A Two-Stage Low Rank Approach for Calibrationless Dynamic Parallel Magnetic Resonance Image Reconstruction

Transcription

1 DOI /s A Two-Stage Low Rank Approach for Calibrationless Dynamic Parallel Magnetic Resonance Image Reconstruction Likun Hou 1 Hao Gao 2 Xiaoqun Zhang 3 Received: 23 November 2015 / Revised: 31 March 2016 / Accepted: 14 May 2016 Springer Science+Business Media New York 2016 Abstract Parallel magnetic resonance imaging (MRI) is an imaging technique by acquiring a reduced amount of data in Fourier domain with multiple receiver coils. To recover the underlying imaging object, one often needs the explicit knowledge of coil sensitivity maps, or some additional fully acquired data blocks called the auto-calibration signals (ACS). In this paper, we show that by exploiting the between-frame redundancy of dynamic parallel MRI data, it is possible to achieve simultaneous coil sensitivity map estimation and image sequence reconstruction. Specially, we introduce a novel two-stage approach for dynamic parallel MRI reconstruction without pre-calibrating the coil sensitivity maps nor additionally acquiring any fully sampled ACS. Numerical experiments demonstrate that, the performance of the proposed approach is better than other state-of-the-art approaches for calibrationless dynamic parallel MRI reconstruction. Keywords Dynamic magnetic resonance imaging Sensitivity maps estimation Low rank plus sparsity Forward backward splitting method Hao Gao was partially supported by the NSFC (# ), the 973 Program (#2015CB856004), and the Shanghai Pujiang Talent Program (#14PJ ). L. Hou and X. Zhang were partially supported by NSFC (# ) and Sino-German center grant (GZ1025), China Postdoc Science Foundation (# 2014M551392), and 973 Program (# 2015CB856004). B Xiaoqun Zhang xqzhang@sjtu.edu.cn Likun Hou houlk@sjtu.edu.cn Hao Gao hao.gao.2012@gmail.com 1 Institute of Natural Sciences, Shanghai Jiao Tong University, Shanghai, China 2 School of Biomedical Engineering and Department of Mathematics, Shanghai Jiao Tong University, Shanghai, China 3 Institute of Natural Sciences and School of Mathematical Sciences and MOE-LSC, Shanghai Jiao Tong University, Shanghai, China

2 1 Introduction Magnetic resonance image (MRI) is an imaging modality that has been widely adopted in various medical applications. In MRI, data samples are collected in the spatial-frequency domain that is usually referred as the k-space. Currently, the most prevailing approach for MRI data scanning is perhaps the parallel imaging technique. Different from traditional MRI scanning methods, parallel MRI uses multiple receiver coils to acquire the MRI data. Those receiver coils are spatially distributed such that they are able to receive MRI signals simultaneously with overlapped sensitivity maps. The overlapping nature of sensitivity maps provides data redundancy, thus enables accelerated data acquisition via subsampling the k-space data. Regarding to the incompleteness of the acquired data, various parallel MRI reconstruction methods have been proposed. Those methods can roughly be classified into two categories: sensitive encoding based methods and auto-calibration based methods. Sensitivity encoding based approaches, like SENSE [18] and SMASH[21], commonly require the explicit knowledge of the sensitivity maps associated with the set of receiver coils (as well the the object to be scanned). Thus, a pre-scan procedure is usually needed for estimating the set of sensitivity maps when applying such techniques, which is time-consuming and could be less accurate due to the change of scanning conditions. In comparison, auto-calibration based methods, like GRAPPA [9] and SPRiT [16], do not need a pre-scan procedure for estimating the set of sensitivity maps. Instead, this kind of approaches require a densely/fully sampled region located at center of the acquired k-space, which is usually referred as the auto-calibration signals (ACS). However, for dynamic parallel MRI, repeatedly acquiring the ACS could be time-consuming as well. This naturally leads us to the problem of calibrationless dynamic parallel MRI reconstruction which does not require either the explicit knowledge of the coil sensitivity maps or any densely sampled ACS. SAKE [20] is an approach for parallel MRI reconstruction that neither requires the set of sensitivity maps as a prior knowledge (like SENSE) nor any fully sampled region centred at the k-space domain for auto-calibration (like GRAPPA, SPIRiT and EPSPIRiT). However, it is a single frame method that does not maximally utilize the temporal correlation of data. The main objective of this paper is to introduce a new approach for calibrationless dynamic parallel MRI reconstruction. Different from previous (calibrationless) approaches that explore structural information within a single frame, the proposed approach explore the correlations between different data frames in order to gain robustness as well as visual quality improvement, and it suits particularly well when the subsampling rate is high for each acquired parallel MRI data frame. The rest of this paper is organized as follows. In Sect. 2, we will give a brief introduction to the problem of parallel MRI reconstruction, and review several works that are closely related to the proposed approach of this paper. In Sect. 4, we will present the details of the proposed approach.next in Sect.5, we conduct several numerical experiments to validate the effectiveness of the proposed approach. Finally in Sect. 6, we will point out a few potential research directions and conclude the paper. 2 Problem Formulation and Related Work 2.1 Problem Formulation, Notations and Terminology We briefly introduce the data acquisition process of parallel MRI first. In parallel MRI, the information of the scanned object is gathered by several receiver coils simultaneously in k-

