Nonlinear microwave imaging using Levenberg-Marquardt method with iterative shrinkage thresholding

Summary form only given. Development of microwave imaging methods applicable in sparse investigation domains is becoming a research focus in computational electromagnetics (D.W. Winters and S.C. Hagness, IEEE Trans. Antennas Propag., 58(1), 145-154, 2010). This is simply due to the fact that sparse/sparsified domains naturally exist in many applications including remote sensing, medical imaging, crack detection, hydrocarbon reservoir exploration, and see-through-the-wall imaging. In this work, for the first time, a nonlinear scheme is proposed for solving the two-dimensional inverse electromagnetic scattering problem on sparse domains. The nonlinearity and ill-posedness of the problem are tackled using LevenbergMarquardt (A. Franchois and C. Pichot, IEEE Trans. Antennas Propag., 45(2), 203-215, 1997) and iterative shrinkage (I. Daubechies, et. al., Commun. Pure Appl. Math, 57(11), 1413-1457, 2004) algorithms, respectively. The Levenberg-Marquardt scheme, first, expresses the scattering equations as a nonlinear function of contrast to construct a “forward model”. Then, using this forward model and its Frechet derivative within a Newton-like iterative scheme, a sequence of linear ill-posed systems is constructed. These ill-posed systems are regularized using weighted zeroth/first penalty term. The resulting minimization problem is solved using an iterative shrinkage algorithm. However, naive application of these algorithms results in very slow convergence rates. To this end, a two-step iterative thresholding (TWIST) is used (J. M. BioucasDias and M. A. Figueiredo, IEEE Trans. Image Process., 16(12), 2992-3004, 2007). TWIST increases the convergence rate of shrinkage iterations by using updates weighted between the current iteration and the previous one. The weights are selected depending on the largest and smallest singular values of the discretized Frechet derivative. Note that TWIST applies the thresholding to the Newton step. Additionally, to maintaining the sparsity of the solution by “clearing” the background ripples generated within the Newton path, an extra thresholding is applied to the solution at each Newton iteration. This helps in increasing accuracy of the solution and convergence rate of the Newton iterations. It should also be noted here that as Newton iterations evolve, the weight of the penalty term (and consequently the TWIST thresholding level) is decreased. This also helps to improve the efficiency of the method since the size of the Newton step decreases quadratically with increasing iterations. Numerical results, which demonstrate the efficiency, accuracy, and applicability of the proposed nonlinear inversion algorithm, will be presented. Results will be compared to those obtained by other schemes using smoothness promoting regularization methods.