To develop a purely data-driven postprocessing method capable of removing B0 fluctuation-induced ghosting artifacts in long-TE gradient-echo scans.

We formulate the navigator-less reconstruction as an optimization problem, which involves finding the minimum value of the objective function (Eq. 1). We seek the unknown phase values Φ that are associated with low values of the image quality measure φ, which we choose to be the entropy function. More precisely, we compute the entropy of the spatial intensity variations in the SOS-combined image. The matrices that are used to perform the finite difference operations in the x and y direction are denoted by G_x and G_y, respectively. The phase values Φ are applied to the acquired images u_c (for each coil element c) using the diagonal matrix A, whose elements are the complex exponentials exp(iΦ_t), with t being the repetition index. Further, F denotes a discrete Fourier transform matrix.

$$\hat{\boldsymbol{\varPhi}}=\underset{\boldsymbol{\varPhi}}{\arg\min}\;\varphi((\mathbf{G_{\mathbf{x}}+G_{y}})SOS(\mathbf{\mathbf{F}^{\mathsf{H}}A}_{\boldsymbol{\varPhi}}\mathbf{u}_{c}))+\lambda\left\Vert \mathbf{G}\mathbf{\boldsymbol{\boldsymbol{\varPhi}}}\right\Vert ^{2} $$

In this formulation, the objective function is invariant to circular shifts of the image in the phase-encoding direction because such circular shifts amount to phase ramps (composed of recovered phases Φ) in the frequency domain. The problem of unnecessary circular shifts can be avoided by adding a regularization term which penalizes strong variations of the recovered phases. The parameter λ controls the strength of the regularization (we set it to 0.1). The resulting non-linear optimization problem is solved in 80 iterations of the LBFGS² algorithm. We implemented the operations from Eq. 1 on the GPU in CUDA, bringing the computation time (for each slice) down to a few seconds.

To evaluate the performance of the proposed method we acquired long-TE GRE images of the brain of a healthy volunteer after obtaining informed consent and approval by the local ethics committee. Data was acquired at 9.4T using a custom-built head coil (16 transmit / 31 receive channels)⁴ . We acquired 9 slices of the ventral portions of the brain where field variations are relatively severe, mainly due to breathing-related motion. The GRE sequence included a non-phase-encoded navigator (or phase-stablization) scan after each imaging readout. Sequence parameters were as follows: TR=356 ms, TE = 30 ms, nominal flip angle = 45º, matrix = 512x512, resolution = 0.4x0.4 mm², slice thickness = 1 mm.

Figure 1 shows a comparison between uncorrected images with images corrected for B0 fluctuations using a conventional navigator-based approach as well as the proposed autofocusing-based method. Ghosting artifacts in the uncorrected data are more severe in slice 6 (shown on the bottom), which is positioned lower than slice 3 (top). In both slices, autofocusing and navigator-based correction techniques are able to significantly improve image quality. Apart from some flow-related artifacts, ghosting is completely removed and the images resulting from both techniques are practically indistinguishable from one another. In fact, the differences between the autofocusing and navigator-based approaches amount to the minute high-frequency details as illustrated in Figure 2.

Figure 3 compares the phase offsets retrieved by our autofocusing algorithm with the navigator-based measurement. Since there is a sign as well as a global phase offset ambiguity, we adjusted the sign and subtracted the mean from both phase series before plotting them. Although, there are a few differences in the recovered phase values, the general pattern of oscillations (caused by breathing) is the same.

The problem of finding the correct phase offsets is similar to the well-studied autofocusing-based motion correction^1,3, and can be seen as its special case. We formulate and solve the optimization problem, where we seek the latent phase offsets in the Fourier domain that are associated with a minimal value of the image quality measure that is evaluated in the spatial domain. This way we avoid the need for extra non-phase-encoded navigator scans and related increase in sequence complexity, and, in some cases, scan time. The experimental results demonstrate our method is capable of removing the ghosting artifacts, and that the quality of the outcome images is similar to navigator-based reconstructions. To conclude, the proposed method is a valid alternative to using navigators with only a slight increase in postprocessing time.