Parallel Imaging Reconstruction from Undersampled K-space Data via Iterative Feature Refinement

Jing Cheng^{1}, Leslie Ying^{2}, Shanshan Wang^{1}, Xi Peng^{1}, Yuanyuan Liu^{1}, Jing Yuan^{3}, and Dong Liang^{1}

The reconstruction problem of SparseSENSE can be
formulated as$$$\min_I\sum_{j=1}^N‖F_P (S_j I)-k_j ‖_2^2 +λ‖I‖_{TV}$$$ with the total variation (TV) as one example of sparsity-promoting
regularizer. To solve this problem, an auxiliary variable *u* is introduced and the alternating minimization strategy is used. Then, the original problem is decomposed to a TV-based sparsity-promoting denoising problem in Eq. (1.a) and a Tikhonov regularization problem in Eq. (1.b). The framework of IFR-SENSE is illustrated in Eq.(2). We used conjugate gradient (CGSENSE) to
reconstruct the initial image from the partial k-space data. After TV-based
denoising, we got the noise-reduced image *u*. As pointed in [7], the sparsity-promoting term not only suppresses
the noise-like signal but also some fine small features, i.e. details in
denoising. In other words, the discarded image $$$v=I-u$$$ contains not only noise-like artifacts but
useful features. To address this problem, we
propose to use a feature descriptor *T *to extract
the useful features from* v* as $$$T\otimes v$$$ and get the feature refined image $$$I_{t}=u+T\otimes v$$$. *I _{t}* in Eq.(2.b) was then projected back into k-space using initial sensitivity maps. The fully acquired central k-space data can be
inserted back to update the reconstructed k-space. And the sensitivity maps were recalculated from updated k-space
data set. To improve spatial resolution of

$$\begin{cases}u^k=argmin_u‖I^k -u^k ‖_2^2+λ_{TV} ‖u^k ‖_{TV} &(1.a)\\I^{k+1}=argmin_I∑_{j=1}^N‖F_P (S_j I^{k+1} )-k_j ‖_2^2 +μ‖I^{k+1} -u^k ‖_2^2 & (1.b)\end{cases}$$

$$\begin{cases}u^k=argmin_u‖I^k -u^k ‖_2^2+λ_{TV} ‖u^k ‖_{TV} &(2.a)\\I_t^k=u^k+T^k⊗(I^k -u^k) & (2.b)\\I^{k+1}=argmin_I∑_{j=1}^N‖F_P (S_j I^{k+1} )-k_j ‖_2^2 +μ‖I^{k+1} -u^k ‖_2^2 & (2.c)\end{cases}$$

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[5] Pruessmann KP et al. SENSE:Sensitivity encoding for fast MRI. MRM. 42:952-962, 1999.

[6]Huang F et al. A rapid and robust numerical algorithm for sensitivity encoding with sparsity constraints: self-feeding sparse SENSE. MRM. 64:1078-1088,2010.

[7] Osher S et al. An iterative regularization method for total variation-based image restoration. Multiscale Modeling and Simulations. 4:460-489, 2005.

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[9]Liu Q et al. Adaptive image decomposition by improvedbilateral filter. Int. J. Comput. Appl, vol. 23, 2011.

Fig 1. Flowchart of the proposed IFR-SENSE method.

Fig 2. Reconstructed results with 32-channel data from
Possion mask. (a) The reference image, (b) and (c) are the reconstruction by
the SFSS and the proposed method, respectively. (e)(f) The difference maps of
(b) and (c). (d) The estimated feature descriptor in the last iteration.

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

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