Improved Nyquist ghost removal for single-shot spatiotemporally encoded (SPEN) MRI with joint rank constraint

Congyu Liao^{1}, Ying Chen^{1}, Hongjian He^{1}, Song Chen^{1}, Hui Liu^{2}, Qiuping Ding^{1}, and Jianhui Zhong^{1}

For
data with multi-channel receivers, each column of the SPEN dataset can be reconstructed
by joint regularized linear least squares estimation ^{3} as: $$\hat{\bf X}_{i}=argmin\parallel{\bf FX}_{i}-{\bf M}_{i}\parallel_{2}+\parallel\triangledown{\bf X}\parallel_{2}, i=1,2,...,N_{x},\ \ \ \ \ \ \ \ \ \ \ \ \ [1]$$ where $$${{\bf X}_{i}\in \mathbb{C}^{N_{f}\times N_{c}} }$$$ is
the reconstructed matrix corresponding to the *i*th column of the image which
contains the data of N_{c} channels and a
vector of N_{f} number for each channel, N_{f} is the
bandwidth-time product of the chirp pulse, N_{x} is the column number (the
acquisition number along readout (RO) dimension), $$${{\bf M}_{i}\in \mathbb{C}^{N_{y}\times N_{c}} }$$$ is
the SPEN dataset after 1D inverse Fourier transform along RO, Ny is the number
contained in each column (echo number), $$$\triangledown$$$ is
the 1st order finite difference operator to enforce the spatial smoothness
constraint along SPEN encoding direction, and **F **
is the SPEN
encoding matrix digitized spatially into N_{f} voxels.

After
column-by-column reconstruction, the reconstructed image can be formulated as: **X**=[**X**_{1},**X**_{2},..,**X**_{Nx}]. Considering the phase inconsistency between even and
odd echoes, the ghost-free images from solely even and odd echoes (**X**_{even} and **X**_{odd}) can be reconstructed
using Eq. [1] by cutting N_{y} data rows into two parts and the their phase maps can
be obtained, denoted as **P**_{even}
and **P**_{odd} respectively. In this study, before phase subtraction, an
additional rank-1 constraint was implemented on **X**_{even} and **X**_{odd}
to remove the phase inconsistency between them. Denote **I**= [**X**_{even}(:); **X**_{odd}(:)] as the two-column matrix
rearranged from **X**_{even} and
**X**_{odd}, and the rank of **I** is 2. **I** can be reduced to
a rank-1 matrix using the singular value Decomposition (SVD). As a rank-1
matrix, **I** can be represented as **I**=**UV**,
in which **U** is the one-column vector
and **V** is the 1*2 row vector. After this, only the constant term
remains in the phase difference between the two elements of **V** and the spatial variation terms
contained in their phase inconsistency can be removed.

The ghost
removal procedure is separated into the following steps: (1) reconstruct the
images from even and odd
echoes using Eq. [1]. (2) Regrouping **X**_{even}
and **X**_{odd} into **I**, and then calculate the SVD of **I**. (3) Enforce rank-1 constraint on **I**, and then get the new phase difference.
(4) Utilize the phase estimation to compensate the phase inconsistency between
even and odd images, and calculate the phase compensated data **M**_{even} and **M**_{odd}. (5) Iterate the
procedures of (1)-(4) twice. (6) Combine **M**_{even}
and **M**_{odd} to obtain the
final ghost corrected image using Eq. [1] again. (7) Reshape the results of all
channels into one matrix, implementing SVD to reduce the influence induced by
fat signal contamination before multi-channel combination.

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

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