An Adaptive Directional Haar Framelet-Based Reconstruction Algorithm for Parallel Magnetic Resonance Imaging

Yan-Ran Li, Raymond H. Chan, Lixin Shen, Yung-Chin Hsu, Wen-Yih Isaac Tseng
2016 SIAM Journal of Imaging Sciences  
Parallel magnetic resonance imaging (pMRI) is a technique to accelerate the magnetic resonance imaging process. The problem of reconstructing an image from the collected pMRI data is ill-posed. Regularization is needed to make the problem well-posed. In this paper, we first construct a 2-dimensional tight framelet system whose filters have the same support as the orthogonal Haar filters and are able to detect edges of an image in the horizontal, vertical, and ±45 o directions. This system is
more » ... erred to as directional Haar framelet (DHF). We then propose a pMRI reconstruction model whose regularization term is formed by the DHF. This model is solved by a fast proximal algorithm with low computational complexity. The regularization parameters are updated adaptively and determined automatically during the iteration of the algorithm. Numerical experiments for in-silico and in-vivo data sets are provided to demonstrate the superiority of the DHF-based model and the efficiency of our proposed algorithm for pMRI reconstruction. 1 MRI (pMRI) image reconstruction from the undersampled data. The most well-known ones include sensitivity encoding (SENSE) [37] and generalized autocalibrating partially parallel acquisitions (GRAPPA) [18] . Both methods provide good results with nearly identical reconstruction quality [1] . SENSE has been made commercially available for clinical purposes. Many clinical applications already benefit from the capabilities of SENSE in terms of increased imaging speed, effectively reduced blurring, and increased spatial resolution provided by pMRI [44, 17] . For pMRI reconstruction by SENSE, accurate estimation of coil sensitivity is required, but it is difficult to be determined because of the complex geometry of the coils and the noise in the coil images. As a consequence, the reconstructed image by SENSE often suffers from artifacts like noise amplification [42], Gibbs effect [48] and aliasing [1] . To overcome these problems, regularization techniques that do not need modifications in hardware or data acquisition, are widely adopted for SENSE-based reconstruction model. In [31] , Tikhonov regularization was used due to the existence of a closed-form solution while the regularization parameter was set automatically by using the L-curve method. With Tikhonov regularization, bias is often introduced due to the poor quality of reconstruction image, in particular, at high reduction factors. Recently, the smooth prior used in Tikhonov regularization was replaced by non-smooth edge-preserving prior. The resulting edge-preserving regularization, like total-variation (TV) regularization [2, 46, 23] or wavelet-based regularization [8], has been proposed for pMRI reconstruction problem. In a comparison with Tikhonov regularization, edge-preserving regularization makes a noticeable improvement in preserving sharp edges of reconstructed images and removing the noise or artifacts. The accuracy of sensitivity maps is crucial for pMRI reconstruction since the aliasing artifacts, caused by inaccurate sensitivity maps, can hardly be reduced by any kind of reconstruction methods. Hence, sensitivity estimation method is as important as the reconstruction method in SENSE. In [47], a method that jointly estimates the sensitivities and SENSE reconstruction was proposed to refine the sensitivities iteratively so that the SNR of reconstruction is improved and the image artifact is reduced. In [24], a framework that allows the integration of a-priori information, such as a TV or total generalized variation penalty [3] , in an iteratively regularized Gauss-Newton method, was proposed with a goal of joint estimation of images and coil sensitivities. The selection of a regularization parameter in regularization method is critical in order to achieve a reconstructed image with acceptable quality. The regularization parameter in [2, 46, 23, 8] was not set automatically; therefore, limiting their practical applications for the pMRI reconstruction. GRAPPA [18] is a k-space method that interpolates the missing data in the k-space for each coil. It does not need to explicitly know the coil sensitivities, but requires to estimate an interpolation kernel using the auto calibration signal (ACS) data. To acquire the correct interpolation kernel for reconstructing high quality MRI images, the authors in [50, 36] regularize the GRAPPA-based interpolation kernel and estimate it by an iterative scheme. The method in [45] applies sparsity-promoting regularizers on the GRAPPA-based calibration kernel and the coil image in each channel. It jointly updates the kernel and the images iteratively. The relationship between SENSE and GRAPPA was clarified and the gap between them was bridged in [43] . Both approaches restrict the solution to a subspace spanned by the sensitivities. By combining the advantages from both SENSE and GRAPPA, a hybrid reconstruction method to pMRI was then proposed. Recently, compressive sensing (CS) has been applied to pMRI problems in order to accelerate imaging speed and to improve the quality of MRI images [33] . The ℓ 1 -SPIRiT [34] is an algorithm for auto calibrating parallel imaging and permits an efficient implementation with clinically-feasible runtimes by using compressive sensing. The state-of-the-art ℓ 1 -ESPIRiT algorithm proposed in [43] uses the wavelet regularization and sensitivities estimated from the calibration matrix as in GRAPPA. Sparse dictionary learning with compressive sensing was proposed in [38] to reconstruct MR images from highly undersampled kspace data. This work complements the existing TV and wavelet-based regularization methods for pMRI reconstruction. The main contributions of the work are described as follows:
doi:10.1137/15m1033964 fatcat:mwzurcvxxrft3bud5gqq76pbxu