Kernelized Low-Rank: Improve Low-Rank with Adaptive Nonlinear Kernel for Dynamic MRI
Enhao Gong1, Tao Zhang1, Joseph Cheng1, Shreyas Vasanawala2, and John Pauly1

1Electrical Engineering, Stanford University, Stanford, CA, United States, 2Radiology, Stanford University, Stanford, CA, United States

Synopsis

Low-Rank methods are widely applied to improve reconstruction for Dynamic Contrast Enhanced (DCE) MRI by imposing linear spatial-temporal correlation in global, local or multiple scales. This assumption does not fully capture the highly nonlinear spatial-temporal dynamics of DCE signals. We proposed a generalized Kernelized-Low-Rank model, assumed Low-Rank property after nonlinear transform and solved it by Regularizing singular-values with Adaptive Nonlinear Kernels. The proposed method captures the spatial-temporal dynamics as a sparser representation and achieves more accurate reconstruction results. Kernelized-Low-Rank model can be easily integrated to provide significant improvements to Global Low-Rank, Locally Low-Rank, LR+S and Multi-scale LR models.

Purpose

Dynamic-Contrast-Enhanced MRI(DCE-MRI) is a powerful MR technique that provides both anatomy and perfusion information. Low-Rank(LR)/Partially-Separable[1,2,3] models are widely applied in DCE-MRI for acceleration and improved sparse-reconstruction by exploiting spatial-temporal correlation. Existing LR methods model the spatial-temporal signal as a sparse and linear combination of basis-signals. However, these linear models cannot fully capture the complex contrast enhancement and may lead to inaccurate contrast dynamics in reconstruction. Dictionary-Learning based methods[4-5] tackle this issue using over-complete dictionaries, which needs expensive computation and training data. Here we propose a generalized LR model with Adaptive-Nonlinear-Kernels. This Kernelized-Low-Rank(KLR) model assumes the LR property in a nonlinear-transform domain instead of the original spatial-temporal domain. The proposed method captures the spatial-temporal dynamics with sparser nonlinear representations, and achieves more accurate reconstruction results.

Theory

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).

Method

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).

Results

In-vivo experiments(Fig-2) demonstrated sharper structural delineation by the proposed KLR method. As shown in Fig-3, simulation with DCE-phantom validated that KLR out-performs other low-rank methods in reconstruction. In addition, KLR showed more accurate recovery of temporal dynamics(shown in Fig-3c and Fig-3d).

Discussion and Conclusion

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.

Acknowledgements

We acknowledge the support from NIH R01 EB009690, R01 EB019241, P41 EB015891, and GE Healthcare.

References

[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.

Figures

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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