A contrast source inversion method for reconstructing objects in an inhomogeneous background medium

We present a contrast source inversion (CSI) algorithm using the finite-difference approach for reconstructing the unknown material properties of an inhomogeneous object immersed in a known inhomogeneous background medium from the scattered field data. Similar to the CSI algorithm using the integral equation (IE) approach, the unknown contrast source and the unknown contrast are updated alternately to reconstruct the scatterer without requiring the solution of the full forward problem at each iteration step in the inversion process. In the present method, we use a finite-difference (FD) frequency domain method incorporated with a PML absorbing boundary condition. The latter enables us to implement the reconstruction of an inhomogeneous object embedded in an unbounded inhomogeneous background medium. This approach makes this algorithm more versatile than the IE-based CSI algorithm, which is only efficient for handling a homogeneous background medium. An attractive feature of introducing the finite-difference operator into the algorithm is that the stiffness matrix of such an operator is only dependent on the background medium, which is invariant throughout the inversion process. Therefore, in two-dimensional (2D) configurations, where the size of the stiffness matrix is manageable, this finite-difference operator only needs to be inverted once and the results can be reused in successive iterations of the inversion. Numerical experiments show that this local (FD-based) CSI algorithm has excellent performance for both homogeneous and inhomogeneous background media.