k-t SPARSE: High frame rate dynamic MRI exploiting spatio-temporal sparsity
k-t SPARSE: High frame rate dynamic MRI exploiting spatio-temporal sparsity
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发表时间：
2006
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通讯作者：
M. Lustig;Juan M. Santos;D. Donoho;J. Pauly
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作者：
M. Lustig;Juan M. Santos;D. Donoho;J. Pauly
M. Lustig， J. M. Santos， D. L. Donoho， J. M. Pauly电气工程，斯坦福大学，斯坦福，CA，美国，统计学，斯坦福大学，斯坦福，CA，美国最近提出了使用欠采样随机摄动螺旋和非线性重建来利用图像空间稀疏性的快速成像方法[1,2]。这些方法的灵感来自于稀疏信号恢复的理论结果[1-5]，这些结果表明，从随机欠采样的频率数据中可以恢复稀疏或可压缩的信号。我们提出了一种基于类似思想的高帧率动态成像方法，现在利用动态MRI图像序列（动态场景）的空间和时间稀疏性。我们通过对相位编码在时间上的随机排序，对k-t空间随机进行欠采样（图1）。我们通过最小化受数据保真度约束的变换动态场景的L1范数来重建。与之前提出的线性方法不同[7,8]，我们的方法不需要已知的时空结构，也不需要训练集，只需要动态场景具有稀疏表示。我们在模拟数据和体内非门控笛卡尔平衡ssfp心脏MRI中展示了7倍的帧率加速。动态磁共振图像在空间和时间上具有高度冗余性。通过使用线性变换（如小波、傅立叶等），我们可以只用几个稀疏变换系数来表示一个动态场景。空间-频率-时间空间（k-t空间）的采样不足会导致空间-时间-频率空间（x-f空间）的混叠。随机欠采样引起的混叠伪影与均匀欠采样引起的相干伪影相反是不相干的。更重要的是，这些伪影在稀疏变换域中是不相干的。利用[1-5]中的非线性重构方案，我们可以恢复稀疏变换系数，从而恢复动态场景。我们利用稀疏性，通过解决约束优化问题，约束我们的重建具有稀疏表示，并与测量数据一致：最小化||Ψm||1，服从：||Fm - y||2 < e。这里m是动态场景，Ψ将场景转换为稀疏表示，F是随机相位编码有序傅里叶矩阵，y是测量的k空间数据，e控制重建对测量数据的保真度。E通常设置为噪声级。方法针对动态心脏成像，提出在空间维度上使用小波变换，在时间维度上使用傅立叶变换。小波稀疏化医学图像b[1]，而傅里叶变换稀疏化平滑或周期性的时间行为。此外，对于随机k-t采样，混叠在这个特定的变换域中是极不相干的。为了验证我们的方法，我们考虑了一个具有周期性心脏运动的模拟动态场景。模拟随机相位编码有序笛卡尔采集（见图2），TR=4ms， 64像素，共获取1024个相位编码（4.096秒）。采用非线性共轭梯度L1重构方案，以15FPS的帧率（4倍加速因子）重构数据。结果与滑动窗口重构（长度为64个相位编码）进行了比较。为了进一步验证我们的方法，我们考虑了笛卡尔平衡- ssfp动态心脏扫描（TR=4.4， TE=2.2, α=60°，分辨率=2.5mm，切片=9mm）。1152个随机有序相位编码（5sec），使用L1方案以7倍加速度（25FPS）收集和重建。结果与滑动窗口（64相位编码）重建结果进行了比较。实验在1.5T GE Signa扫描仪上进行，使用5英寸表面线圈。图2和3显示了模拟的幻影和实际的动态心脏扫描重建。注意，即使在4到7倍的加速度下，所提出的方法也能够恢复运动，保留空间频率并抑制混叠伪影。该方法可以很容易地扩展到任意轨迹，也可以很容易地与其他加速方法（如相位约束部分k空间和SENSE[1]）集成。在目前的Matlab实现中，我们能够在一个小时内重建一个64x64x64的场景。这可以通过使用新提出的重建技术来改善[5,6]。先前提出的线性方法[7,8]利用已知或可测量的时空结构。该方法的优点是信号不需要有已知的结构，只需要稀疏性，这在动态医学图像中是一个非常现实的假设[1,7,8]。因此，不需要训练集。参考文献[1]Lustig等人，第13期ISMRM 2004:p605 [1] Lustig等人，“快速MR血管造影…”接受SCMR06 ' [3] Candes等，“稳健不确定性原则”。手稿。[10]张晓华，“压缩传感”。手稿。[10] Candes et al.“从随机投影中实际信号恢复”手稿。b[6] M. Elad，“为什么简单的收缩仍然相关？”手稿。[7]曹等人…中华医学杂志，2003,11(5):1031- 1042。[cn] Madore等。中华医学杂志，1999;42(5):813-28。图2：模拟的动态数据。(a)截面的变换域是真正稀疏的。(b)地面真值截面。(c)随机相位编码顺序L1重构，4倍加速(d)滑动窗口（64）随机相位编码顺序重构。图1:(a)顺序相位编码排序。(b)随机相位编码排序。k-t空间是随机采样的，这使得使用L1重构可以恢复稀疏的时空动态场景。
M. Lustig, J. M. Santos, D. L. Donoho, J. M. Pauly Electrical Engineering, Stanford University, Stanford, CA, United States, Statistics, Stanford University, Stanford, CA, United States Introduction Recently rapid imaging methods that exploit the spatial sparsity of images using under-sampled randomly perturbed spirals and non-linear reconstruction have been proposed [1,2]. These methods were inspired by theoretical results in sparse signal recovery [1-5] showing that sparse or compressible signals can be recovered from randomly under-sampled frequency data. We propose a method for high frame-rate dynamic imaging based on similar ideas, now exploiting both spatial and temporal sparsity of dynamic MRI image sequences (dynamic scene). We randomly under-sample k-t space by random ordering of the phase encodes in time (Fig. 1). We reconstruct by minimizing the L1 norm of a transformed dynamic scene subject to data fidelity constraints. Unlike previously suggested linear methods [7, 8], our method does not require a known spatio-temporal structure nor a training set, only that the dynamic scene has a sparse representation. We demonstrate a 7-fold frame-rate acceleration both in simulated data and in vivo non-gated Cartesian balanced-SSFP cardiac MRI . Theory Dynamic MR images are highly redundant in space and time. By using linear transformations (such as wavelets, Fourier etc.), we can represent a dynamic scene using only a few sparse transform coefficients. Inadequate sampling of the spatial-frequency -temporal space (k-t space) results in aliasing in the spatial -temporal-frequency space (x-f space). The aliasing artifacts due to random under-sampling are incoherent as opposed to