3 subsampling... mul plica on Image of the scanned object subsampling Sensi vity maps Coil (coded) images k-space data Fig. 1 Illustration of the acquired data for parallel MRI space. Each receiver coil posses some localized sensitivity profile, so that the data acquired by each coil is essentially a multiplication of the full image by a sensitivity map associated with that coil. Then, the acquired data of parallel MRI is nothing but the Fourier coefficients of the coil-coded images. To accelerate the scanning speed, there is also a subsmapling procedure, so that the whole data acquisition process can be mathematically described as follows K i = EF(S i I ), i = 1, 2,...,Nc, (1) where K i is the data gathered in coil i, E is the subsampling operator, F is the Fourier transform, S i is the sensitivity map of receiver coil i, is the component-wise product (i.e. Hadamard product), I is the ground truth image of the scanned object, and Ncis the number of coils. A simple instance of this process (with Nc = 2) is illustrated in Fig. 1 (where we have omitted the procedure of subsampling in k-space domain for simplicity). It is seen from the data gathering process that, the acquired parallel MRI data resulted from a single scan is essentially 3D, with the first two dimensions stand for spatial information of the scanned object, and the third dimension stands for indices of the receiver coils. For dynamic parallel MRI, the object usually undergoes a series of scan sequentially, which results in multiple frames of parallel MRI data. In this case, the acquired data is essentially 4D, with the last dimension standing for temporal direction. In this paper, we use K to denote the gathered dynamic parallel MRI data in k-space, where unacquired sample points are filled with zeros. Thus, K is assumed to be a 4D complex array with size Nx Ny Nc Nt,whereNx Ny is the size of acquired data in Fourier domain at each coil of each frame, and Nc is the number of coils for each frame of data, and Nt is the number of frames. The set of sensitivity maps is consisted of a number of Nc matrices {S i, i = 1, 2,...,Nc} of size Nx Ny.LetI t be the image of the scanned object at time t, then the parallel MRI data frame K t at time t is a 3D array composed of Nc layers such that K i,t := {E t F(S i I t ) : i = 1, 2,...,Nc}, t = 1, 2,...,Nt (2) where E t is the subsampling operator at time t (which retains those acquired sample points while setting unacquired sample points to have zero values). These sensitivity maps are

4 complex valued, and they are usually supposed to be spatially smooth and satisfy the following identity at each pixel Nc S i 2 = 1. (3) i=1 It is easy to see that if the full (multi-coil) data is given in the image domain, then the ground truth image I is the sum-of-square (SOS) of these coil coded images, thus the recovery procedure can be implemented without requiring the explicit knowledge of coil sensitivity maps. However, if the acquired data is subsampled, the image of the scanned object cannot be recovered so easily when the knowledge of coil sensitivity maps is entirely missing. Essentially, the main task for parallel MRI reconstruction is to recovery the latent image (sequence) as well as coil sensitivity maps from the subsampled data acquired in k-space. This is a typical bilinear system that poses many ambiguities/difficulties for both theoretical analysis and computation. The main focus of this paper is to explore the correlations among different data frames acquired dynamically for the aim of data recovery. In particular, for a sequence of 2D images (or data frames) {A t : 1 t Nt}, weuse A to denote the nuclear norm of the matrix A such that column t of A is exactly the vectorization of A t, i.e. the column vector formed by concatenating all columns of A t consecutively. 2.2 Related Work Now we are ready to introduce several works that are most closely related to the proposed one of this paper. As we have suggested previously, these approaches work either on image domain or on k-space domain. Among them, image domain based approaches are usually more easier to understand mathematically with better interpretability, so we start with a typical image domain approach called sensitivity encoding,orsense for short. SENSE By assuming explicit knowledge of coil sensitivity maps, SENSE [18] treats MRI reconstruction as a regular linear inverse problem. It is seen from (1) that, if those S i sare known and the subsampling rate is less than the number of receiver coils, then the number of unknowns is less than the size of the acquired data set, thus it is highly probable that one can get I explicitly by solving the least square alternative of (1) without any extra assumption on I. However, it is often better that we incorporate some regularization techniques to stabilize the reconstruction as well as to remove noise and possible artefacts (see e.g.[13,15,23]). GRAPPA Different from SENSE, GRAPPA [9] treats parallel MRI reconstruction as an interpolation problem in k-space domain. In order to apply GRAPPA, one needs a fullysampled regional data in the k-space domain, i.e. the auto-calibration signal (ACS). A set of interpolation kernels (often called the GRAPPA kernels) can then be derived from the autocalibration region, and the reconstruction of parallel MRI is done in a coil-by-coil fashion using block-wise interpolation. In more detail, to recover a missing sample x i (p) in the i th coil at the k-space position p, we first choose a block of k-space data from all coils around position p (filling with zeros when missing entries are encountered), then we sub-sample the chosen k-space data block to get the acquired data only and further stretch it into a column vector denoted by y i,p,thenx i (p) is recovered via x i (p) = y i,p w i,p,

