Data continuation for the explicit solution of an inverse biomagnetic problem

This paper considers the problem of data continuation for an explicit identification method for electric dipoles situated in a conductor, and shows that the magnetic field measured over a part of a closed surface (which encloses the conductor) can be analytically continued to the rest of the surface. Such a continuation problem has a unique solution for an arbitrarily shaped conductor and observation surface. The paper proposes two straightforward methods for the continuation - the spherical harmonic expansion and the integral equation approach - and presents continuation examples for the case of a single dipole in a spherical conductor with the data measured on a part of a spherical observation surface. The spherical harmonic expansion method works well when the field is continued relatively far from the conductor and when the measured data are clean; however, the spherical harmonic expansion fails when the data are noisy. On the other hand, the integral equation approach, although it suffers from numerical integration errors, shows a much better continuation with quite noisy data. (A method to reduce the numerical integration errors is proposed.) The latter approach is also much more accurate than the spherical harmonic expansion for continuing the field in the proximity of the conductor. Finally, the paper discusses the applicability of both methods to an explicit identification method.