To reduce error in dynamic parallel imaging by estimating and removing high-variance low-frequency signal components prior to un-aliasing reconstruction, without additional data or sampling requirements

In reconstruction of k-t under-sampled data, DC (baseline) signal estimates are often used to estimate sensitivities¹ or regularize un-aliasing reconstructions². When used as prior information in un-aliasing procedures (e.g. by subtracting it from the sampled data, and reconstructing the residuals), it can improve conditioning of the problem and reduce residual aliasing. The DC-term is frequently used because it is easy to estimate as the temporal mean across the under-sampled data; by definition, however, it fails to capture any of the dynamic signal content. As errors in parallel imaging from residual aliasing scale with signal power, removing as much true signal as possible prior to un-aliasing reconstruction minimizes these artefacts. In Xu et al., an adaptive regularization was proposed using a generalized series model (SPEAR)³ that capture more dynamic signal variance. However, this approach requires additional sampling constraints and complex model fitting. Here we propose a simple method for estimating temporally adaptive prior information directly from the under-sampled data, obtained via singular value decomposition (SVD) on a sliding-window k-t estimate.

Motivated by the observation that the highest variance components of a dynamic MRI dataset are often low-frequency, we compute the SVD of a sliding window estimate and keep the first $$$r$$$ components as a rank-$$$r$$$ estimate of the underlying data ($$$\hat{X}_{r}$$$). If the spectral power of the true components falls within the bandwidth of the sliding window filter (Fig. 1), they are estimated with high fidelity. $$$\hat{X}_{r}$$$ can then be subtracted from the measured data to remove as much true signal variance prior to un-aliasing reconstruction:

$$\hat{X}=\hat{X}_{r}+G(E(X-\hat{X}_{r}))\tag{1}$$

where $$$E$$$ is the k-t sampling operator, $$$G$$$ denotes the un-aliasing linear operator (in this case GRAPPA⁴), and $$$\hat{X}$$$ is the un-aliased final data estimate. Like SPEAR, this method generates an aliased residual x-f space that is non-zero everywhere (Fig. 2), and so is free from DC-nulling temporal filtering artefacts⁵. Additionally, Ding et al. showed that the SVD-filtered sliding window estimate $$$\hat{X}_{r}$$$ can be used to robustly estimate dynamic sensitivities or train GRAPPA kernels, with no need for additional reference or auto-calibration data⁶. However, the method by Ding et al. focused solely on producing the best possible time-dependent sensitivity or GRAPPA kernel estimates, and did not discuss the utility of using the SVD-filtered estimate directly to reduce residual aliasing. We evaluated this approach on retrospectively under-sampled cardiac cine (R=4,6 128x128 and 100 time-points) and resting FMRI data (R=4, 64x64 and 500 time-points) using standard k-t lattice sampling. Coil sensitivities were based on 32-channel measurements made from real data, compressed down to 8 virtual coils for reconstruction using the SVD, and GRAPPA kernels (5x4) trained on the entirety of $$$\hat{X}_{4}$$$ were used for all comparisons within each dataset, to remove variability due to the GRAPPA operator.

Figure 3 plots the normalized RMSE for various reconstructions against the rank of the sliding window estimate, in general showing reduction in reconstruction error as rank increased. The error for a DC-subtracted reconstruction (dashed lines) is virtually identical to the rank-1 reconstruction error. In all cases, error begins to increase as the rank of the sliding-window estimate is increased beyond a certain point. While the selection of optimal rank for the sliding window estimate depends on both the spectral power distribution in the data and the sliding window width (i.e. under-sampling factor), the remaining comparisons were performed at a conservative rank-4 cutoff. Figure 4 shows a single column through the R=6 cine data over time, comparing the ground truth to reconstructions using DC-, rank-1 and rank-4 subtraction. Residual aliasing errors apparent in the DC/rank-1 reconstructions are largely absent in the rank-4 reconstruction. Similarly, Figure 5 shows RMSE reconstruction error across all voxels of the frequency domain information, clearly showing residual aliasing errors centered at the characteristic $$$\pm f_{max}/4$$$ and $$$-f_{max}/2$$$ aliasing harmonics in the rank-1 data, significantly reduced using a rank-4 estimate.We have demonstrated a simple, adaptive and reference-free method for reducing residual aliasing in accelerated k-t image reconstruction. While low-rank modelling has become popular in dynamic imaging, the method presented here only requires that the highest variance components are sufficiently band-limited to be well characterized using a sliding window filter, and makes no assumption otherwise on the rank or structure of the data to be reconstructed.