5 where w i,p is a weight vector that has the same size of y i,p, specific to the data acquiring pattern around k-space position p in coil i. In literature those w i,p s are often referred as GRAPPA kernels, and they are computed from the ACS. The original GRAPPA algorithm is non-iterative, while the reconstruction quality could be improved by some iterative algorithms (see e.g. [24]). SPIRiT and ESPIRiT It is seen that GRAPPA kernels must vary from position to position because of the change of local data acquiring patterns, which lacks shift-invariance. In SPIRiT [16], a set of shift-invariant features for representing the generic block-structures in k-space domain is derived. In more detail, by prefixing the block-size w w Nc and scanning over all possible data blocks in the ACS, a structure data matrix A can be constructed, where each column of A represents a vectorized version of a block in the auto-calibration region. Then, principle component analysis is applied to A to obtain its principle components. Suppose there are a total numberof P principle components, we may reshape them into block shape and denote the reshaped principle components by B 1, B 2,..., B P.ThenwetreatB j, j = 1,...,P as convolution kernels and convolve them with the k-space data, which results in P convolution matrices B 1, B 2,...,B P.LetB =[B 1 ; B 2 ;...; B L ], SPIRiT recovers the full k-space data z by solving the following minimization problem z = arg min z (Id B B)z 2 2, s. t. z C = y C (4) where Id is the identity matrix, and C is the subset of of the k-space grid that corresponds to the acquired sample points only. It is seen from (4) that SPIRiT essentially computes a full k-space data whose blocks are closest to the space spanned by {B j : j = 1, 2,...,L} in l 2 sense. In other words, the optimal solution z approximately lies in the null-space of the operator Id B B. ESPIRiT [22] further explores this null-space representation, and formulate sensitivity map recovery as an eigenvalue decomposition problem. ESPIRiT is seen as an approach bridging the gap between the SENSE and GRAPPA (SPIRiT). SAKE Auto-calibration based approaches could be ineffective when there is no ACS in the acquired k-space data. SAKE overcomes this difficult by recovering the missing entries in the auto-calibration region via structure matrix completion, and it essentially explores the low-rank nature of the data matrix which is exactly the same data matrix mentioned in SPIRiT that is derived from the auto-calibration region. It is noted that, if the auto-calibration region of k-space are subsampled randomly and missing entries are filled with zeros, then the low-rank structure of the data matrix of formed from the auto-calibration region is quite probably ruined. By assuming that we have a fairly good estimation on the rank of the data matrix formed from the auto-calibration region of the latent full k-space data, SAKE works essentially by alternatively enforcing the low-rank property of the structure matrix and preserving the known data samples. Once a data preserving low-rank structure data matrix is obtained, a conjugate operation can be applied to transform the data matrix into the data samples in the auto-calibration region. By now SAKE requires least input for parallel MRI data reconstruction, yet it is single-frame based and tends to fail when the subsampling rate is high. Low-rank-plus-sparse decomposition In parallel MRI, the data is usually acquired dynamically in a frame-by-frame fashion, i.e. the acquired data is consisted of a sequence of k-space data frames that are highly correlated. Thus, it is natural to explore the data redundancy along

6 the temporal direction and utilize the coherence among difference data frames to enhance the performance of data reconstruction, see e.g. [10,12,14]. Specially, one may consider the low-rank-plus-sparse (LPS) model [3]. In more detail, let I t be the latent image of the scanned object at time t,1 t Nt, by stretching all I t s into column vectors and concatenate them sequentially in to a matrix G according to the order of their inherited time index, then G can be written as the following decomposition G = L + S, where L is a low-rank matrix, and S is a sparse matrix (the sparsity of S could be measured under certain appropriate invertible transform T, i.e. S itself might be dense while T (S) contains few nonzero entries). The LPS model has been adopted by Gao et al. [7,8], Otazo et al. [17] for dynamic parallel MRI reconstruction. If we use D to denote the linear operator that relates the latent image sequence to the acquired k-space data samples K, then as suggested in [17], the latent image sequence G L + S could be recovered by solving the following minimization problem min L,S 1 2 D(L + S) K 2 F + λ L L + λ S T (S) 1. (5) Note that the operator D is the composition of three operators E t FS i as presented in (2). Thus, the LPS model suggested in (5) also requires the explicit knowledge of coil sensitivity maps, which is the same as SENSE. In fact, the first term in (5) is same as the SENSE model for data recovery. However, the LPS model takes into account the correlation of the sequential data frames, so it can afford higher subsampling rate for each individual data frame, in which case the SENSE approach might become less effective even under regularization. 3 Motivation It is seen from the previous literature review that, obtaining the set of coil sensitivity maps is of crucial importance for most parallel MRI reconstruction methods. Different from recovering the full parallel MRI data, the estimation of coil sensitivity maps usually requires much less information thanks to their spatial smoothness. In fact, if the physical pre-scan procedure is abandoned, then one only needs to acquire the ACS instead of the full data so as to derive the set of coil sensitivity maps with high precision. Notice that the centered part of the MRI data corresponds to low frequencies that reflect smooth variations in image domain, we may arrive at a rough conclusion that a fair estimation of the set of coil sensitivity maps can be obtained from a coarse approximation of the true parallel MRI data by omitting some high-frequency details. How to obtain a good coarse approximation of the full dynamic parallel MRI data when it is subsampled? It is known that for dynamic imaging, the acquired image frames are usually highly correlated. In other words, there is high rate of data redundancy, which makes it possible for full data recovery from subsampled one. As applied to the case of dynamic parallel MRI imaging, this means we can subsample the k-space data at higher rate while maintaining the possibility for full k-space data recovery. High subsampling rate is important for MRI since it will accelerate the data acquiring process significantly. Such effort has already been made in several previous works (see e.g. [7,17]), yet in order to guarantee high-quality data recovery, they often need assume that the set of coil sensitivity maps is known as a prior (either explicitly or implicitly).

