Compressed sensing based dynamic MR image reconstruction by using 3D-total generalized variation and tensor decomposition: k-t TGV-TD – BMC Medical Imaging

In Magnetic Resonance Imaging (MRI), imaging speed is limited by slow acquisition of full k-space using magnetic field gradients [1]. Minimizing the k-space recording time without compromising image quality has been a main thrust of MR imaging research. With the advent of compressed sensing (CS) theory [2, 3], MR image reconstruction with sparsity-promoted regularization (e.g., 1-based regularization), termed as CS-MRI [4,5,6,7,8,9,10], has gained popularity for its high imaging speed. The effective exploitation of the signal sparsity enables the MR image reconstruction from far fewer k-space samples possible than conventional methods require, thus CS-MRI can significantly reduce the scan time. The compressed sensing theory has been successfully applied to both static and dynamic magnetic resonance imaging (dMRI) reconstructions [11,12,13,14].

In CS-MRI, the method used to sparsify the MR image plays an important role in the image reconstruction. The most used sparsity bases are predefined mathematical transforms, such as discrete cosine transform (DCT), and discrete wavelet transform (DWT). Recently, the singular value decomposition (SVD) method has been used as a data-adaptive sparsity basis in CS-MRI reconstruction [15, 16], and it has been found that the SVD-based method could significantly accelerate the reconstruction process and achieve better image quality than those commonly used sparsifying transforms (DCT and DWT). Majumdar et al. proposed to exploit the nuclear norm regularization to implement the CS-MRI reconstruction, and the results showed that the proposed reconstruction method was faster than other methods [6]. In addition, the linear combination of Total Variation (TV) and wavelet sparse regularization, known as TV-L1 problem, is very popular in many CS-MRI models [5, 6, 17], which can be considered as processing the MR image to be sparse by both the specific transform and finite-differences at the same time. Due to the stair-case artifacts caused by the conventional TV-based regularization [18, 19], several generalizations and extensions of TV have been introduced to improve the CS-MRI reconstruction accuracy, such as Total Generalized Variation (TGV) [18,19,20], Higher Degree Total Variation (HDTV) [21]. Nonlocal Total Variation (NLTV) [22,23,24] is another effective way to address the issue of stair-case artifacts. Although effective in practice, it involves higher computational complexity than the conventional TV method.

For dynamic MR image reconstruction, Ji, et al. adopted the difference between the reconstructed image and the reference image to represent the spatial sparsity [25]. However, when compared with the reference frame, the sparsity of the difference image got worse with the increase of the subsequent frame distance. To solve this problem, Majumdar took the difference between two adjacent sub-images as a sparse representation of the reconstructed MR image [13]. In addition, Usman put forward the concept of a sparse group of dynamic MRI, utilizing both MRI signal itself sparsity and the group structure information between signals [26, 27], which can effectively improve the image reconstruction quality. Moreover, a novel blind compressed sensing frame work was proposed to recover dynamic magnetic resonance images from undersampled measurements [28, 29], which has been proved to provide superior reconstruction performance in comparison to existing low rank and compressed sensing schemes. Recently, k-t SLR (k-t Sparisity and Low-Rank) method has been proposed to accelerate dynamic MRI by exploiting sparsity and low rank properties of the image data [30, 31]. To exploit the low-rank structure, the k-t SLR method reshaped the 3D dataset into a large 2D matrix through a two-step process: vectorize the 2D images in a dynamic sequence first and then concatenate them to form a matrix. In most of the existing dynamic CS-MRI methods, 2D/1D transforms were applied to solve the 3D dynamic problem, which, by treating the 3D data as a series of 2D images, unfolded the 3D dataset into a 2D matrix to explore the spatiotemporal redundancy [30,31,32,33]. In addition, Majumdar [34, 35] acted the dynamic MR image reconstruction problem as a least squares minimization regularized by lp-norm as the sparsity penalty and Schatten-q norm as the low-rank penalty sparsity, which can yield much better reconstruction results than k-t SLR method. However, reshaping a high-order tensor into a matrix or vector may neglect the inherent data redundancy, thus greatly degrading the reconstructed image quality. To promote the signal sparsity representation by exploring the redundancy of the high-dimension data format, Yu et al. proposed tensor decomposition-based sparsifying transform, that is, high-order Singular Value Decomposition (HOSVD) [36], which can outperform the conventional sparse recovery methods for high-dimensional cardiac imaging reconstruction accuracy given the same amount of k-space data set [37].

In this paper, we will further improve the HOSVD based CS-MRI method to synergistically integrate 3D-TGV algorithm and HOSVD-based Tensor Decomposition, termed as k-t TGV-TD method. In the proposed method, the low rank structure of the 3D dynamic cardiac MR data is performed by the HOSVD method, and the localized image sparsity is achieved by the 3D-TGV method. Meanwhile, the Fast Composite Splitting Algorithm (FCSA) method [6], combining the variable splitting with operator splitting techniques, is employed to solve the low-rank and sparse problem [38]. Two different cardiac MR datasets (cardiac perfusion and cine MR datasets) are used to evaluate the performance of the proposed method.

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