coherent conditions to tensor coherent conditions. We first prove the theorem belonging to the case of Fourier‐type basis under certain coherent conditions. Then, we prove that our results hold for any ortho‐normal basis meeting the conditions. Our work covers the existing t‐SVD‐based tensor completion problem as a special case. We conduct numerical experiments on random tensors and dynamic magnetic resonance images (d‐MRI) to demonstrate the performance of the proposed methods.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.