Rui Tian^{1}, Franciszek Hennel^{1}, and Klaas P Pruessmann^{1}

Super-resolution MRI with 1D phaseless encoding achieves high-resolution with immunity to shot-dependent phase fluctuation by simultaneously acquiring multiple k-space bands. We now explore a 2D extension of this technique to facilitate more k-space sampling strategies. Two distinct encoding schemes were analyzed and tested with EPI acquisition. By properly adjusting the overlapping of the mixed k-space bands, the 2D phaseless encoding could also be combined with the spiral acquisition. The amplitude modulation caused by band overlapping was eliminated by an inverse filter during reconstruction. The overlapped bands regions were also exploited to provide information about unexpected bands errors for post-processing corrections.

By translating the philosophy of structured illumination microscopy (SIM) to MRI, the phaseless encoding has been applied in one dimension to reconstruct a super-resolution (SR) MRI image from several low-resolution acquisition cycles with different shifts of a sub-pixel tagging pattern [1][2]. This allows multi-shot scanning without sensitivity to motion-related phase fluctuations [3]. The extension of this technique to two dimensions should allow further shortening of sampling time in each cycle (and reduction of off-resonance effects) and provide isotropic resolution enhancement with non-Cartesian trajectories, e.g. spirals.

Multiple k-space regions can be mixed by a tagging pattern in various ways. For example, a rapid repetition of two orthogonal tags [1] excites a 3x3 array of k-space “tiles” and may provide a 3-fold resolution enhancement in both directions. Alternatively, the same enhancement can be achieved by a rotation, and, optionally, scaling of a 1D tag which mixes three tiles at a time. The goal of this study is to demonstrate and compare different 2D phaseless encoding schemes with respect to their compatibility with various k-space trajectories, the complexity of the SR reconstruction, the minimum scan time and the signal-to-noise efficiency. We also present optimized post-processing steps which improve the robustness of this method.

Experiments were carried out on a 3T Achieva scanner (Philips Netherlands). Three encoding schemes were implemented:

1. Two consecutive orthogonal sinusoidal tags with three shifts each, covering a rectangular 3x3 pattern of k-space tiles (Fig.1A).

2. A single 1D sinusoidal tag with 4 rotations and 3 shifts with each direction, covering a circularly arranged pattern (Fig.1B).

3. The same pattern as (2) but with diagonal directions scaled to achieve rectangular arrangement as in (1) (Fig.1C).

In all cases, the overlapping of the k-space tiles was programmable and typically set from 10% to 15% depending on the k-space trajectory.

The reconstruction followed the similar principles of the 1D SR reconstruction [3], with the modified tag shifts cycling and the adapted corrections for tagging distortions in the scheme No.1. The SNR between the schemes No.1 and No.3 was examined by quantifying the noise propagation through the reconstruction matrix and tested experimentally with the GRE-EPI sequence in-vivo.

The spiral acquisition was conducted by both the schemes No.2 and No.3 to compare the efficiency in the overlapping of the round-shaped tiles. An inverse filter was calculated based on the known effective k-space window to eliminate the amplitude modulation. The phase of the overlapped k-space pixels contributed from neighboring bands were compared to estimate any unexpected constant phase shifts, with the presumption that these pixels should be identical in the absence of any errors.

The acquisition cycles for all the schemes above were reconstructed properly, for which the results are summarized:

1. The adapted reconstruction for the 2D phaseless encoding achieved the expected resolution enhancement without shot-dependent phase fluctuation (Fig.2).

2. Covering the same rectangular pattern of k-space tiles, the scheme with the rotational 1D tag was analyzed to have 1.81 times higher SNR than the one with the two orthogonal tags, which was also experimentally measured with 1.85 times higher SNR verifying our theories.

3. For the rotational schemes, the circularly arranged pattern naturally fits the spiral acquisition better than the rectangular arrangement, due to the less minimum overlapping of k-space tiles to prevent empty holes between resolved bands (Fig.3).

4. The apodization window on the low-resolution scans was replicated on the top of all resolved k-space tiles, which led to a complicated effective k-space window and thus amplitude modulation. However, the amplitude modulation was easily corrected by the inverse filter (Fig.4).

5. A set of constant phase offsets on resolved neighboring bands were estimated successfully and used to effectively remove the ringing artifacts through the post-processing correction (Fig.5).

1. Ropele, S., Ebner, F., Fazekas, F., & Reishofer, G. (2010). Super-resolution MRI using microscopic spatial modulation of magnetization. Magnetic Resonance in Medicine, 64(6), 1671-1675. doi:10.1002/mrm.22616

2. Hennel, F., & Pruessmann, K. P. (2016). MRI with phaseless encoding. Magnetic Resonance in Medicine, 78(3), 1029-1037. doi:10.1002/mrm.26497

3. Hennel, F., Tian, R., Engel, M., & Pruessmann, K. P. (2018). In-plane “superresolution” MRI with phaseless sub-pixel encoding. Magnetic Resonance in Medicine. doi:10.1002/mrm.27209

4. Tian, R., Hennel, F., & Pruessmann, K. P. (2018, June). “Exploring the Limits of Super-resolution MRI with Phaseless Encoding.” In proceedings of the joint annual meeting of ISMRM-ESMRMB 2018, Paris, France, abstract 2671

Figure 1. The three encoding schemes for the 2D phaseless encoding with corresponding preparation sequence plots. The column A: the rectangular arrangement of the 3x3 k-space tiles based on two consecutive orthogonal sinusoidal tags. The column B: the circularly arranged pattern with rotational 1D SR experiments. The column C: the rectangular arrangement with rotational 1D SR experiments.

Figure 2. The 2D phaseless encoded reconstruction for the rectangular arrangement of k-space tiles with EPI acquisition, by the scheme with the two orthogonal tags (the upper row) and the one with the rotational 1D tag (the bottom row). Note the absence of ghosting in the high-resolution diffusion image (H) which demonstrates the immunity of our method to phase fluctuations.

Figure 3. The 2D phaseless encoding with the spiral acquisition, by the rectangular arrangement (B, C, F, G) and the circularly arranged pattern (D, H). The low-resolution reference was also provided in (A, E).

Figure 4. The inverse filter was calculated from the known effective k-space window for correcting the amplitude modulation (A). Two examples for correcting the amplitude modulation for the rectangular arrangement and the circularly arranged pattern of k-space tiles were demonstrated in (B, C, D) and (E, F, G) respectively.

Figure 5. The unexpected homogenous tagging shifts were estimated from the overlapped k-space pixels contributed from neighboring bands(A). The correction based on the estimated phase errors significantly improved the image quality in both two phantom experiments (the bottom row with B, C, D, E).