Biomagnetic inverse solution using combined-norm and pointwise normalization

In this paper, we propose a method of solving the biomagnetic inverse problem consisting of two approaches, the first of which is pointwise normalization. In the conventional normalization technique, each variable is normalized individually. This variablewise normalization is appropriate for scalar fields, but not for vector fields where one vector on a grid point is represented by several variables. Hence, pointwise operation is needed for vector fields. The second is a combination of norms. Use of the ℓ₂-norm as the cost function of an optimization problem is known to lead to spatially spread solutions, while the ℓ₂-norm leads to sparse solutions. To control sparseness of solutions, we propose to use an internal division of the ℓ₂-norm and pointwise normalized ℓ₂-norm. The optimization problem constructed as above can be recast as a second-order cone program (SOCP), a nonlinear convex problem. The problem can be solved using recently developed efficient interior-point methods. Computer simulations showed that the sparseness of estimators obtained with the proposed method reflects both the ratio of internal division and the sparseness of true sources. Regularization of normalization and relaxation of constraint conditions in the presence of noise are also presented.