1290 IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 32, NO. 7, JULY 2013 Highly Undersampled Magnetic Resonance Image Reconstruction Using Two-Level Bregman Method With Dictionary Updating Qiegen Liu*, Shanshan Wang, Kun Yang, Jianhua Luo, Yuemin Zhu, and Dong Liang Abstract—In recent years Bregman iterative method (or related I. INTRODUCTION augmented Lagrangian method) has shown to be an efficient opti- mization technique for various inverse problems. In this paper, we propose a two-level Bregman Method with dictionary updating for AGNETIC resonance imaging (MRI) is an essential highly undersampled magnetic resonance (MR) image reconstruc- M medical diagnostic tool which provides clinicians tion. The outer-level Bregman iterative procedure enforces the with important anatomical information in the absence of ion- sampled k-space data constraints, while the inner-level Bregman method devotes to updating dictionary and sparse representation izing radiation. However, despite its superiority in obtaining of small overlapping image patches, emphasizing local structure high-resolution images and excellent depiction of soft tissues, adaptively. Modified sparse coding stage and simple dictionary MRI still has its own limitations; specifically, one property updating stage applied in the inner minimization make the whole accompanying MRI is that its scanning time is linearly related algorithm converge in a relatively small number of iterations, to the amount of data it acquires. As reported in [1], increased and enable accurate MR image reconstruction from highly un- dersampled k-space data. Experimental results on both simulated scan duration may introduce some potential issues such as MR images and real MR data consistently demonstrate that the physiological motion artifacts and patient discomfort. There- proposed algorithm can efficiently reconstruct MR images and fore, it is necessary to reduce the acquisition time. On the other present advantages over the current state-of-the-art reconstruc- hand, the reduction of the acquisition time may result in quality tion approach. degradation of MR images due to the undersampling, which compromises its diagnostic value. In this sense, accurate re- construction from highly undersampled k-space data is of great Index Terms—Augmented Lagrangian, Bregman iterative necessity for both quick MR image acquisition and clinical method, dictionary updating, image reconstruction, magnetic diagnosis. Compressed sensing (CS) theory, as a fundamental resonance imaging (MRI), sparse representation. and newly developed methodology in information society, has provided a crucial theoretical foundation for quick MR image acquisition. Specifically, the application of CS to MRI is known Manuscript received December 15, 2012; revised March 18, 2013; accepted as CS-MRI [2]–[6]. March 25, 2013. Date of publication April 02, 2013; date of current version June The basis for CS to work is sparsity, namely the image 26, 2013. This work was supported in part by High Technology Research De- has a sparse representation in certain domain. Normally, the velopment Plan of China under 2006AA020805, in part by the NSFC of China under 30670574 and 61262084, in part by Shanghai International Cooperation transforms which allow the image to have a sparse representa- Grant under 06SR07109, in part by Region Rhone-Alpes of France under the tion are named as sparsifying transforms. Total variation (TV) project Mira Recherche 2008, in part by the joint project of Chinese NSFC and wavelet transform are two such transforms [1], [6]–[9] (under 30911130364) and French ANR 2009 (under ANR-09-BLAN-0372–01), and in part by China Scholarship Council under Grant 2011623084. Asterisk in- frequently employed in CS recovery problems. For instance, dicates corresponding author. Lustig et al. [1] focused on MR image reconstruction with TV *Q. Liu is with the Department of Electronic Information Engineering, Nan- penalty and the wavelet transform of Daubechies. Trzasko et chang University, Nanchang 330031, China (e-mail: [email protected]). D. Liang is with the Paul C. Lauterbur Research Centre for Biomedical al. [8] proposed a homotopic -minimization strategy, instead Imaging, Shenzhen Key Laboratory for MRI, Shenzhen Institutes of Advanced of -minimization, to reconstruct the MR image. The work of Technology, Chinese Academy of Sciences, Shenzhen 518055, China (e-mail: [10] presented an edge guided compressive sensing reconstruc- [email protected]). S. Wang is with the School of Biomedical Engineering, Shanghai Jiao Tong tion (EdgeCS) method, which alternatively performs TV-based University, Shanghai 200240, China, and with Biomedical and Multimedia In- CS reconstruction and edge detection with each step