Sampling theory for neuromagnetic detector arrays

The sampling theorem for wave-number-limited multivariable functions is applied to the problem of neuromagnetic field mapping. The wave-number spectrum and other relevant properties of these fields are estimated. A theory is derived for reconstructing neuromagnetic fields from measurements using sensor arrays which sample either the field component B _z perpendicular to the planar grid of measurement points, or the two components partial B _z/partial x and partial B _z/partial y of its gradient in the xy plane. The maximum sensor spacing consistent with a unique reconstruction is determined for both cases. It is shown that, when two orthogonal components of the gradient are measured at every site of the measurement grid, the density of these sensor-pair units can be reduced, without risk of aliasing, to half of what is necessary for single-channel sensors in an array sampling B _z alone. Thus the planar and axial gradiometer arrays are equivalent in the sampling sense provided that the number of independent measurements per unit area is equal.