Low-rank plus sparse tensor reconstruction for high-dimensional cardiac MRI
Rebecca Ramb1, Michael Zenge2, Li Feng1, Matthew Muckley1, Christoph Forman3, Leon Axel1, Dan Sodickson1, and Ricardo Otazo1

1Department of Radiology, New York University School of Medicine, New York, NY, United States, 2Siemens Medical Solutions USA, Inc., Malvern, PA, United States, 3Siemens Healthcare GmbH, Erlangen, Germany


The recently proposed general low rank tensor framework enabled a paradigm change, where data acquisition and image reconstruction are represented in a higher-dimensional space. The overall data space is sampled only as different states randomly coincide, which leads to data gaps. These gaps can introduce challenges in spatiotemporal fidelity for only low-rank- or only sparsity-based reconstructions. Here, a $$$\mathcal{L}+\mathcal{S}$$$ tensor decomposition is investigated, which offers a more robust solution as the sparse component captures updates on top of the overall dynamics represented in the low-rank component. A free-breathing, T1-sensitive cardiac MRI with real-time Cartesian data acquisition over multiple cardiac and inversion recovery phases is employed to investigate potentials for comprehensive cardiac MRI, including for instance late gadolinium scar cine imaging.


Rapid continuous data acquisition and multidimensional image reconstruction has the potential to disentangle motion and contrast dynamics, which promises to reduce challenges in cardiac MRI and to use scan time more efficiently, e.g.$$$~$$$[1,2]. Whereas previous work employed sparsity constraints along each dimension separately$$$~$$$[2], tensors were more recently proposed to represent high-dimensional images in a unified way and assuming a low-rank tensor representation$$$~$$$[3,4,5]. However, low-rank tensor representation or sparsity constraints alone might be challenging, if rank-assumptions are violated or data gaps impair reconstruction fidelity. In this work, we investigate a low-rank plus sparse ($$$\mathcal{L}+\mathcal{S}$$$)$$$~$$$[6,7] tensor completion approach, which can represent multidimensional images more flexibly as the superposition of a common low-rank background tensor and dynamic innovations. This approach is applied to free-breathing, T1-sensitive cardiac MRI with real-time data acquisition over multiple cardiac and inversion recovery phases. This is motivated to enable late gadolinium enhancement (LGE) scar Cine imaging for efficient clinical practice.


Repeated RF inversion was implemented in a 2D Cartesian real-time balanced steady state free precession prototype sequence$$$~$$$[8] which supports variable density incoherent sampling. 15$$$~$$$readout lines were acquired per time frame resulting in a frame rate of 43$$$~$$$ms and RF inversion was applied every 30$$$~$$$time frames (Fig.$$$~$$$1). Experiments were performed in two healthy volunteers on a clinical 3T MRI scanner (Magnetom Skyra, Siemens Healthcare GmbH, Erlangen, Germany) equipped with body and spine local RF receive arrays with up to 30$$$~$$$active elements. Continuously acquired 2D data were manually sorted into an undersampled 4D data tensor with 20$$$~$$$cardiac motion and 91$$$~$$$inversion recovery phases as dynamic dimensions. To this end, cardiac cycles were extracted from the real-time series and arranged as 'diagonals' in the space spanned by cardiac and inversion recovery phases (illustrated in Fig.$$$~$$$1). $$$\mathcal{L}+\mathcal{S}$$$ tensor reconstruction was performed by solving the following optimization problem:$$arg\!\min_{\mathcal{L},\mathcal{S}}{\left\|{\mathbf{E}\left(\mathcal{L}+\mathcal{S}\right)-\mathcal{Y}}\right\|}_2^2+\sum_{i=2}^{3}\lambda_i{\left\|{L_{(i)}}\right\|}_{\star}+\sum_{i=2}^{3}\mu_i{\left\|{\mathbf{T}S_{(i)}}\right\|}_{1},$$where $$$\mathcal{L},\mathcal{S}\in\mathbb C^{N_xN_y{\times}N_{ph}\times N_{TI}}$$$ represent the low-rank and sparse terms, respectively, of the image tensor $$$\mathcal{M}=\mathcal{L}+\mathcal{S}\in\mathbb{C}^{N_xN_y{\times}N_{ph}{\times}N_{TI}}$$$ (schematic in Fig.$$$~$$$2), $$$\mathcal{Y}\in \mathbb{C}^{N_x^{smp}N_y^{smp}{\times}N_{ph}^{smp}{\times}N_{coils}{\times}N_{TI}^{smp}}$$$ is the data tensor, $$$\mathbf{E}$$$ is the encoding operator, $$$\mathbf{T}$$$ is a sparsifying transform for the sparse component, and $$$\lambda_i,\mu_i$$$ are regularization parameters that weight the contributions of the nuclear-norm and $$$\ell$$$1-norm terms with respect to $$$\ell$$$2-norm data consistency. The subscript notation $$$L_{(i)}$$$ and $$$S_{(i)}$$$ refers to unfoldings along the $$$i$$$th dimension$$$~$$$[9]. For comparison, image reconstruction was also performed separately for only the low-rank (L-only) or only the sparsity (S-only) constraint, incorporating either the nuclear or $$$\ell$$$1-norm regularizer.


Figure$$$~$$$2 demonstrates the $$$\mathcal{L}+\mathcal{S}$$$ tensor decomposition for selected inversion recovery phases of the same cardiac phase. The low-rank background tensor $$$\mathcal{L} $$$ captures the main overall inversion recovery dynamics, whereas the sparse component $$$\mathcal{S}$$$ contains the innovations related to contrast changes on top of the main dynamics. This leads to an improved spatiotemporal resolution compared to the L-only or S-only reconstructions (Fig.$$$~$$$3). In particular, the L-only approach results in significant degradation, which becomes further apparent in the inversion recovery curve at the myocardium region (Fig.$$$~$$$4). Animation of one complete cardiac cycle at different contrasts (Fig.$$$~$$$5) shows slight residual aliasing also for $$$\mathcal{L}+\mathcal{S}$$$.