7 In this paper, we show that by exploring the redundancy of data frames in dynamic parallel MRI, it is possible to achieve simultaneous coil sensitivity map estimation and full k-space data reconstruction without auto-calibration or explicitly knowledge of the set of coil sensitivity maps. There are two key steps for achieving this objective: firstly we propose a low-rank based model for reconstructing the data sequence in a coarse manner, then we derive a more intuitive and mathematically more direct approach for estimating the set of coil sensitivity maps from the coarsely reconstructed parallel MRI data. Once the set of coil sensitivity maps is estimated, a refined reconstruction of the dynamic parallel MRI data could be obtained by applying the regular LPS model. We want to emphasize that, although another approach for calibrationless parallel MRI reconstruction, namely SAKE, can achieve similar objective of this paper. However, the taste of the proposed approach and SAKE is quite different. Firstly, SAKE is essentially a single frame based approach with its emphasize on exploring the incoherent structure within the data provided in a single frame, while our approach tries to explore the data redundancy among the acquired multiple data frames. Secondly, the set of coil sensitivity maps estimated from SAKE is usually less accurate due to the use of single frame data, and it is still unclear how to improve the precision of SAKE by using multiple frames of data; in contrast, the proposed model utilizes naturally multiple frame information and tends to result in less reconstruction error. 4 The Proposed Approach for Dynamic Parallel MRI Reconstruction 4.1 The Main Work Flow The proposed approach of this paper is motivated by the LPS model derived in [17] for dynamic parallel MRI reconstruction. However, instead of focusing on the LPS structure of the matrix obtained from the latent image sequence {I t : t = 1, 2,...,Nt} obtained by first stretching all latent images into column vectors and then concatenating together, we go deep to each coil and consider the LPS structure of the matrix obtained from the coil-coded latent image sequence {S i I t : t = 1, 2,...,Nt}, i = 1, 2,...,Nc. By going deep into the coil level, we propose a novel calibrationless dynamic parallel MRI reconstruction method which is consisted of the following two major steps: Step 1: Estimate the set of coil sensitivity maps by recovering the full parallel MRI data coarsely in a coil-by-coil fashion based on exploring the within-coil-between-frame structure of the data sequence; Step 2: Recover the full parallel MRI data more precisely with the help of the estimated sensitivity maps. Note that the above scheme essentially contains two stage of MRI reconstruction: the first stage aims providing a good estimation of coil sensitivity maps via coarse reconstruction, while the second stage refines the reconstruction using the coil sensitivity maps provided by the first stage. Details of the above scheme will be explained in the following subsections.

