The effect of spatial sampling on the resolution of the magnetostatic inverse problem

In magnetoencephalography, linear minimum norm inverse methods are commonly employed when a solution with minimal a priori assumptions is desirable. These methods typically produce spatially extended inverse solutions, even when the generating source is focal. Various reasons have been proposed for this effect, including intrisic properties of the minimum norm solution, effects of regularization, noise, and limitations of the sensor array. In this work, we express the lead field in terms of the magnetostatic multipole expansion and develop the minimum-norm inverse in the multipole domain. We demonstrate the close relationship between numerical regularization and explicit suppression of spatial frequencies of the magnetic field. We show that the spatial sampling capabilities of the sensor array and regularization together determine the resolution of the inverse solution. For the purposes of stabilizing the inverse estimate, we propose the multipole transformation of the lead field as an alternative or complementary means to purely numerical regularization.