Singular value analysis of the Jacobian matrix in microwave image reconstruction

For non-linear inverse scattering problems utilizing Gauss-Newton methods, the Jacobian matrix encodes rich information concerning the system performance and algorithm efficiency. In this paper, we perform an analytical evaluation of a single-iteration Jacobian matrix based on a previously derived nodal adjoint representation. Concepts for studying linear ill-posed problems, such as the degree-of-ill-posedness, are used to assess the impact of important system parameters on the expected image quality. Analytical singular value decomposition (SVD) of the Jacobian matrix for a circular imaging domain is derived along with the numerical SVD for optimizing imaging system configurations. The results show significant reductions in the degree-of-ill-posedness when signal frequency, antenna array density and property parameter sampling are increased. Specifically, the decay rate in the singular spectrum of the Jacobian decreases monotonically with signal frequency being approximately 1/3 of its 0.1 GHz value at 3 GHz, is improved with antenna array density up to about 35 equally-spaced circumferentially positioned elements and drops significantly with increased property parameter sampling to more than twice the amount of measurement data. These results should serve as useful guidelines in the development of design specifications for an optimized hardware installation.