Boundary Element Method for an Inverse Problem in Magnetic Resonance Imaging Gradient Coils
Liviu Marin; Henry Power; Richard W. Bowtell; Clemente Cobos Sanchez; Adib A. Becker; Paul Glover; and Arthur Jones

Source CMES: Computer Modeling in Engineering & Sciences, Vol. 23, No. 2, pp. 149-174, 2008
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Keywords Inverse problem, regularization, divergence-free boundary element method, magnetic resonance imaging, electromagnetism, coil design.
Abstract We investigate the reconstruction of a divergence-free surface current distribution from knowledge of the magnetic flux density in a prescribed region of interest in the framework of static electromagnetism. This inverse problem is motivated by the design of gradient coils for use in magnetic resonance imaging (MRI) and is formulated using its corresponding integral representation according to potential theory. A novel boundary element method (BEM) which employs linear interpolation on quadratic surfaces and also satisfies the continuity equation for the current density, i.e. a divergence-free BEM, is presented. Since the discretised BEM system is ill-posed and hence the associated least-squares solution may be inaccurate and/or physically meaningless, the Tikhonov regularization method is employed in order to retrieve accurate and physically correct solutions.
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