8 4.2 Coarse Parallel MRI Data Recovery For the proposed approach, a coarse recovery of parallel MRI data is performed in a coil-bycoil fashion in spatial domain. In more detail, let {I t : t = 1, 2,...,Nt} be the latent image sequence of the scanned object, then the data sequence in coil i is generated by composing with the corresponding sensitivity map S i, so that we obtain {S i I t : t = 1, 2...,Nt}, where denotes the Hadamard product. Treating each I t (S i I t ) as a column vector, we use G (G i ) to denote the matrix obtained by concatenating the sequence {I t : t = 1, 2,...,Nt} ({S i I t : t = 1, 2...,Nt}, respectively). As we have mentioned previously, the matrix G (approximately) obeys the LPS decomposition model, i.e. G = L + S, where L is a low-rank matrix, and S is sparse matrix. The matrix G i can be seen as applying the same sensitivity map S i to each column of G, thus it follows that G i = diag(s i )L + diag(s i )S. (6) As can be seen from the above model, the LPS decomposition model still holds true when restricting to the acquired data in each coil. However, the localization of each sensitivity map, say S i, brings along two important changes as well: firstly, the matrix diag(s i )L tends to have lower rank compared with L; secondly, the matrix diag(s i )S tends to have fewer nonzero entries compared with S and is thus even sparser. This observation suggests that, if the regular low-rank-plus sparse model is adopted for recovering the data sequence within coil i, then we need to impose larger regularization weights on the rank-control term of diag(s i )L and sparsity control term of diag(s i )S, otherwise the recovery will become unstable and erroneous. However, larger regularization weights also tend to bring loss of details to the recovered results. Besides, the (approximate) rank of diag(s i )L is likely to change when the coil index i changes, thus additional effort needs to be made for parameter tuning with respect to each coil, which brings instability when the regular LPS model is applied. For reasons as stated previously, it is unlikely that we can reconstruct the coil image sequence quite accurately by merely applying the LPS model. Therefore at this stage we only focus on coarse recovery of the data in each coil, which is mainly for removing the aliasing artefacts caused by subsampling. Although the aforementioned LPS model might be applied to achieve the objective of coarse data recovery, in implementation we have found that the recovered results are usually not satisfying enough for subsequent stages. In order to overcome the difficulties mentioned above, we proposed a new model to independently recover the data sequence within each coil. To introduce the proposed model, let us get back to the LPS model (6) in certain coil, say coil i.aswehavediscussedpreviously, the localization property of diag(s i ) makes diag(s i )S even sparser than S, so we ignore this sparse components, and only focus on the approximate low-rank prior of the latent coil image sequence plus some additional stabilization term(s). In designing the stabilization term, we have found that the regular low-rank assumption does not effectively take into account the sequential information of the coil image sequence. For instance, there should be relatively smooth transit between consecutive data frames, and thus their similarity should be emphasized. However, the regular low-rank model fails to do so and only treat all frames equally. Thus, we also adopt a new stabilization term by exploring the smooth transition between consecutive data frames. We note that this temporal stabilization term was also

9 adopted for dynamic SPECT reconstruction in [6]. In more detail, for the ith coil, let D := EF be the composition of Fourier transform and sub-sampling operator, then the coil image sequence recovery model can be written as 1 min L 2 DL Ki 2 F + λ L L + λ M 2 t L 2 F, (7) where K i denotes the acquired k-space data K in the ith coil, λ L and λ M are two positive constants. In practice, we find that the introduction of the stabilization term λ M 2 t L 2 F allows us to use a smaller regularization weight λ L for recovering the coil image sequence, thus more details are preserved and the recovery quality is much improved compared with the regular LPS model. Besides, this single λ L is shared by all coils and thus free us from the tedious work of coil-by-coil parameter tuning. The main advantage of the above reconstruction procedure is that it can be implemented independently for each coil, and is thus highly parallelizable. Moreover, the model in (7) contains only a single non-smooth term, which is simpler than the regular LPS model and can be solved more quickly in practise. The most important thing is, although the proposed model in (7) is still aims at coarse recovery, the recovered results of it are often enough to obtain some good estimation on the set of coil sensitivity maps. This is key for subsequent steps. 4.3 Full k-space Data Recovery Using Estimated Sensitivity Maps In this step, we will recover the images using estimated sensitivity maps after obtaining a coarse reconstruction of the full parallel MRI data. There are many possible ways for estimating the set of coil sensitivity maps from this coarsely reconstructed data. One popular approach is to crop a set of auto-calibration data centred at the k-space of the coarsely reconstructed data, and then apply ESPIRiT to compute the set of coil sensitivity maps. However, the computational cost of ESPIRiT could be high when coil dimension of the acquired data is big. In this paper, we take an alternative approach which is computationally much cheaper. We assume that the coil sensitivity maps are invariant in the temporal dimension. To address the issue of phase-ambiguity between the set of coil sensitivity maps and the image of the scanned object, we may well assume that the image of the scanned object is real-valued. Then, let E i,t be the coarse image in the ith coil at time t recovered by the model (7), then the sensitivity map in coil i can readily be computed by S i E i,t, R t where R t is the SOS image of the coarsely recovered data in frame t, i.e. R t = Nc E i,t 2. i=1 For better estimation of S i, we may take the average of the above estimation with respect to time, and set Ŝ i = 1 Nt E i,t. (8) Nt R t t=1

