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Flexible spatial encoding strategy using receive coil aggregates for Halbach magnet array based magnetic resonance imaging
Dong Wei Lu1, Zhi Hua Ren1, and Shao Ying Huang1

1EPD Pillar, Singapore University of Technology and Design, Singapore, Singapore

Synopsis

To make a MRI system portable, a practical approach is applying Halbach magnet array and nonlinear spatial encoding strategy. Here, the rotation of a magnet array for imaging is replaced by electrically forming RF receive coil aggregates with phase delay. For the resultant system with a new encoding matrix, Truncated-Singular-Value-Decomposition with an optimal regularization parameter is proposed which reconstructs images with good quality. An accelerated L-curve method is proposed to obtain the optimal regularization parameter. Results show that the proposed strategy provides considerable improvement of the image quality compared to existing method, e.g. Kaczmarz iteration, without rotating the magnet array.

INTRODUCTION

Current clinical Magnetic Resonance Imaging (MRI) system is bulky, thus researchers are looking for imaging approaches with significant size reduction to let MRI possible in a wider range of working conditions. One realistic approach of a portable MRI system is based on a rotating Halbach magnet array and nonlinear spatial encoding strategy1,2. In this abstract, the rotation of a magnet array for imaging is replaced by electrically forming RF receive coils aggregates with phase delay. The physical setup leads to a new encoding matrix. Truncated Singular Value Decomposition (TSVD) with an optimal regularization parameter is proposed to solve the system, which reconstructs images with good quality. The optimal regularization parameter is selected by the L-curve method with accelerated calculations. With the same signal-to-noise (SNR) level, the proposed approach provides a significant improvement of the reconstructed image quality.

METHODS

Fig. 1(a) shows the inhomogeneous B0 field generated by a Halbach array for imaging. Eight RF loops are used. The signal array is given as $$$\mathbf{s=Em}$$$ where $$$\mathbf{m}$$$ is the image and $$$\mathbf{E}$$$ is the encoding matrix containing the phase encoded by B0 and the amplitude encoded by the coil sensitivity. Four coils chosen from the eight coils can be turned on/off simultaneously and they are called supercoils. The total number of supercoils is $$$C(8,4)=70$$$. The phase delay of the successive coils is set to be $$$π/4$$$. The B1 field maps of one of the supercoils is shown in Fig. 1(b). The encoding matrix for this portable system is ill-conditioned. The effect of noise can be magnified severely and the image may not be reconstructed by Least Mean Square (LMS) method. By discarding small singular values of the encoding matrix which can magnify the effect of noise, TSVD method is practical to solve such an ill-conditioned problem. The encoding matrix can be decomposed as3: $$$\mathbf{E=U\sum V^{T}}$$$, $$$\mathbf{\sum} = diag(\sigma_1, \sigma_2,....\sigma_m)$$$. The pseudo-inverse matrix can be reconstructed as:

$$\mathbf{m = E^+s = \sum_{i=1}^{k} v_iu_i^T}\sigma_i^{-1}, 1\leq k<m$$

The number $$$k$$$ is a regularization parameter. In TSVD method, we only use $$$\sigma_i(1\leq i\leq k <m)$$$ which are relatively large singular values of the encoding matrix and represent the main components of MR signal for image reconstruction. With the system resulted by using supercoils, the regularization parameter can be chosen by L-curve method4. L-curve describes the relationship between $$$\mathbf{log||m||}$$$ and $$$ \mathbf{log||Em-s|}|$$$. The L-curve’s turning point can be an optimal number for the regularization parameter. For obtaining a L-curve, a direct calculation takes days. Here, the pseudo-inverse matrix equation is applied which shortens the calculation time to be less than one hour for a matrix size of 5476×35910.

RESULTS

In this numerical simulation, the sampling number is 513 and the matrix size is 5476×35910. The rotation of magnet array is not needed and supercoils are used. TSVD, Kaczmarz iteration (KI), and LMS were applied for the reconstruction. Fig. 2 shows the reconstructed images by different methods when SNR is 60dB and the L-curve. The normalized root mean square error (NRMSE) of the reconstructed images using these methods at different SNR’s are tabulated and compared in Table. 1.

DISCUSSION

Fig. 2(a)-(c) shows the results using supercoils. As shown, at the same SNR, a center blurry area is observed in the image reconstructed using KI whereas TSVD shows the best image quality. The results show that LMS method is not suitable for the reconstruction here. Because of large condition number of encoding matrix, random error could be magnified greatly in the solution. Fig. 2 (d) shows the L-curve. The first turning point of the L-curve represents the singular value that the error of the solution begins to decrease. The second turning point should be selected to get a stable solution because at this point, all the large singular values of the encoding matrix can be included in the image reconstruction. In Table. 1, for SNR up to 40dB, the NRMSE of TSVD is lower than that of KI method. It is observed that, the NRMSE of iteration method decrease more slowly than TSVD as SNR increases. For the supercoil setup, by using TSVD, the NRMSE is relatively low at the same SNR and meanwhile, the blurred area can be mitigated.

CONCLUSION

It is shown that in a portable MRI scanner, the image can be reconstructed by electrically forming receive RF coils aggregates with phase delay and the rotation of Halbach magnet array is no longer needed. In the resultant system, TSVD with an optimized regularization parameter (enabled by an accelerated L-curve method) reconstructs images with an improved quality with lowered NRMSE compared to that of Kaczmarz iteration. Moreover, by TSVD, the central blurred area of reconstructed image is successfully mitigated.

Acknowledgements

No acknowledgement found.

References

1. Cooley, C. Z., et al. Two-dimensional imaging in a lightweight portable MRI scanner without gradient coils. Magnetic Resonance Medicine. 2015;73(2): 872-883. 2. Cooley, C. Z. Portable low-cost magnetic resonance imaging. Ph.D. dissertation, USA, Massachusettts Institute of Technology, Sep. 2014. 3. D. W. Lu, S. Y. Huang. A TSVD-based Approach for Flexible Spatial Encoding Strategy in Portable Magnetic Resonance Imaging (MRI) System. In 39th PIERS, Nov. 2017. 4. P. C. Hansen. The L-curve and its use in the numerical treatment of inverse problems. Computational Inverse Problems in Electrocardiology. 1999: 119-142.

Figures

Table. 1 The NRMSE of reconstructed image by different methods

Fig. 1 The simulation configuration (a) Eight loop RF coils are arranged clockwisely and four of them are chosen to work at the same time. The direction of inhomogeneous B0 field is along the x-axis. The phantom is a human head model, (b) The B1-map of a supercoil (Coil1, Coil3, Coil5 and Coil7). The x and y componet of B1 would take effect in signal receiving and image reconstruction. The phase delay of successive coils is π/4 . The field strength can be changed by changing the phase delay.

Fig. 2 The reconstructed image at SNR=60dB (a) by Kaczmarz iteration. Although the shape of the phantom can be seen, there exists blurry area in the central part of the image; (b) by LMS method. The phantom can not be reconstructed at all because of the disturbance of noise; (d) by TSVD. Most details of the phantom can be observed and the central blurry area is mitigated significantly. (d) The L-curve for the encoding matrix (the number of sampling is 513 and the matrix size is 5476×35910) when SNR is 60 dB. The second turning point is chosen to be the regularization parameter, k=4747.

Proc. Intl. Soc. Mag. Reson. Med. 26 (2018)
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