Application of ℓp-regularized least squares for 0≤p≤1 in estimating discrete spectrum of relaxations for electromagnetic induction responses

The EMI response of a target can be accurately modeled by a sum of real exponentials. However, it is difficult to obtain the model parameters from measurements when the number of exponentials in the sum is unknown and the summands are highly correlated. Traditionally, the time constants and residues are estimated by nonlinear iterative search. Most of these methods, however, often suffer from (a) sub-optimal solutions that are far from the truth and (b) complex parameters that do not have physical meaning. In this paper, the estimation problem is reformulated into a linear system by enumerating the relaxation parameter space. The model parameters are then estimated through a modified Lp-regularized least squares algorithm for 0 <= p <= 1. Using tests on synthetic data and laboratory measurement of known targets the proposed method is shown to provide satisfactory and stable estimates of the model parameters. From the lab data, we demonstrate that the proposed method can correctly estimate the relaxation time constants of a target, which is of great value as the time constants are position and orientation invariant and can be used as features in target discrimination. We also propose an empirical method to select the regularization parameter to the Lp-regularization problem. The resulting selection rule has the logged regularization parameter in a linear relationship with the noise level. This simple relationship allows the proposed method be applied in practice with short computation time while being robust.