10 However, there is no guarantee that the set of coil sensitivity maps computed through (8) is spatially smooth, thus we may apply some smoothing technique, like Tikhnov type of regularization to enhance the compatibility of the estimated coil sensitivity maps with this smoothness prior. In more detail, the estimated sensitivity maps are obtained by solving the following set of minimization problems 1 S i = arg min X 2 X Ŝ i ν 2 X 2 2, i = 1,...,Nc (9) where Ŝ i is the the maps estimated directly in (8). Finally, a normalization procedure is needed so as to make sure that the sensitivity maps obtained would satisfy equation (3), i.e. we set where S = Nc S i S i 2 at each spatial position. S i, i = 1, 2,...,Nc, (10) S i=1 After obtaining the set of coil sensitivity maps, we are ready to recover the full k-space data with higher precision. In more detail, the Fourier space subsampling operator EF and coil sensitivity maps S can be put together to form the regular data acquisition operator D.At this stage, we directly apply the LPS model in [17] for full parallel MRI data reconstruction, i.e. we solve the minimization problem as suggested in (5). 4.4 Algorithms In the proposed approach, we need to solve three optimization problems, namely (7), (9) and (5). Among those three problems, the LPS model is identical to the model proposed in [17], so we adopt the algorithms therein, i.e. the iterative soft-thresholding (IST) algorithm (see also [5]). The coil image sequence recovery model in (7) can be solved by classical Proximal forward backward splitting (PFBS) algorithms [4] or its accelerated version (see, e. g. [1,19]). We note that, since (7) is a real-valued function with complex variable L, we need to compute the Wirtinger gradient (see e.g. [11]) instead of traditional gradient of the objective function. The explicit solver for (7) is summarized in Algorithm 1. It is easy to see from the above algorithm that the Lipschitz constant of f is directly related to the spectral norm of the operator D D + λm t t.since D D + λm t t D D +λm t t = λ M, thus in Algorithm 1 we should set β λ M. Besides, the backward step suggested in Algorithm 1 is essentially based on singular value thresholding (see e.g. [2]). Thecoil map smoothing model in (9) is an unconstrained quadratic minimization problem and can be solved directly with ease. In fact, it is easy to see that (9) has the following closedform solution S i = (Id νδ) 1 Ŝ i, where Idis the identity operator and Δ is the (discrete) Laplacian operator. Note that Id νδ is diagonalizable under FFT when periodic boundary condition is assumed, so its inversion (Id νδ) 1 can be computed with ease. Finally, we summarize the proposed approach in Algorithm 2.

11 Algorithm 1 Numerical solver for (7) Require: Subsampled k-space data sequence K i in coil i Parameter Setting: Choose λ L > 0andλ M > 0. Abbreviations: Let and f (L) = 1 2 DL K i 2 F + λ M 2 t L 2 F h(l) = λ L L. where is spatial gradient operator and t is the temporal gradient operator. Initialization: SetL (0) = F 1 K i, k = 0 for k = 0, 1, 2...do Step (1). compute Step (2). apply gradient descent f (L (k) ) = ( D D + λ M t t )L (k) D K i, L (k) = L (k) 1 β f (L(k) ), where β>0 is no less than the global Lipschitz constant of f, Step (3). compute the singular value decomposition of L (k) (by treating temporal frames of L (k) as column vectors) [U, S, V ]=svd( L (k) ) where such that U and V are both unitary matrices, and S is a diagonal matrix with nonnegative entries such that L (k) = USV. Step (4). apply soft-thresholding to (the diagonal entries of) S such that where T γ (x) = (T γ (x 1 ), T γ (x 2 ),...)with Step (5).update end for S = T λm /β(s), T γ (x j ) = x j x j max (0, x j γ), for j = 1, 2,... L (k+1) = U SV. 5 Experiments In this section we conduct several numerical experiments to test the performance of the proposed approach. We compare the results of the proposed approach to those of SAKE, which currently serves as the state-of-the-art approach for calibrationless parallel MRI reconstruction. However, SAKE is single-frame based and the proposed approach is multiple-frame based, i.e. it explores correlation among multiple data frames. In order to gain robustness for SAKE and make the comparison fairer, in this experimental section we also adopt a

12 Algorithm 2 The main work flow of the proposed approach Require: : Subsampled k-space data frame sequence K with size Nx Ny Nc Nt, where unacquired entries are filled with zeros. Ensure: : Image sequence and coil sensitivity maps G ={I t : t = 1,...,Nt} {S i : i = 1,...,Nc}. Step (1) For i = 1, 2,...,Nc, apply Algorithm 1 to coarsely recover the image domain data contained in coil i; Step (2) Compute the estimated coil sensitivity maps { S i : i = 1,...,Nc} via (7), and then apply smoothing (9) and normalization (10) to obtain the refined coil sensitivity maps {S i : i = 1,...,Nc}. Compute the refined image sequence G = L + S by solving the LPS model suggested in (5). mixed strategy as follows: for sensitivity map estimation we use the module in SAKE, while for MRI image sequence recovery we use the LPS model in [17]. We name this mixed approach as SAKE + LPS in this paper. However, to implement SAKE + LPS, firstly one needs to choose a single data frame for estimating the set of coil sensitivity maps via SAKE. Since there is no clue to which choice of frame will give the best estimation, we simply go through all data frames, and pick out the one that results in the minimum reconstruction error. Quantitative image quality assessment was performed using the metrics of root mean square error (RMSE) in this paper. A detail description of experiments on two test data sets and the performance of the aforementioned three approaches are present in the following subsections. 5.1 Experiment 1 Our first experiment is carried out on a sequence of dynamic cardiac perfusion parallel MRI data taken from [17]. As suggested therein, this data were acquired in a volunteer with a modified TurboFLASH pulse sequence on a whole-body 3 T scanner using a 12-element matrix coil array. The acquired data sequence is composed of 40 temporal frames with image matrix size , and other key parameters with respect to this dataset are: FOV= mm 2, slice-thickness = 8 mm, flip angle = 10 o, TE/TR = 1.2/2.4ms, spatial resolution = mm 2, and temporal resolution = 07 ms. This full data are subsampled at the rate of 8 using certain variable-density random undersampling pattern along y-direction in k-space for each temporal frame. We can see in Fig. 2 the visualization of the input data together with its subsampling pattern. The parameters of the proposed method for both data sets are summarized in Table 1.The reconstructed results of all three approaches are shown in Fig. 3, together with corresponding error maps. It is seen that the reconstruction error of the proposed approach is relatively small, which demonstrate the effectiveness of the proposed approach. In comparison, the reconstruction results of SAKE has much larger error compared with the proposed approach. This is very likely to be caused by the inadequacy of acquired data for each single frame. In contrast, the results produced by SAKE + LPS have much smaller error compared with the results of SAKE. This is not a surprise since the LPS model explores correlations among different data frames and its thus more robust in the case of data inadequacy. However, the results of SAKE + LPS are still not as good as those of the proposed approach. This could

