Kernelized Low-Rank: Improve Low-Rank with Adaptive Nonlinear Kernel for Dynamic MRI

Enhao Gong^{1}, Tao Zhang^{1}, Joseph Cheng^{1}, Shreyas Vasanawala^{2}, and John Pauly^{1}

**1. Kernelized-Low-Rank(KLR) Model**

Due to non-uniform contrast changes and motion, dynamic MRI signals may not be completely rank-deficient as desired with linear Singular-Value-Decomposition(SVD). However, the spatial-temporal signal $$$X$$$ can be better represented (more rank-deficient) after certain nonlinear transforms $$$T$$$ with nonlinear kernel $$$K$$$.

Based on the relationship between SVD and Principle-Component-Analysis (PCA), we proposed to use Kernel-PCA^{[6]} to compute singular-values and corresponding LR representation of nonlinear-transformed data $$$T_K(X)$$$. In Kernel-PCA the decomposition is achieved more efficiently by conducting Eigen-Decomposition
of the kernel-transformed Covariance Matrix of $$$T_K(X)$$$ which is $$$cov(T_K(X))=K(cov(X))=K(X^T X)$$$.

For the nonlinear transform, we apply Radial-Basic-Function (rbf) kernels as proof-of-concept: $$$K(X^TX)_{i,j}=\exp{\left(-\gamma\Vert x_i-x_j\Vert^2\right)}$$$. The parameter $$$\gamma$$$ (default 0.5) is adaptively tuned to different scale to make the representation more rank-deficient.

**2. KLR Projection**

The original Linear-LR Projection $$$P_{LR}^r$$$ is formulated by thresholding singular-values (SVT):

$$P_{LR}^r(X)=\arg\min_x\left[\frac{1}{2}\Vert T_K(X)-x\Vert_F^2\ s.t.\ Rank(T_K(x))\leq r \right]=\sum_{i=1}^r\sigma_i (X)u_iv_i^T$$

The fundamental idea of the proposed KLR Projection $$$P_{KLR}^r$$$ is to conduct Linear-LR Projection with SVT on the nonlinear transformed data, which is achieved efficiently by thresholding Eigenvalues (EVT) after Kernel-PCA. Then the KLR projection is achieved by finding the best match in the original spatial-temporal domain based on the EVT results in the nonlinear-transform domain. Similar idea (Pre-image method) has been used for image denoising applications^{[6]}:

$$P_{KLR}^r(X)=\arg\min_x\frac{1}{2}\Vert T_K(x)-P_{LR}^r(T_K(X))\Vert_F^2$$

**3. KLR based DCE Reconstruction**

Next, we re-formulated the DCE
reconstruction by regularizing the rank of nonlinear-transformed data, with $$$y$$$ as acquired *k*-space samples,
$$$F_u$$$ as undersampling-Fourier operator including coil-sensitivities.

$$min_x\Vert y-F_ux\Vert_2^2\ s.t.Rank(T_K(x))\leq R$$

Or equivalently,

$$min_x\Vert y-F_ux\Vert_2^2+\lambda\Vert T_K(X)\Vert_*$$

The SVT in nonlinear-transform domain is achieved efficiently using EVT with Kernel-PCA as described previously.

Linear-LR is a special case of the formulation above when $$$K$$$ is a linear-kernel. Nonlinear representations are shown to provide sparser and more accurate representations(Fig-1.3).

**1. Algorithm**

The optimization was solved using Iterative Projections(Fig-1):

**1) Data Consistency** is enforced using weighted-ESPIRiT (wESPIRiT)^{[7] }method to “soft-gate” the acquired *k*-space data based on estimated motions from navigators. The data is then inverse-Fourier transformed into image domain.

**2) Sensitivity for Parallel-Imaging: ** Sensitivity-maps are estimated with ESPIRiT^{[8]} and used to combine multi-channel sensitivity-weighted images(POCS-SENSE).

**3) KLR Projection **is developed to improve conventional Linear-LR SVT projection. As described previously, here the SVT is conducted on the nonlinear-transformed signals, achieved using the KLR approximation formula with Kernel-PCA, EVT and Pre-image method^{[6]}. The computational complexity of KLR-EVT with Kernel-PCA is the same as original SVT in Linear-LR.

The KLR Projection can be applied to replace the original linear SVT in Global-Low-Rank(GLR), Locally-Low-Rank(LLR)^{[3]} as well as Low-Rank+Sparse(LR+S)^{[2]} model and Multi-scale-Low-Rank(Multi-scale-LR)^{[9]}.

**2. Experiment settings**

To evaluate the performance, a free breathing DCE dataset of a 3-year-old female was acquired on a 3T GE-MR750 scanner with 32-channel cardiac array using a RF-spoiled-GRE-sequence (VDRad^{[10]} undersampling acquisition, Matrix Size:320x180x100, S/I 280mm FOV, R/L 224 mm FOV, 60% partial-echo, 178.51 sec for 18 contrast phases, R-factor 8.5). We integrated KLR with GLR and LLR to demonstrate the improvements(Fig-2). All the methods were performed with the same settings(Coil-Compression[11], ESPIRiT sensitivity-maps and wESPIRiT "soft-gating" and the same thresholding parameter previously tuned for Linear-LR).

