Analysis of Degrees of Freedom in Scattered Fields for Nonlinear Inverse Scattering Problems

In the study of nonlinear inverse scattering problems (ISPs), the traditional method for estimating the number of degrees of freedom (NDoF) of the scattered field does not consider the effect of the scatterer contrast. In this work, we study the relationship between NDoF and scatterer contrast by modifying the radiation matrix. Specifically, three numerical methods for calculating the NDoF are presented. The first NDoF is defined as the number of principle singular values of the external radiation matrix, which is used as a benchmark. The second NDoF is defined based on the modified radiation matrix that incorporates the contrast of domain of investigation (DoI) with the external radiation matrix. By exploring the relationship between the second NDoF and contrast, an approximate upper bound of the second NDoF is obtained, that is, the third NDoF. The accuracy of the proposed NDoF is verified by interpolation experiments on both synthetic and experimental data using frequency-domain zero-padding method. Results show that the second NDoF is more accurate than traditional NDoF for high-contrast scatterers, and the third NDoF can be easily estimated by the traditional NDoF plus a positive constant, which provides a simple way to estimate the upper bound of NDoF.