Haifeng Wang^{1}, Yuchou Chang^{2}, Leslie Ying^{3}, Xin Liu^{1}, and Dong Liang^{1}

Nonlinear GRAPPA is a kernel-based approach for improving parallel imaging reconstruction, by reducing noise-induced error. Virtual coil conception has been applied into the reconstruction process for parallel acquisitions, by generating virtual coils containing conjugate symmetric k-space signals from actual multiple-channel coils. In this work, we proposed a hybrid method to combine nonlinear GRAPPA and virtual coil conception for incorporating additional image- and coil-phase information into the reconstruction process. The experiments of in vivo human brain data show that the proposed method can reduce more noise and artifacts than the traditional GRAPPA and original Nonlinear GRAPPA methods.

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Fig 1. Comparison of the mathematical equations
of GRAPPA, Nonlinear GRAPPA (NL-GRAPPA) and Nonlinear GRAPPA with virtual coil conception
(VCC-NL-GRAPPA). Their linear combinations of GRAPPA only has linear terms from
real coils; NL-GRAPPA has constant terms, linear and nonlinear terms from signals
of real coils; VCC-NL-GRAPPA has constant terms, linear and nonlinear terms from signals
of real coils and symmetric-complex conjugate signals of virtual coils.

Fig 2. Reconstruction comparison of GRAPPA,
NL-GRAPPA and VCC-NL-GRAPPA at R = 3. Here, the net reduction factors (NetR) are all 2.4.
The difference map (Diff.) of VCC-NL-GRAPPA at R = 3 shows over 25 percent
smaller mean-square-error (MSE) than others.

Fig 3. Reconstruction comparison of GRAPPA,
NL-GRAPPA and VCC-NL-GRAPPA at R = 5. Here, the net reduction factors (NetR) are all 3.0.
The difference map (Diff.) and its amplification (5×) of VCC-NL-GRAPPA presents over 25 percent
smaller mean-square-error (MSE) than others at R = 5.

Fig 4. Mean-square-error comparison of GRAPPA,
NL-GRAPPA and VCC-NL-GRAPPA at the outer reduction factors of 2, 3, 4 and 5.
The MSE with references presents that VCC-NL-GRAPPA has the smallest MSE value
of three methods at different outer reduction factors.