In addition, a simulation dataset (DCE-phantom, retrospectively under-sampled with VDRad) was used to validate the spatial-temporal recovery performance for these methods(Fig-3).

Here we developed Kernelized-Low-Rank(KLR) model to further improve LR reconstruction for DCE-MRI. Generalized from Linear-LR, the proposed method better captures the nonlinearity of spatial-temporal signals by modeling them into sparser nonlinear representations. Simulation and in-vivo experiments demonstrated that KLR outperforms linear LR methods.

KLR can be easily integrated into LR methods including GLR, LLR, LR+S and Multi-scale-LR, to significantly improve reconstruction performance.

[1] Haldar, Justin P., and Zhi-Pei Liang. "Spatiotemporal imaging with partially separable functions: A matrix recovery approach." Biomedical Imaging: From Nano to Macro, 2010 IEEE International Symposium on. IEEE, 2010.

[2] Otazo, Ricardo, Emmanuel Candès, and Daniel K. Sodickson. "Low-rank plus sparse matrix decomposition for accelerated dynamic MRI with separation of background and dynamic components." Magnetic Resonance in Medicine 73.3 (2015): 1125-1136.

[3] Zhang, Tao, et al. "Fast pediatric 3D free-breathing abdominal dynamic contrast enhanced MRI with high spatiotemporal resolution." Journal of Magnetic Resonance Imaging 41.2 (2015): 460-473.

[4] Nakarmi, Ukash, et al. "Dynamic magnetic resonance imaging using compressed sensing with self-learned nonlinear dictionary (NL-D)." Biomedical Imaging (ISBI), 2015 IEEE 12th International Symposium on. IEEE, 2015.

[5] Lingala, Sajan Goud, and Mathews Jacob. "Blind compressive sensing dynamic MRI." Medical Imaging, IEEE Transactions on 32.6 (2013): 1132-1145.

[6] Mika, Sebastian, et al. "Kernel PCA and De-Noising in Feature Spaces." NIPS. Vol. 4. No. 5. 1998.

[7] Cheng, Joseph Y., et al. "Free-breathing pediatric MRI with nonrigid motion correction and acceleration." Journal of Magnetic Resonance Imaging (2014).

[8] Uecker, Martin, et al. "ESPIRiT—an eigenvalue approach to autocalibrating parallel MRI: where SENSE meets GRAPPA." Magnetic Resonance in Medicine71.3 (2014): 990-1001.

[9] Ong, Frank, and Michael Lustig. "Beyond Low Rank+ Sparse: Multi-scale Low Rank Matrix Decomposition." arXiv preprint arXiv:1507.08751 (2015).

[10] Cheng, Joseph Y., et al. "Variable-density radial view-ordering and sampling for time-optimized 3D Cartesian imaging." Proceedings of the ISMRM Workshop on Data Sampling and Image Reconstruction, Sedona, Arizona, USA. 2013.

[11] Zhang, Tao, et al. "Coil compression for accelerated imaging with Cartesian sampling." Magnetic Resonance in Medicine 69.2 (2013): 571-582.

Figure1. Flowchart of the Iterative Projection algorithm with the proposed Kernelized-Low-Rank model. Step 1: wESPIRiT k-space consistency update. Step 2: ESPIRiT Sensitivity-Map and iFFT based transform to image domain. Step 3: Replace conventional singular-value-soft-thresholding (SVT) with the proposed Kernelized-Low-Rank projection, in which SVT is conducted in nonlinear transform domain and then project back to image domain. Nonlinear transform leads to sparser Low-Rank representation.

Figure 2. Comparison of optimization performance on in-vivo DCE dataset. The proposed algorithm is compared with conventional Linear-LR methods. By applying the proposed Kernelized-Low-Rank (KLR) method, the solution of optimization is greatly improved with shaper detailed and less artifacts. A) Axial view, comparison between Global-Low-Rank (GLR), Locally-Low-Rank (LLR), GLR+KLR and LLR+KLR. B) Coronal View, LLR and LLR+KLR.

Figure 3. Reconstruction and temporal recovery performance on simulated data with DCE-Phantom. a)Reconstruction and error-maps show KLR achieves better reconstruction. b)Phantom Image and the sampled positions of simulated organs. c)Compared with Global-Low-Rank(GLR) and Locally-Low-Rank(LLR), Kernelized-GLR(GLR+KLR) and Kernelized-LLR(LLR+KLR) reconstructs more accurate temporal dynamics. d)Table shows optimal temporal recovery Root-Mean-Square-Error(RMSE) achieved with GLR+KLR and GLR+LLR.

Proc. Intl. Soc. Mag. Reson. Med. 24 (2016)

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