Colour level set regularization for the electromagnetic imaging of highly discontinuous parameters in 3D

Abstract

In this paper, we propose a novel reconstruction scheme for the low-frequency near-field electromagnetic imaging of high-contrast conductivity distributions inside shielded regions using the system of Maxwell's equations in 3D. In our novel scheme, we focus on estimating the shape characteristics of the electrical conductivity profile inside these regions from low-frequency electromagnetic data measured at external locations for a single frequency. We introduce a color level set regularization scheme which is a shape-based approach focusing on the simultaneous reconstruction of several shape-like distributions of different conductivity values in the same region of interest. We also introduce a topological perturbation scheme alongside the color level set regularization that is used to avoid a certain type of local minima which is characteristic for this simultaneous multi-value shape-based reconstruction. Using two numerical experiments focusing on a three-value reconstruction problem related to the imaging of shielded boxes or cargo containers, we compare this novel approach with results obtained from standard voxel-based reconstruction schemes on the one hand and the more established two-value shape based approach on the other hand. We demonstrate that, depending on the particular situation of the imaging setup, this three-value (or in general multiple-value) shape-based reconstruction technique has the potential to provide superior reconstruction results in many situations, in particular regarding reconstruction of the correct shapes. We also discuss particular challenges of this novel methodology.

Stochastic Optimization Methods for Parametric Level Set Reconstructions in 2D through-the-Wall Radar Imaging

“…Certainly, an adaptive selection of step sizes, e.g., being reduced with increasing iteration number, recommends itself here. Such an approach has been applied in a different application of low-frequency electromagnetic imaging in [26,27] with very good success. However, a similar more detailed study of adaptive line search techniques for the application in stochastic shape optimization schemes for through-the-wall radar imaging is beyond the scope of this paper.…”

“…In stochastic frameworks, standard gradient-based line search criteria such as the Wolfe or Armijo condition are not possible to use since the full gradient of the cost is not easily available [24]. Therefore, we adopt a specific backtracking procedure where no additional run of our forward simulator is required [25][26][27]. Instead, it focuses on obtaining a controlled speed of the underlying shape evolution, measured by the number of pixels that change value in one given time step.…”

A regularization method based on level-sets for the problem of crack detection from electrical measurements

“…For n = 0 set G (0) ε,α (φ, ψ) := G ε,α (φ, ψ) as in (24), where (φ 0 , ψ 0 ) is some initial guess. The iterative regularization method under consideration consists in minimizing the family of functionals…”

“…[3,8,16,22,29]). Crack reconstruction approaches includes shape optimization methods [3,12,23,24,34], reciprocity principle [32], probe method [22], factorization method [6,7,19], asymptotic analysis [9], among others.…”

Parametric Level-sets Enhanced To Improve Reconstruction (PaLEnTIR)

“…One important limitation of existing PaLS models (indeed most all level set methods) is the ability to recover media with one two contrast values using a single level set function. Existing level-set based approaches for multi-contrast, multi-object problems, such as the colour level set [33], vector level set [80], and binary level set [48], either use N level sets for N objects [12,54,80] or log N level sets for N objects [33,48,73]. As a result, with the existing methods, as the number of objects with different contrast values increases, the required number of level sets also increases.…”