13 frame 10 frame 20 frame 30 Fig. 2 Illustration of the selected frames of the dynamic perfusion parallel MRI data in Experiment 1. Top row ground truth images (obtained via SOS). Middlerowthe subsampling patterns for corresponding frames. Bottom row the observed aliased images (obtained via SOS) Table 1 The parameters set for the proposed method Data block sizes Model (7) Model (9) Model (5) ACS size Window size λ L λ M ν λ L λ S Data Data potentially be caused by the relative high subsampling rate in all data frames, which renders SAKE less effective in recovering the auto-calibration regions and coil sensitivity maps. Again, to better illustrate the performance of all three approaches, we summarize the errors of all reconstruction results in Table Experiment 2 The second experiment is carried out on a set of dynamic cardiac function data. 1 The data were collected from a CINE 2D breath-held fully sampled fully balanced steady state free precession (TrueFisp/bSSFP) sequence. It is consisted of 20 temporal frames with image matrix size The data is subsampled at the rate 14, i.e. only (around) 7 % of k- space data is acquired. Instances of the adopted subsampling pattern are shown Fig. 4 together with the visualization of the input data. 1 Data source -

14 Reconstruction 1 Error (x5) SAKE (frame by frame) 0 Reconstruction 1 Error (x5) SAKE+LPS 0 Reconstruction 1 Error (x5) Proposed 0 Fig. 3 Reconstruction results of the first testing data set (cardiac perfusion data). For each approach, the upper row shows the reconstruction results of the selected frames while lower row shows their difference with the reference data frames (ground truth)

15 Table 2 Reconstruction error for the testing dataset in Experiment 1 Frame 10 Frame 20 Frame 30 Average (40 frames) SAKE (frame by frame) SAKE + LPS Proposed The bold text indicates the smallest reconstruction error of the three methods frame 5 frame 10 frame 15 Fig. 4 Illustration of the sampling pattern for the selected frames of the dynamic parallel MRI data for Experiment 2. For the selected frames, we show the ground truth images in the top row, the subsamping patterns in the middle row, and the observed aliased images in the bottom row The reconstruction results of all three approaches are shown in Fig. 5. It is seen the reconstruction error of the proposed approach is relatively small for the object of interest. In comparison, the reconstruction results of both SAKE and SAKE + LPS have larger error. Again, the SAKE + LPS approach performs better than SAKE thanks to the exploration of correlation among different data frames using LPS. The numerical (relative) error of all reconstruction results is summarized in Table 3, which is consistent with the result of visual comparison. 5.3 Discussion on Stability and Computation Cost It is seen that the overall performance of the proposed approach is better than SAKE + LPS, when the subsampling rate of acquired data frames is relatively high. Moreover, in the experiments we have found that the performance of SAKE + LPS strongly depends on the right choice of data frame for applying the SAKE procedure. When the choice of data frame changes, the quality of reconstruction results can fall dramatically. To illustrate this fact, we take out to first 10 frames of the input data of Experiment 2, apply SAKE independently

16 Reconstruction 1 Error (x5) SAKE (frame by frame) 0 Reconstruction 1 Error (x5) SAKE+LPS 0 Reconstruction 1 Error (x5) Proposed 0 Fig. 5 Reconstruction results of the second testing data set. For each approach, the upper row shows the reconstruction results of the selected frames while lower row shows their difference with the ground truth data on those data frames, and summarize corresponding errors of the SAKE + LPS approach in Fig. 6. It is seen that the those errors resulted varies a lot from frame to frame, and many are far away from the smallest error summarized in Table 3. In our implementation, the best choice of data frame for SAKE + LPS can only be identified via exhaustive enumer-

