Cheng Y-C N, Eagan T P, Chmielewski T, Flock J, Kang M-C, Kidane T K, Shvartsman Sh M, Brown R W
Department of Physics, Case Western Reserve University, Cleveland, Ohio 44106, USA.
MAGMA. 2003 Jul;16(2):103-11. doi: 10.1007/s10334-003-0009-5. Epub 2003 Aug 7.
A method for finding closed-form solutions for the normal mode frequencies of systems with circulant (2 pi/n) symmetry was investigated. This method is particularly useful for questions of degeneracy that arise when one considers parallel imaging techniques like SENSE and SMASH in MRI. It is applicable to systems that include birdcage coils as well as planar coils with the appropriate rotational symmetry. A proof is given that complete degeneracy of all normal mode frequencies is impossible when all mutual inductive couplings are included. We tested the method against measurements made on a planar coil array and on an 8-element birdcage coil. The inclusion of the co-rotating end-ring mode changes the fundamental symmetry of the system from circulant to 'bordered circulant.' Closed-form solutions for the normal mode frequencies of a bordered circulant system are also given.
研究了一种用于寻找具有循环（2π/n）对称性系统的正常模式频率闭式解的方法。该方法对于在考虑MRI中诸如SENSE和SMASH等并行成像技术时出现的简并性问题特别有用。它适用于包括鸟笼线圈以及具有适当旋转对称性的平面线圈的系统。给出了一个证明，即当包括所有互感耦合时，所有正常模式频率完全简并是不可能的。我们根据在平面线圈阵列和8元鸟笼线圈上进行的测量对该方法进行了测试。共旋转端环模式的纳入将系统的基本对称性从循环变为“带边循环”。还给出了带边循环系统正常模式频率的闭式解。