coherent artifacts in equispaced under sampling. More importantly the artifacts are incoherent in the sparse transform domain. By using the non-linear reconstruction scheme in [1-5] we can recover the sparse transform coefficients and as a consequence, recover the dynamic scene. We exploit sparsity by constraining our reconstruction to have a sparse representation and be consistent with the measured data by solving the constrained optimization problem: minimize ||Ψm||1 subject to: ||Fm – y||2 < e. Here m is the dynamic scene, Ψ transforms the scene into a sparse representation, F is randomized phase encode ordering Fourier matrix, y is the measured k-space data and e controls fidelity of the reconstruction to the measured data. e is usually set to the noise level. Methods For dynamic heart imaging, we propose using the wavelet transform in the spatial dimension and the Fourier transform in the temporal. Wavelets sparsify medical images [1] whereas the Fourier transform sparsifies smooth or periodic temporal behavior. Moreover, with random k-t sampling, aliasing is extremely incoherent in this particular transform domain. To validate our approach we considered a simulated dynamic scene with periodic heart-like motion. A random phase-encode ordered Cartesian acquisition (See Fig. 2) was simulated with a TR=4ms, 64 pixels, acquiring a total of 1024 phase encodes (4.096 sec). The data was reconstructed at a frame rate of 15FPS (a 4-fold acceleration factor) using the L1 reconstruction scheme implemented with non-linear conjugate gradients. The result was compared to a sliding window reconstruction (64 phase encodes in length). To further validate our method we considered a Cartesian balanced-SSFP dynamic heart scan (TR=4.4, TE=2.2, α=60°, res=2.5mm, slice=9mm). 1152 randomly ordered phase encodes (5sec) where collected and reconstructed using the L1 scheme at a 7-fold acceleration (25FPS). Result was compared to a sliding window (64 phase encodes) reconstruction. The experiment was performed on a 1.5T GE Signa scanner using a 5inch surface coil. Results and discussion Figs. 2 and 3 illustrate the simulated phantom and actual dynaic heart scan reconstructions. Note, that even at 4 to 7-fold acceleration, the proposed method is able to recover the motion, preserving the spatial frequencies and suppressing aliasing artifacts. This method can be easily extended to arbitrary trajectories and can also be easily integrated with other acceleration methods such as phase constrained partial k-space and SENSE [1]. In the current, Matlab implementation we are able to reconstruct a 64x64x64 scene in an hour. This can be improved by using newly proposed reconstruction techniques [5,6]. Previously proposed linear methods [7,8] exploit known or measured spatio-temporal structure. The advantage of the proposed method is that the signal need not have a known structure, only sparsity, which is a very realistic assumption in dynamic medical images [1,7,8]. Therefore, a training set is not required. References [1] Lustig et al. 13th ISMRM 2004:p605 [2] Lustig et al. ” Rapid MR Angiography ...” Accepted SCMR06’ [3] Candes et al. ”Robust Uncertainty principals". Manuscript. [4] Donoho D. “Compressesed Sensing”. Manuscript. [5] Candes et al. “Practical Signal Recovery from Random Projections” Manuscript. [6] M. Elad, "Why Simple Shrinkage is Still Relevant?" Manuscript. [7] Tsao et al.. Magn Reson Med. 2003 Nov;50(5):1031-42. [8] Madore et al. Magn Reson Med. 1999 Nov;42(5):813-28. Figure 2: Simulated dynamic data. (a) The transform domain of the cross section is truly sparse. (b) Ground truth crosssection. (c) L1 reconstruction from random phase encode ordering, 4-fold acceleration (d) Sliding window (64) reconstruction from random phase encode ordering.. Figure 1: (a) Sequential phase encode ordering. (b) Random Phase encode ordering. The k-t space is randomly sampled, which enables recovery of sparse spatio-temporal dynamic scenes using the L1 reconstruction.