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Monostatic imaging of an extended target with MCMC sampling

Jiho Hong, Sangwoo Kang, Mikyoung Lim

TL;DR

This work tackles monostatic inverse scattering for planar extended targets in 2D by formulating a Bayesian MCMC approach that leverages a shape-derivative–based basis tailored to the measurement configuration. The authors derive a first-order far-field derivative with boundary densities and construct a target-specific perturbation basis, then optimize the initial disk parameters and perform posterior sampling via systematic-scan Hastings to recover the boundary. Numerical experiments show the new basis outperforms Fourier-based parametrizations, enabling accurate reconstruction of small perturbations and successful recovery of an extended target from diagonal (monostatic) data under noise. The method provides a practical, robust framework for shape reconstruction in monostatic imaging with potential broad applicability in non-invasive sensing.

Abstract

We consider the imaging of a planar extended target from far-field data under a monostatic measurement configuration, in which the data is measured by a single moving transducer, as frequently encountered in practical application. In this paper, we develop a Bayesian approach to recover the shape of the extended target with MCMC sampling, where a new shape basis selection is proposed based on the shape derivative analysis for the measurement data. In order to optimize the center and radius of the initial disk, we use the monostatic sampling method for the center and the explicit scattered field expression for disks for the radius. Numerical simulations are presented to validate the proposed method.

Monostatic imaging of an extended target with MCMC sampling

TL;DR

This work tackles monostatic inverse scattering for planar extended targets in 2D by formulating a Bayesian MCMC approach that leverages a shape-derivative–based basis tailored to the measurement configuration. The authors derive a first-order far-field derivative with boundary densities and construct a target-specific perturbation basis, then optimize the initial disk parameters and perform posterior sampling via systematic-scan Hastings to recover the boundary. Numerical experiments show the new basis outperforms Fourier-based parametrizations, enabling accurate reconstruction of small perturbations and successful recovery of an extended target from diagonal (monostatic) data under noise. The method provides a practical, robust framework for shape reconstruction in monostatic imaging with potential broad applicability in non-invasive sensing.

Abstract

We consider the imaging of a planar extended target from far-field data under a monostatic measurement configuration, in which the data is measured by a single moving transducer, as frequently encountered in practical application. In this paper, we develop a Bayesian approach to recover the shape of the extended target with MCMC sampling, where a new shape basis selection is proposed based on the shape derivative analysis for the measurement data. In order to optimize the center and radius of the initial disk, we use the monostatic sampling method for the center and the explicit scattered field expression for disks for the radius. Numerical simulations are presented to validate the proposed method.
Paper Structure (11 sections, 3 theorems, 69 equations, 8 figures)

This paper contains 11 sections, 3 theorems, 69 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Layer potential operators
  3. Shape derivative analysis for the far-field pattern
  4. MCMC sampling scheme with a new shape basis
  5. Shape basis functions associated with the measurement configuration
  6. MCMC sampling scheme
  7. Numerical simulation
  8. Performance for different shape parameters
  9. Shape reconstruction of extended targets
  10. Conclusion
  11. Far field asymptotic for disks

Key Result

Theorem 3.1

Fix $k$ and direction vectors $\hat{\mathbf{x}},\mathbf{d}\in S^1$. Let $\Omega$ be given by (Om:deform) with $h=\varepsilon h_0$, where $h_0$ is a reference shape deformation function and $\varepsilon$ is a small parameter. For the incident field as $u^{\textrm{inc}}(\mathbf{x}) = e^{{\rm i}k\mathb

Figures (8)

  • Figure 1.1: (a) describes the MSR matrix, where the numbers of incident waves and measurement directions are both $N$; (b) indicates the measurement data in monostatic configuration.
  • Figure 1.2: Monostatic measurement configuration. The dots on the big circle describes the directions of a moving transducer which is assumed to be infinitely far from the target $\Omega$. The direction $\mathbf{d}$ of the plane wave is always the opposite direction of the direction $\hat{\mathbf{x}}$ of the receiver.
  • Figure 5.1: We illustrate the initial shapes for the MCMC sampling described in Figure \ref{['fig:compare:disknellipse']}.
  • Figure 5.2: Result for 73 Fourier basis functions (up) and 72 new basis functions (down). On the left and middle, we illustrate the target $\partial\Omega_j$ (filled) for $j=1$ (left) and $j=2$ (middle) and the reconstruction (dashed) by taking mean of the last $1000$ accepted coefficients. On the right, we plot the graph of $d_J(\Omega^{(nm)},\Omega_2)$ against the iteration number $m$.
  • Figure 5.3: Graph of the index function in MSM Kang:2022:MSM for the target domain ${\Omega}_3$ using the monostatic data with noise of SNR values $\infty$, $20$dB and $5$dB from left to right.
  • ...and 3 more figures

Theorems & Definitions (4)

  • Theorem 3.1
  • proof
  • Lemma A.1
  • Theorem A.2