Application of Multiplicative Regularization to the Finite-Element Contrast Source Inversion Method

Multiplicative regularization is applied to the finite-element contrast source inversion (FEM-CSI) algorithm recently developed for microwave tomography. It is described for the two-dimensional (2D) transverse-magnetic (TM) case and tested by inverting experimental data where the fields can be approximated as TM. The unknown contrast, which is to be reconstructed, is represented using nodal variables and first-order basis functions on triangular elements; the same first-order basis functions used in the FEM solution of the accompanying field problem. This approach is different from other MR-CSI implementations where the contrast variables are located on a uniform grid of rectangular cells and represented using pulse basis functions. The linear basis function representation of the contrast makes it difficult to apply the weighted
$L_{2}$
-norm total variation multiplicative regularization which requires that gradient and divergence operators be applied to the predicted contrast at each iteration of the inversion algorithm; the use of finite-difference operators for this purpose becomes unwieldy. Thus, a new technique is introduced to perform these operators on the triangular mesh.