SPARSE ELECTROMAGNETIC IMAGING USING NONLINEAR LANDWEBER ITERATIONS

Abdulla Desmal, Hakan Bagci
2015 Electromagnetic Waves  
A scheme for efficiently solving the nonlinear electromagnetic inverse scattering problem on sparse investigation domains is described. The proposed scheme reconstructs the (complex) dielectric permittivity of an investigation domain from fields measured away from the domain itself. Least-squares data misfit between the computed scattered fields, which are expressed as a nonlinear function of the permittivity, and the measured fields is constrained by the L 0 /L 1 -norm of the solution. The
more » ... e solution. The resulting minimization problem is solved using nonlinear Landweber iterations, where at each iteration a thresholding function is applied to enforce the sparseness-promoting L 0 /L 1 -norm constraint. The thresholded nonlinear Landweber iterations are applied to several two-dimensional problems, where the "measured" fields are synthetically generated or obtained from actual experiments. These numerical experiments demonstrate the accuracy, efficiency, and applicability of the proposed scheme in reconstructing sparse profiles with high permittivity values. 78 Desmal and Bagcı permittivity. This drawback can be overcome by developing "complete" nonlinear inversion schemes that do not linearize the forward scattering equations. Examples of these schemes include Newtontype methods such as the inexact Newton [21-23], distorted Born [24], and Levenberg-Marquardt [25] methods. Additionally, other schemes making use of the nonlinear conjugate gradient [26] [27] [28] [29] or the steepest descent [30, 31] are widely used for image reconstruction with strong scatterers. The ill-posedness is alleviated by constructing the inverse problem in the form of a minimization problem for the data misfit between the measured and the computed scattered fields, which is constrained by adding a regularization/penalty term [2, 3] . Data misfit is represented in the L 2 -norm (i.e., leastsquares fit) and the penalty term can be the L 0 -, L 1 -, or L 2 -norm of the solution [2, 3]. The L 2 -norm penalty term, which has been the more commonly used one, promotes smoothness in the solution [3] . On the other hand, solving the minimization problem regularized with the L 0 /L 1 -norm penalty term promotes sharpness and sparseness, i.e., discontinuities in the recovered solution are detected more accurately [32, 33] . It should be noted here that minimization problems with the sparseness-promoting L 0 -and L 1 -norm penalty terms have often been studied in linear or linearized ill-posed problems. Their use in electromagnetic inverse problems has been limited to only the Born iterative [19, 20] and inexact Newton [21] methods even though sparse domains are very common in non-destructive testing, through-wall, and radar imaging [6-9, 19-21, 34]. In this work, to enable efficient and accurate electromagnetic imaging of the sparse domains involving strong scatterers, nonlinear least-squares data misfit between the measured scattered fields and the computed ones, which are expressed directly as a nonlinear function of the permittivity, is constrained by the sparseness-promoting L 0 -and L 1 -norm penalty terms. The resulting minimization problem, which is also referred as sparsity-constrained nonlinear Tikhonov problem, is solved using nonlinear Landweber (NLW) iterations [35, 36] , where at each iteration a thresholding function is applied to enforce the sparsity constraint. Unlike inexact Newton [21] and Born iterative [19, 20] methods with sparsity constraints, the proposed scheme avoids generation of a sequence of linear sparse optimization problems and requires only one regularization parameter, which directly penalizes the nonlinear problem, to be set. Consequently, it simplifies the task of heuristic parameter "tweaking", which is oftentimes very cumbersome for existing inversion algorithms. Application of the proposed scheme to several twodimensional (2-D) problems, where the "measured" fields are synthetically generated or obtained from actual experiments, demonstrates that the proposed scheme (i) produces sharper and more accurate reconstruction of permittivity profiles in sparse domains (in comparison with schemes which use L 2norm regularization) and (ii) maintains its convergence during the reconstruction of profiles with higher permittivity values (in comparison with schemes which make use of (iterative) linearization of the nonlinear problem). The rest of the paper is organized as follows. Section 2.1 presents the nonlinear 2-D electromagnetic scattering equations, Section 2.2 describes a scheme to discretize the scattering equations and constructs the nonlinear forward solver, Section 2.3 focuses on the sparsity-constrained minimization problem and its solution using the thresholded NLW iterations, and Section 2.4 describes a frequency hopping scheme to be used together with the NLW iterations under excitations with multiple frequencies. Section 3 presents numerical experiments, which demonstrate the efficiency, accuracy, and applicability of the proposed scheme in reconstructing the permittivity profiles in sparse domains with strong scatterers. Finally, Section 4 concludes with a summary and future research directions. FORMULATION Nonlinear Scattering Equations Let S represent the support of the 2-D investigation domain residing in an unbounded homogenous background medium (Figure 1) . The permeability and the (complex) permittivity of the investigation domain and the background medium are represented by {μ 0 , ε(r)} and {μ 0 , ε 0 }, respectively. Here, ε(r) is the unknown to be determined. The investigation domain is surrounded by a transmitter and N R receivers (Figure 1 ). Let ω and E inc u (r), u ∈ {x, y, z} represent the frequency of the transmitter and the three components of the incident electric field it generates. Upon excitation, electric current density with three components J u (r), u ∈ {x, y, z} is induced on S. These current density components satisfy J u (r) = jωε 0 τ (r)E u (r). Here, E u (r) are the three components of the total electric field and τ (r)
doi:10.2528/pier15052806 fatcat:ewx5tfi3srgtxhfccd27mipnlm