17 Table 3 Reconstruction error for the testing dataset of Experiment 2 Frame 5 Frame 10 Frame 15 Average (20 frames) SAKE (frame by frame) SAKE + LPS Proposed The bold text indicates the smallest reconstruction error of the three methods Fig. 6 Plot of SAKE + LPS reconstruction error with respect to different choices of data frames for applying SAKE. The scatters circles are those average RMSE values. It is seen from this plot that if wrong frame index is chosen, the reconstruction error (averaged among all frames) could become very large Average RMSE Optimal Error SAKE+LPS Frame index ation. 2 In this sense, the performance of the proposed approach is more stable, since its coil sensitivity map estimation procedure efficiently integrates information within all data frames. The main computation of the proposed approach rests on two steps: the coarse data recovery step and the refined data recovery step using the LPS model. The former one possess more computational cost since the two step have the same data acquisition operations while the former one needs to perform a larger number of singular value decompositions. In Table 4, we present the overall computation time for the three approaches with MATLAB implementation on a work station with two Intel Xeon E CPU. The first approach applies SAKE independently on each frame, and it takes the most of time as the sensitive map is estimated on every frame. Our method is slower than SAKE + LPS as there is one additional step to estimate the coarse images. We also note that the current implementation is not optimized as the coarse data recovery step can be implemented in parallel (coil-bycoil) and an accelerated version PFBS algorithm can be applied to solve (7). In practice, when parallel computing is applied, the computational time for coarse data recovery can be greatly reduced, thus the overall computation time is comparable to solving the LPS model. 2 One might intuitively think that the the data frame with maximum number of samples in the auto-calibration region should be the best choice, but that is not true. For instance, in the second experiment of Sect. 5, the data frame with maximum number of samples in the auto-calibration region is frame 13, while exhaustive enumeration suggests frame 2 is the best choice, and the later one leads to significantly lower reconstruction error compared with the former one.

18 Table 4 The CPU time (s) of the all data frame reconstruction for the two experiments SAKE (frame by frame) SAKE + LPS Proposed Experiment Experiment Conclusion and Future Work In this paper, we have proposed a novel two-stage approach for calibrationless dynamic parallel MRI reconstruction. The key of the proposed approach is the first phase, which recovers the full k-space data coarsely by exploring the redundancy of data sequence within each coil, and then use the coarsely reconstructed data to estimate the set of coil sensitivity maps. The coarse data recovery procedure is highly parallelizable (multiple coils could be processed at the same time), and the model is easy to implement. The results of numerical experiments suggest that the overall performance of the proposed approach is better than the other calibrationless approaches for dynamic parallel MRI reconstruction. The proposed approach suits particularly well for cases when the subsampling rate of all data frames is high. Besides, the reconstruction quality of the proposed approach is much stable, and it does not rely upon special choice of data frames like SAKE + LPS. Exploring redundancy and inter-correlation is key for recovering missing entries of various data structures. Dynamic parallel MRI datasets are naturally 4D tensors, and within in each dimension exist different degree of data redundancy rate and incoherent relations. In this paper we merely focus on exploring the data redundancy in the time dimension. It is quite probable that the reconstruction results could be further improved and unified as one stage approach by deeper exploration of data representations with respect to other three dimensions, like low-rank tensor. However, by approaching that way the computational load must increase as well, thus we must develop practical tools for resolving this controversial issue, and strike a good balance between reconstruction quality and computational intensity. Acknowledgments We would thank the authors of [17] and[16,20,22] for making their codes, demos and experimental datasets free for academic use. References 1. Beck, A., Teboulle, M.: A fast iterative shrinkage-thresholding algorithm for linear inverse problems. SIAM J. Imaging Sci. 2(1), (2009) 2. Cai, J.-F., Candès, E.J., Shen, Z.: A singular value thresholding algorithm for matrix completion. SIAM J. Optim. 20(4), (2010) 3. Chandrasekaran, V., Sanghavi, S., Parrilo, P., Willsky, A.S., et al.: Sparse and low-rank matrix decompositions. In: 47th Annual Allerton Conference on Communication, Control, and Computing, Allerton 2009, pp IEEE (2009) 4. Combettes, P.L., Wajs, V.R.: Signal recovery by proximal forward backward splitting. Multiscale Model.Simul. 4(4), (2005) 5. Daubechies, I., Defrise, M., De Mol, C.: An iterative thresholding algorithm for linear inverse problems with a sparsity constraint. Commun. Pure Appl. Math. 57(11), (2004) 6. Ding, Q., Zan, Y., Huang, Q., Zhang, X.: Dynamic SPECT reconstruction from few projections: a sparsity enforced matrix factorization approach. Inverse Probl 31(2), (2015) 7. Gao, H., Lin, H., Ahn, C.B., Nalcioglu, O.: PRISM: A divide-and-conquer low-rank and sparse decomposition model for dynamic MRI. UCLA CAM Report (2011)

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