Kirsten Koolstra^{1}, Merel de Leeuw den Bouter^{2}, Thomas O'Reilly^{1}, Peter BĂ¶rnert^{1,3}, Rob Remis^{4}, Martin van Gijzen^{2}, and Andrew Webb^{1}

Inaccuracies and temporal fluctuations in field map measurements form a major problem in image reconstruction for permanent magnet based low field MRI systems. These inaccuracies can potentially be corrected by using a joint image reconstruction and field map estimation algorithm. Simulation results show improved image quality when using a new updating scheme compared to standard iterative reconstructions.

*Experimental setup:* A Halbach
magnet was constructed, containing four rings each with 24 2.5×2.5×2.5 cm^{3} cubes of N-52 neodymium
boron iron magnets (producing a 0.06T magnetic field at the center), with
an internal ring of smaller magnets to produce a linear field gradient. The magnetic field was measured on a 5×7.5 mm^{2} grid
using a gaussmeter (Alphalab GM-2).

*Simulation:* A solenoid coil was modelled for RF transmission and reception. The sample was rotated 36 times with angular increments of $$$\theta =10$$$°. For each
rotation, a spin-echo sequence was simulated with RF pulse durations of 20 µs,
and TE=5 ms; 512 data points and a dwell time of 2 µs. A signal model

$$\textbf{S}_\theta=ETCR_\theta \textbf{m} \quad \text{(1)}$$

was used with $$$\textbf{m}$$$ the unknown image (simulated as a Shepp Logan MATLAB phantom), $$$\textbf{S}_\theta$$$ the measured time-domain signal for rotation
angle $$$\theta$$$, $$$R_\theta$$$ the corresponding rotation matrix, $$$C$$$ a diagonal matrix with uniform
spatial coil sensitivity weighted by the coil frequency profile (Q-factor of 13.8 based on an S_{11} measurement) on the diagonal, $$$T$$$ a diagonal matrix including the pulse profile (simulated
via the Bloch equations) and transmit bandwidth, $$$E$$$ the signal encoding matrix with elements $$$E_{pq}=e^{-2\pi iB_q t_p}$$$,
and $$$B_q$$$ the main field difference (with respect to the transmitter frequency) in Hz in pixel $$$q$$$. White Gaussian noise was
added to the signals such that the SNR was 10.

*Image reconstruction:* The image is reconstructed by solving
the non-linear problem

$$\hat{\textbf{m}}=\min \frac{\mu}{2}|| \sum_{\theta} \textbf{S}_\theta - ETCR_\theta \textbf{m} ||_2^2 + \frac{\lambda}{2} TV(\textbf{m}) \quad \text{(2)}$$

where $$$TV$$$ is a total variation operator and $$$\mu$$$ and $$$\lambda$$$ regularization parameters. Substitution of the perturbations $$$\textbf{m}^R=\textbf{m}^R+\Delta \textbf{m}^R $$$, $$$\textbf{m}^I=\textbf{m}^I+\Delta \textbf{m}^I $$$ and $$$\textbf{B}=\textbf{B}+\Delta \textbf{B}$$$ in Eq. (1) gives a system for the error terms:

$$\sum_\theta A_\theta^H \left[ \begin{matrix} \textbf{S}_\theta^R -\textbf{a}_1 \\ \textbf{S}_\theta^I - \textbf{a}_2 \end{matrix} \right] = \sum_\theta A_\theta^H A_\theta \left[ \begin{matrix} \Delta \textbf{m}^R \\ \Delta \textbf{m}^I \\ \Delta \textbf{B} \end{matrix} \right] \quad \text{(3)} $$

$$$\textbf{S}_\theta^R/\textbf{S}_\theta^I$$$ are the real/imaginary parts of the sampled time signal at rotation angle $$$\theta$$$. $$$A_\theta$$$ describes the linearized data model for the error terms and $$$\textbf{a}_1/\textbf{a}_2$$$ describe the simulated time domain signal based on the estimated image $$$\textbf{m}$$$. The final image is reconstructed by the algorithm shown in Figure 1.

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