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Near-field imaging of locally perturbed periodic surfaces

Xiaoli Liu, Ruming Zhang

TL;DR

This work addresses inverse scattering from locally perturbed $2\pi$-periodic surfaces by combining a Floquet-Bloch transform with a two-step reconstruction: first obtain a rough periodic-surface guess via a sampling method, then refine both the periodic profile $\zeta$ and the local perturbation $p$ using a Newton-CG optimization. The forward problem is formulated with a sound-soft boundary, wrapped in a Bloch-transformed variational framework that encodes surface information in coefficient matrices; the inverse problem leverages the scattering operator and its Fréchet derivative, enabling adjoint-based updates. A key insight is that, for certain incident fields, the perturbation’s influence on the scattered field is separable enough to permit sequential imaging: recover $\zeta$ from less-informative data, then recover $p$ from data rich in perturbation information, with numerical examples confirming accurate reconstructions. The approach provides a robust and practical pathway for near-field imaging of complex periodic surfaces with local defects, with potential applications in nondestructive testing and surface inspection where both global and local features must be recovered from limited measurements.

Abstract

This paper concerns the inverse scattering problem to reconstruct a locally perturbed periodic surface. Different from scattering problems with quasi-periodic incident fields and periodic surfaces, the scattered fields are no longer quasi-periodic. Thus the classical method for quasi-periodic scattering problems no longer works. In this paper, we apply a Floquet-Bloch transform based numerical method to reconstruct both the unknown periodic part and the unknown local perturbation from the near-field data. By transforming the original scattering problem into one defined in an infinite rectangle, the information of the surface is included in the coefficients. The numerical scheme contains two steps. The first step is to obtain an initial guess, i.e., the locations of both the periodic surfaces and the local perturbations, from a sampling method. The second step is to reconstruct the surface. As is proved in this paper, for some incident fields, the corresponding scattered fields carry little information of the perturbation. In this case, we use this scattered field to reconstruct the periodic surface. Then we could apply the data that carries more information of the perturbation to reconstruct the local perturbation. The Newton-CG method is applied to solve the associated optimization problems. Numerical examples are given at the end of this paper to show the efficiency of the numerical method.

Near-field imaging of locally perturbed periodic surfaces

TL;DR

This work addresses inverse scattering from locally perturbed -periodic surfaces by combining a Floquet-Bloch transform with a two-step reconstruction: first obtain a rough periodic-surface guess via a sampling method, then refine both the periodic profile and the local perturbation using a Newton-CG optimization. The forward problem is formulated with a sound-soft boundary, wrapped in a Bloch-transformed variational framework that encodes surface information in coefficient matrices; the inverse problem leverages the scattering operator and its Fréchet derivative, enabling adjoint-based updates. A key insight is that, for certain incident fields, the perturbation’s influence on the scattered field is separable enough to permit sequential imaging: recover from less-informative data, then recover from data rich in perturbation information, with numerical examples confirming accurate reconstructions. The approach provides a robust and practical pathway for near-field imaging of complex periodic surfaces with local defects, with potential applications in nondestructive testing and surface inspection where both global and local features must be recovered from limited measurements.

Abstract

This paper concerns the inverse scattering problem to reconstruct a locally perturbed periodic surface. Different from scattering problems with quasi-periodic incident fields and periodic surfaces, the scattered fields are no longer quasi-periodic. Thus the classical method for quasi-periodic scattering problems no longer works. In this paper, we apply a Floquet-Bloch transform based numerical method to reconstruct both the unknown periodic part and the unknown local perturbation from the near-field data. By transforming the original scattering problem into one defined in an infinite rectangle, the information of the surface is included in the coefficients. The numerical scheme contains two steps. The first step is to obtain an initial guess, i.e., the locations of both the periodic surfaces and the local perturbations, from a sampling method. The second step is to reconstruct the surface. As is proved in this paper, for some incident fields, the corresponding scattered fields carry little information of the perturbation. In this case, we use this scattered field to reconstruct the periodic surface. Then we could apply the data that carries more information of the perturbation to reconstruct the local perturbation. The Newton-CG method is applied to solve the associated optimization problems. Numerical examples are given at the end of this paper to show the efficiency of the numerical method.

Paper Structure

This paper contains 14 sections, 4 theorems, 63 equations, 7 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. Direct Scattering Problem
  3. Mathematical Model
  4. Floquet-Bloch transform
  5. Approximation of the scattering problems with periodic surfaces
  6. Inverse Problem and the Newton-CG Method
  7. Scattering operator and its properties with respect to rough surfaces
  8. Discretization for locally perturbed periodic surfaces
  9. Numerical implementation
  10. Sampling method
  11. Iteration method
  12. Numerical results
  13. Sampling method
  14. Newton's method

Key Result

Theorem 2

For $|r|<1$, given any incident field $u^i$ and the function $f$ defined by eq:sca3 belongs to the space $H_r^{-1/2}(\Gamma_H)$, the variational problems eq:sca_var has a unique solution $u\in \widetilde{H}^1_r(\Omega^p_H)$.

Figures (7)

  • Figure 1: Mathematical model for the scattering problem.
  • Figure 2: (a): the first surface; (b): the second surface.
  • Figure 3: (a): the first surface; (b)&(c): the reconstructions.
  • Figure 4: (a): the first surface; (b)&(c): the reconstructions.
  • Figure 5: Real- and imaginary-part of the incident field $u^i$.
  • ...and 2 more figures

Theorems & Definitions (11)

  • Remark 1
  • Theorem 2
  • Remark 3
  • Theorem 4
  • Remark 5
  • Remark 6
  • Remark 7
  • Theorem 8
  • Theorem 9
  • Remark 10
  • ...and 1 more