Single-shot spatiotemporally encoded (SPEN) MRI is a novel technique developed from single-shot EPI with ability to provide images with reduced off-resonance induced distortions ¹. Similar to single-shot EPI, Nyquist ghost exists in SPEN images due to the phase inconsistency between even and odd echoes. Conventional correction method utilizes the phase difference maps between the two images obtained from solely even and odd echoes respectively which is 2D polynomial fitted to compensate the phase inconsistency ². However, its performance would be compromised by the presence of unsuppressed fat signal. In this study, a rank constraint based ghost correction method is proposed for better Nyquist ghost removal in such cases.

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 ith 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=[X₁,X₂,..,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.

The SPEN brain data were acquired on a Siemens 3T Prisma scanner with 16-channel head coil. The sampling data matrix M was 64*64, and N_f was 256. Figure 1 shows the phase of the even-echo-image and odd-echo-images and their difference before (a) and after (b) imposing rank-1 constraint on the data of a single channel. It can be seen that the spatial variations in the phase difference map is removed after applying the rank-1 constraint. Figure 2 shows the reconstructed image without correction (a), by conventional phase compensation method (b) and our proposed method (c). It can be seen that the artifact induced by the residual alias of fat signal in Fig.2 (c) is reduced compared with the result obtained from conventional method (red arrow in Fig. 2). In addition from calculation of the labeled signal and noise areas (red and blue boxes in Fig. 2), we can see that the SNR obtained from the proposed method is 69.55, which is about 2.90 times higher than that obtained from conventional method (23.97).The rank-1 constraint can help improve the performance of Nyquist ghost correction with higher SNR and reduced artifact.