A new regularization scheme for inverse scattering

Abstract

The reconstruction of the complex permittivity profile of inhomogeneous objects from measured scattered field data is a strongly nonlinear and ill-posed problem. Generally, the quality of the reconstruction from noisy data is enhanced by the introduction of a regularization scheme.Starting from an iterative algorithm based on a conjugate gradient method and applied to the nonlinear problem, this paper deals with a new regularization scheme, using edge-preserving (EP) potential functions. With this a priori information, the object to reconstruct is modelled with homogeneous areas separated by borderlike discontinuities. The enhancement is illustrated throughout some examples with noisy synthetic data.

Edge-Preserving Regularization Scheme Applied to Modified Gradient Method to Reconstruct Two-Dimensional Targets From Data Laboratory-Controlled

“…Previously, this regularizing term was implemented within an halfquadratic technic [5,11,8]. Here, according to [7], we do not consider this approach but directly implement our cost functional as…”

“…Different regularization schemes can be considered; in this paper the authors investigate an edgepreserving approach [6]. This regularizing method has already shown its usefulness for image enhancement [6] and image reconstruction [11,8] using a conjugate gradient algorithm. As far as the authors know, this regularization approach has not been applied to the Modified Gradient Method (MGM) nor validated against experimental data.…”

“…Moreover, taking into account [7], the edge-preserving regularization scheme is directly implemented without considering a half-quadratic technique as previously done [5,6,8,11].…”

3d Microwave Tomography With Huber Regularization Applied to Realistic Numerical Breast Phantoms

“…In this case a regularization is applied to the permittivity update in each iteration of the optimization: Tikhonov regularization -similar to [11,13] -in [14,34,37], a combination of L1 and L2 norm sparsity promoting regularization in [38] and other compressive sensing schemes in, e.g., [44] and a conjugate gradient for least squares (CGLS) algorithm in [3,33,39,40]. In the present paper, as in [26,42,43,45] for microwave breast imaging and as in, e.g., [19,30,[46][47][48][49][50][51][52] for microwave imaging, we adopt a regularized cost function consisting of the data fit term and a regularization term. The regularization term allows for easy incorporation of a priori information on the complex permittivity profile.…”

“…Various edge-preserving potential functions, which smooth the homogeneous areas of an image while preserving edges, without a priori knowledge of the edge locations, are employed in [50][51][52][53][54] in a spatially structured regularization; for example, discontinuity adaptive (DA) models [57] turn off smoothing less abruptly than line process [53] and weak membrane [54] models; an advantage of the well-known Huber function [58] is that it is nonquadratic but convex [59]: it consists of a quadratic function for arguments below a threshold, for smoothing small scale noise, and of a linear function above the threshold, for preserving discontinuities. In [50,51] we tested different DA functions, including the Huber function, with measured scattered field data for 2D and 3D piecewise-constant lossless dielectric targets in air from Institut Fresnel [60,61]; the Huber function yielded an average reconstruction error of 14.9% for the 2D case (see [50] for the error values and [62, p.78] for a correction in the error formula) -the maximum target-to-background permittivity contrast was 3-to-1; for the 3D case reconstructed permittivities were within 5% of the actual values for the two-spheres and two-cubes targets [51] with maximum target-to-background contrasts of 2.6-and 2.3-to-1, respectively.…”