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 cm3 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 mm2 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 θ=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
Sθ=ETCRθm(1)
was used with m the unknown image (simulated as a Shepp Logan MATLAB phantom), Sθ the measured time-domain signal for rotation angle θ, Rθ 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 S11 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 Epq=e−2πiBqtp, and Bq 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
ˆm=min
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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