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Time-domain direct sampling method for inverse electromagnetic scattering with a single incident source

Chen Geng, Minghui Song, Xianchao Wang, Yuliang Wang

TL;DR

The paper addresses inverse electromagnetic scattering from time-dependent boundary measurements and introduces a time-domain direct sampling method that uses a single incident source. By defining a time-domain imaging functional that encodes retarded-time travel and echo amplitude, and linking it to a frequency-domain indicator via the Fourier-Laplace transform, the authors establish a solid theoretical basis for locating scatterers. They provide forward-model well-posedness through a Lippmann-Schwinger framework and Paley-Wiener theory, derive asymptotic and stability results using modified Bessel and angular-integral lemmas, and prove that the imaging functional concentrates at scatterer locations under suitable limits. Numerical experiments in 2D TM/TE and 3D validate the approach with Gaussian and sawtooth temporals, show robustness to noise, and highlight the practical advantages of using a single, flexible incident source for real-time imaging, while discussing limitations in full shape reconstruction without multiple sources. The method offers an efficient, compatible alternative to time-domain imaging techniques, with potential for real-time EM scattering investigations and dynamic target tracking.

Abstract

In this paper, we consider an inverse electromagnetic medium scattering problem of reconstructing unknown objects from time-dependent boundary measurements. A novel time-domain direct sampling method is developed for determining the locations of unknown scatterers by using only a single incident source. Notably, our method imposes no restrictions on the the waveform of the incident wave. Based on the Fourier-Laplace transform, we first establish the connection between the frequency-domain and the time-domain direct sampling method. Furthermore, we elucidate the mathematical mechanism of the imaging functional through the properties of modified Bessel functions. Theoretical justifications and stability analyses are provided to demonstrate the effectiveness of the proposed method. Finally, several numerical experiments are presented to illustrate the feasibility of our approach.

Time-domain direct sampling method for inverse electromagnetic scattering with a single incident source

TL;DR

The paper addresses inverse electromagnetic scattering from time-dependent boundary measurements and introduces a time-domain direct sampling method that uses a single incident source. By defining a time-domain imaging functional that encodes retarded-time travel and echo amplitude, and linking it to a frequency-domain indicator via the Fourier-Laplace transform, the authors establish a solid theoretical basis for locating scatterers. They provide forward-model well-posedness through a Lippmann-Schwinger framework and Paley-Wiener theory, derive asymptotic and stability results using modified Bessel and angular-integral lemmas, and prove that the imaging functional concentrates at scatterer locations under suitable limits. Numerical experiments in 2D TM/TE and 3D validate the approach with Gaussian and sawtooth temporals, show robustness to noise, and highlight the practical advantages of using a single, flexible incident source for real-time imaging, while discussing limitations in full shape reconstruction without multiple sources. The method offers an efficient, compatible alternative to time-domain imaging techniques, with potential for real-time EM scattering investigations and dynamic target tracking.

Abstract

In this paper, we consider an inverse electromagnetic medium scattering problem of reconstructing unknown objects from time-dependent boundary measurements. A novel time-domain direct sampling method is developed for determining the locations of unknown scatterers by using only a single incident source. Notably, our method imposes no restrictions on the the waveform of the incident wave. Based on the Fourier-Laplace transform, we first establish the connection between the frequency-domain and the time-domain direct sampling method. Furthermore, we elucidate the mathematical mechanism of the imaging functional through the properties of modified Bessel functions. Theoretical justifications and stability analyses are provided to demonstrate the effectiveness of the proposed method. Finally, several numerical experiments are presented to illustrate the feasibility of our approach.

Paper Structure

This paper contains 11 sections, 8 theorems, 98 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Background and motivation
  3. Mathematical modeling and imaging functional
  4. Laplace transform and well-posedness of the forward problem
  5. Analysis of the time-domain direct sampling method
  6. Numerical examples
  7. Numerical results for transverse magnetic (TM) mode in 2D
  8. Gaussian-modulated sinusoidal pulse wave
  9. Smooth sawtooth wave
  10. Numerical results for transverse electric (TE) mode in 2D
  11. Numerical results in 3D

Key Result

Lemma 2.1

Let $\omega\in\mathbb{C}_{\sigma_0}\mapsto f(\omega)\in\mathcal{B}(X,Y)$ be a holomorphic function with values of linear bounded operators between two Banach space $X$ and $Y$, satisfying Let ${ F(t):=\frac{1}{2\pi}\int_{-\infty+\mathrm{i}\sigma}^{+\infty+\mathrm{i}\sigma}\mathrm{ e}^{-i\omega t}f(\omega)\, \mathrm{ d}\omega}$ and ${\mathcal{F}g:=\int_{-\infty}^{+\infty}F(\tau)g(t-\tau) \, \mathr

Figures (8)

  • Figure 1: Plots of the Gaussian modulated pulse function $\chi(t)$ and the corresponding Fourier spectrum with $\lambda=1$ ($\omega\approx1.88\times10^9$).
  • Figure 2: Reconstructions of three small rectangular scatterers for the TM mode using the time-domain direct sampling method $\mathcal{I}(z_l)$ with different wavelengths $\lambda$.
  • Figure 3: Reconstructions of three small rectangular scatterers for the TM mode using the total focusing method $\mathcal{I}_{TFM}(z_l)$ with different wavelengths $\lambda$.
  • Figure 4: Contour plots of the reconstructions for the TM mode with different noise level $\delta$.
  • Figure 5: Contour plots of the reconstructions for the TM mode with different terminal time $T$.
  • ...and 3 more figures

Theorems & Definitions (19)

  • Lemma 2.1
  • Theorem 2.1
  • proof
  • Theorem 3.1
  • proof
  • Remark 3.1
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • ...and 9 more