A Motion Compensating Prior for Dynamic MRI Reconstruction using Combination of Compressed Sensing and Parallel Imaging C¸a˘ gdas¸ Bilen, Ivan Selesnick, Yao Wang Department of Electrical Engineering Polytechnic Institute of NYU Brooklyn, NY, USA Ricardo Otazo, Daniel K. Sodickson Center for Biomedical Imaging NYU School of Medicine New York, NY, USA Abstract—Many areas in signal processing have benefited from the emergence of compressed sensing and sparse reconstruction methods, one of which is magnetic resonance imaging (MRI). Recent studies showed that MRI acquisition can be highly accelerated with the joint use of compressed sensing and parallel imaging methods. It is also suggested that dynamic MRI can be further improved by making use of temporal correlations. Although methods using motion compensation has been proposed to exploit temporal dependence, most of these require reference frames and/or a sub-portion of k-space to be fully sampled. In this paper we propose a new approach to exploit the motion information during compressed sensing reconstruction without any requirement for reference frames, modeled motion or a specific sampling pattern on the k-space measurements. I. I NTRODUCTION Compressed sensing and sparse reconstruction methods have been popular topics of research especially in the last decade. Under certain conditions such as the data being sparse in a domain that is incoherent with measurement domain, compressed sensing enables reliable recovery of signals even if they are measured at a rate under the Nyquist rate [1]. This stimulated research in many different fields in which data acquisition is limited due to constraints. It has been shown that compressed sensing can help acceler- ating magnetic resonance imaging (MRI) data acquisition by subsampling the measurements [2]. The acceleration can be further increased when parallel imaging is used in conjunction with compressed sensing [3], [4]. These methods are quickly adopted in the field of dynamic MRI in which the acquired data is especially limited [5], [6]. Most of the reconstruction methods used in dynamic MRI exploit the temporal correlation in the signal which has been shown to be much more successful than working only on the spatial domain [5], [6]. A similar problem has also been widely studied in the field of video compression, in which video images are compressed using the similarities between different frames in time to achieve high efficiency. Some ideas from video compression algorithms are also used in dynamic MRI reconstruction. In [6], [7] and [8], the sparsity of the residual signal after subtraction of an initial estimate of the signal is utilized. More advanced methods, such as in [9], use motion estimation and compensation to provide better estimates before exploiting the sparsity of the residual. Unfortunately, unlike video compression problems, a good quality reference is usually not available in dynamic MRI to estimate the motion especially at high acceleration rates. Furthermore, due to the entire signal not being available, incorrect initial estimates can also significantly degrade the methods that utilize the sparsity of a residual. In this paper, we propose a method that utilizes the correlation due to motion in the signal while using the compressed sensing framework more efficiently without any need for explicit prior information such as reference frames, or a motion model. The proposed method is shown to perform well even in case of extreme acceleration rates. II. COMPRESSED SENSING FORMULATION IN PARALLEL MRI The measurement in parallel MRI can be formulated as y = FCx + n = F  C T 1 ··· C T Nc  T x + n (1) where F is the 2D spatial Fourier transform, C i ,i =1,...,N c are the diagonal coil sensitivity matrices, y is the multi-coil measurement (k-space) data in time, n is the measurement noise and x is the image signal in time. In compressed sensing, the measurement is undersampled in a random fashion which can be represented as y = Hx + n = F u Cx + n = MFCx + n (2) where M is a subset of the rows of identity matrix and F u = MF is the undersampled Fourier transform matrix. Recovery of x is a general inverse problem which has been studied for decades considering different conditions on H. A. Traditional Compressed Sensing Approach In compressed sensing the unknown signal x in the under- determined system (2) can be estimated as ˜ x = arg min x R(x) s.t. ‖y − Hx‖ 2 2 <ǫ (3) where ‖.‖ k is L k -norm, provided that the regularization term R(x) is properly chosen. The formulation in (3) is called the