Referenceless one-dimensional Nyquist ghost correction in multicoil single-shot spatiotemporally encoded MRI

Unsupervised deep learning model for correcting Nyquist ghosts of single‐shot spatiotemporal encoding

“…In single‐shot experiments, the rapid change of the RO gradients will cause a phase mismatch $$$ {\theta}_{\mathrm{dif}} $$$ between even echo signal $$$ {S}_{\mathrm{even}} $$$ and odd echo signal $$$ {S}_{\mathrm{odd}} $$$, as follows: $$$$ \frac{S_{\mathrm{odd}}}{S_{\mathrm{even}}}={e}^{i{\theta}_{\mathrm{dif}}} $$$$ Affected by the characteristics of the SR matrix $$$ A $$$, the even/odd phase difference can easily result in Nyquist ghosts in the high‐resolution SR image $$$ \hat{\rho} $$$ of $$$ S $$$ 20 . Conventional methods 20,21 usually eliminate the Nyquist ghosts by estimating the phase difference $$$ {\theta}_{\mathrm{dif}} $$$ and correcting it (the details can be found in Figure S1): …”

“…This loss function relies on the fact that the inverse of $$$ A $$$ does not necessarily exist due to the large condition number of $$$ A $$$; consequently, $$$ {A}^{\left(-1\right)}A $$$ is not equal to the identity, producing images with ghosts, if no additional measures are taken to reduce the condition number of $$$ A. $$$ 20,21 As a result, this implies that, due to deviation of $$$ {A}^{\left(-1\right)}A $$$ from identity, increased phase difference between the even and odd echoes will lead to higher error between the corrected image and image obtained after applying $$$ {A}^{\left(-1\right)}A $$$ to the corrected image. This is illustrated in Figure 3C, where the bigger the phase difference between even and odd images, the larger the cycle‐consistency loss will be.…”

“…Affected by the characteristics of the SR matrix A, the even/odd phase difference can easily result in Nyquist ghosts in the high-resolution SR image ρ of S. 20 Conventional methods 20,21 usually eliminate the Nyquist ghosts by estimating the phase difference 𝜃 dif and correcting it (the details can be found in Figure S1):…”

“…As we mentioned previously, one advantage of the SPEN sequence is that it can directly obtain low‐resolution images and do the Nyquist ghost correction without the need for reference scans 17–19 or parallel‐imaging techniques 4 . Previous reports by Seginer et al 20 and Chen et al 21 show that two low‐resolution SPEN images obtained respectively from even echo data and odd echo data of a single SPEN data can provide information on phase inconsistency for implementing 2D Nyquist ghost correction without reference scans. However, it is still difficult for conventional methods to correct the Nyquist ghosts in SPEN images perfectly because of the phase‐wrap problem, especially for low SNR images.…”