The general tensor framework encompasses a paradigm change, where data acquisition and image reconstruction are represented in a higher-dimensional space. However, sampling in the higher-dimensional space usually results in data gaps (Fig.$$$~$$$1) that introduce challenges for low-rank-based or sparsity-based reconstructions. A typical L-only or S-only approach will present a solution that implicitly interpolates the data gaps at the expense of spatiotemporal resolution$$$~$$$(Figs.$$$~$$$3-5). The proposed $$$\mathcal{L}+\mathcal{S}$$$ tensor completion approach offers a more robust solution as data gaps can be represented as outliers in the model, which supports higher spatiotemporal resolutions (Figs.$$$~$$$3-5). One limitation in the current approach is that organ motion is not managed appropriately by $$$\mathcal{L}+\mathcal{S}$$$, which is causing residual aliasing artifacts (Fig.$$$~$$$5). Next steps will include employing a motion model in the $$$\mathcal{L}+\mathcal{S}$$$ reconstruction, as demonstrated in$$$~$$$[10]. Likewise, different sampling patterns and timing parameters, so as to best populate the higher-dimensional space is of further interest. Many clinical protocols comprise inversion recovery for LGE imaging, which requires a pre-scan to determine the best inversion time for targeted contrast behavior. The presented Cartesian acquisition has the potential for establishing efficient comprehensive cardiac exams, including LGE scar Cine imaging or the recently presented high quality T1 assessment [5].


$$$\mathcal{L}+\mathcal{S}$$$ tensor completion can provide a robust, unbiased and flexible approach to multidimensional reconstruction of cardiac phases at different inversion recovery states with high temporal resolution of cardiac and contrast dynamics. The methodology presented here can also be readily applied to multidimensional imaging of different organs such as the liver or kidneys.


We thank the NIH for supporting our work, NIH grant P41EB017183.


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Acquisition parameters of the 2D$$$~$$$Cartesian real-time acquisition covering multiple cardiac cycles with repeatedly applied RF inversion. The sorting result of the k-space data sampled at randomly coinciding pairs of cardiac and inversion recovery phases is further illustrated. Each serial k-space data of 20$$$~$$$cardiac phases of a cardiac cycle forms one 'diagonal' in the space spanned by cardiac and inversion recovery phases. Note that each image in the schematic corresponds to pseudo-randomly undersampled multi-coil k-space data with 15$$$~$$$read-out lines. In this way, a 4D$$$~$$$data tensor (for each of the 30$$$~$$$coil elements) is populated, consisting of two spatial and two dynamic dimensions.

$$$\mathcal{L}+\mathcal{S}$$$ tensor decomposition represents the multidimensional image as a sum of a low-rank tensor $$$\mathcal{L}$$$ that comprises the common background and a sparse component $$$\mathcal{S}$$$ that represents the dynamic information lying on top of the background. The schematic illustrates the decomposition into a low-rank core tensor for $$$\mathcal{L}$$$, using unfoldings [8], and the sparsity transform along different dimensions for $$$\mathcal{S}$$$. For the acquired in vivo data, $$$\mathcal{L}$$$ reflects the main contrast dynamics, as can be seen in the early inversion recovery phases, and $$$\mathcal{S}$$$ captures further contrast variations and the cardiac motion.

Magnitude images at one cardiac phase of three selected inversion recovery phases are displayed for the three different reconstruction approaches, i.e. $$$\mathcal{L}+\mathcal{S}$$$ tensor decomposition, L-only which enforces solely the low-rank constraint and S-only which uses only the sparsity constraint along each dynamic dimension. Among the three approaches, $$$\mathcal{L}+\mathcal{S}$$$ presents the highest spatial and temporal resolution. L-only shows significant temporal blurring (artificial ring in the myocardium). An animation of this figure can be found in Fig.$$$~$$$5.

Magnitude values averaged over a region-of-interest (ROI) comprising either blood pool voxel (a,c) or the myocardium (b,d) are displayed for the $$$\mathcal{L}+\mathcal{S}$$$ reconstruction, the L-only reconstruction (low-rank constraint only) and the S-only reconstruction (sparsity constraint only). Whereas $$$\mathcal{L}+\mathcal{S}$$$ and S-only are in a similar range, magnitude inversion recovery curves for L-only deviate in the myocardium. This indicates that a low-rank constraint might not be sufficient to capture all voxel-wise variabilities in the temporal domain. All three reconstructions are in good agreement for the ROI that consists of blood pool voxels only.

Animation over a complete cardiac cycle of the three reconstructions $$$\mathcal{L}+\mathcal{S}$$$, L-only and S-only, as shown and explained in Fig.$$$~$$$4. All three reconstructions exhibit some level of residual aliasing artifacts arising from the highly undersampled k-space data ($$$15$$$ read-out lines per time frame). L-only reveals strong residual aliasing and blurring. Increasing the weight of the regularizer would increase temporal blurring, decreasing it increases the amount of residual aliasing artifacts. S-only mitigates most of the aliasing artifacts, however, spatiotemporal blurring is evident. Although $$$\mathcal{L}+\mathcal{S}$$$ shows aliasing artifacts, the best spatiotemporal resolution is observed here.

Proc. Intl. Soc. Mag. Reson. Med. 25 (2017)