benefiting formation Technology (BMIT) Research Group, School of Information Tech- from the latest solution of the other. However, since TV prior nologies, The University of Sydney, NSW 2006, Australia (e-mail: sophiaw@it. usyd.edu.au). prefers cartoon-like images which are piecewise constant, it K. Yang is with the Department of Electrical Computer Engineering, National does not apply to MR images to some extent which consist of University of Singapore, 117576 Singapore (e-mail: [email protected]). crucial details for clinical diagnosis. Bredies et al. thus intro- J. Luo is with the College of Aeronautics and Astronautics, Shanghai Jiao Tong University, Shanghai 200240, China (e-mail: [email protected]). duced the total generalized variation (TGV) model for MRI Y. Zhu is with the CREATIS, CNRS UMR 5220, Inserm U 630, INSA Lyon, problems [11], [12]. Unfortunately, although this has improved University of Lyon 1, Lyon, France (e-mail: [email protected]). the reconstruction result, it is still a TV-based regularization, Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. which can be considered as only forcing the reconstructed Digital Object Identifier 10.1109/TMI.2013.2256464 image to be sparse with respect to spatial differences. Other 0278-0062/$31.00 © 2013 IEEE LIU et al.: HIGHLY UNDERSAMPLED MAGNETIC RESONANCE IMAGE RECONSTRUCTION USING TWO-LEVEL BREGMAN METHOD 1291 analytically designed dictionaries, such as the wavelets and the well-known augmented Lagrangian (AL) scheme in some shearlets, also have this intrinsic deficiency, i.e., lacking the particular cases [25], [26]. As a promising iterative mechanism, adaptability to various images. We name the sparsity, which the its significance is soon presented in image deblurring and MR fixed and global transform makes the image possess, as global image reconstruction problems [25], [27]–[33]. sparsity. In this work, we exploit both the strengths of the patch-based In addition to global sparsity, nonlocal similarity is another adaptive dictionaries and the Bregman iteration technique. The popular patch-based sparsity which describes the resemblance main contribution of this paper is the development of a fast and of small image patches in an image. This property has been suc- robust numerical algorithm, named two-level Bregman method cessfully applied in image denoising problems [13]–[15] and with dictionary updating (TBMDU), for MR image reconstruc- many authors have also incorporated this nonlocal information tion. The proposed algorithm consists of a two-level solver em- into the CS recovery problems. For instance, Liang et al. [16] ploying the Bregman technique. One is to estimate the recovery applied the nonlocal total variation (NLTV) regularization to image and the other is to calculate the dictionary and sparse co- reduce the blocky effect introduced by TV regularization. This efficients of image patches. A modified strategy is applied to the method replaces the conventional gradient functional used in sparse coding step of the inner minimization, enabling the effi- TV with a weighted nonlocal gradient function and obtains an ciency of the whole algorithm. improvement in signal-to-noise ratio of reconstruction quality in parallel imaging. The work of [17] further incorporated a II. BACKGROUND AND RELATED WORK semi-nonlocal prior into the homotopic minimization to the reconstruction of breast MR images. Egiazarian et al. [18] A. CS-MRI With Various Regularizers proposed a recursive filtering procedure for CS image recon- With CS theory intensively studied [2], [3] and successfully struction. To excite the algorithm, random noise is injected applied in practical problems, it has become well known that im- in the unobserved part of the spectrum at each iteration, and ages can be reconstructed from very few linear measurements then a spatially adaptive image denoising filter [in particular as long as the image has a sparse representation. To explore the state-of-the-art block-matching and 3-D filtering (BM3D)] the sparsity inherent in MR images, researchers often apply [14] is utilized in the image domain to remove the noise and some sparsifying transforms to convert them into a represen- recover the detail information of the image. In addition, under tation with few values significantly different from zero. Table I the assumption that each image patch can be sparsely repre- is a brief summary of some commonly used sparsity-promoting sented, the K-singular value decomposition (K-SVD) algorithm regularization terms [denoted by ]. For more general situa- proposed by Elad et al. [15], [19] has also been used for MR tion, if a sparsifying transform is defined, it is modeled ideally image reconstruction [20]–[22]. Especially, Ravishankar