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On the application of subspace migration from scattering matrix with constant-valued diagonal elements in microwave imaging

Won-Kwang Park

TL;DR

This work extends subspace migration for microwave imaging by fixing the diagonal elements of the scattering matrix to a constant $C$, enabling imaging when diagonal data are unavailable. The imaging function is shown to decompose into a Bessel-function-based expression that encodes antenna count, geometry, frequency, and material contrast, with the constant $C$ controlling artifacts. A key contribution is a theoretical unique-identification result for small 2D objects under conditions like even $N>8$, uniform circular antenna placement, and low $|C|$, supported by synthetic and experimental data. The findings provide practical guidance for robust SM in realistic measurement scenarios and highlight the importance of small diagonal data in achieving reliable localization. Limitations to 2D and small objects motivate future work toward three-dimensional imaging and large-object identification.

Abstract

We consider the application of a subspace migration (SM) algorithm to quickly identify small objects in microwave imaging. In various problems, it is easy to measure the diagonal elements of the scattering matrix if the location of the transmitter and the receiver is the same. To address this issue, several studies have been conducted by setting the diagonal elements to zero. In this paper, we generalize the imaging problem by setting diagonal elements of the scattering matrix as a constant with the application of SM. To show the applicability of SM and its dependence on the constant, we show that the imaging function of SM can be represented in terms of an infinite series of the Bessel functions of integer order, antenna number and arrangement, and applied constant. This result enables us to discover some further properties, including the unique determination of objects. We also demonstrated simulation results with synthetic data to support the theoretical result.

On the application of subspace migration from scattering matrix with constant-valued diagonal elements in microwave imaging

TL;DR

This work extends subspace migration for microwave imaging by fixing the diagonal elements of the scattering matrix to a constant , enabling imaging when diagonal data are unavailable. The imaging function is shown to decompose into a Bessel-function-based expression that encodes antenna count, geometry, frequency, and material contrast, with the constant controlling artifacts. A key contribution is a theoretical unique-identification result for small 2D objects under conditions like even , uniform circular antenna placement, and low , supported by synthetic and experimental data. The findings provide practical guidance for robust SM in realistic measurement scenarios and highlight the importance of small diagonal data in achieving reliable localization. Limitations to 2D and small objects motivate future work toward three-dimensional imaging and large-object identification.

Abstract

We consider the application of a subspace migration (SM) algorithm to quickly identify small objects in microwave imaging. In various problems, it is easy to measure the diagonal elements of the scattering matrix if the location of the transmitter and the receiver is the same. To address this issue, several studies have been conducted by setting the diagonal elements to zero. In this paper, we generalize the imaging problem by setting diagonal elements of the scattering matrix as a constant with the application of SM. To show the applicability of SM and its dependence on the constant, we show that the imaging function of SM can be represented in terms of an infinite series of the Bessel functions of integer order, antenna number and arrangement, and applied constant. This result enables us to discover some further properties, including the unique determination of objects. We also demonstrated simulation results with synthetic data to support the theoretical result.
Paper Structure (7 sections, 2 theorems, 35 equations, 12 figures, 1 table)

This paper contains 7 sections, 2 theorems, 35 equations, 12 figures, 1 table.

Table of Contents

  1. Introduction
  2. Scattering Parameters and Imaging Function of the SM
  3. Analysis of the Imaging Function
  4. Simulation Results with Synthetic and Experimental Data and Discussions
  5. Simulation Results with Synthetic Data
  6. Simulation Results with Experimental Data
  7. Conclusion

Key Result

Theorem 3.1

Let $\boldsymbol{\theta}_n=\mathbf{a}_n/R=(\cos\theta_n,\sin\theta_n)$, $\mathbf{z}=|\mathbf{z}|(\cos\varphi,\sin\varphi)$, and $\mathbf{z}-\mathbf{x}=|\mathbf{z}-\mathbf{x}|(\cos\phi,\sin\phi)$. If $|k(\mathbf{z}-\mathbf{a}_n)|\gg0.25$ for $\mathbf{z}\in\Omega$, then $\mathfrak{F}(\mathbf{z},C)$ ca where Here, $\mathbb{Z}_0^*=\mathbb{Z}\cup\left\{-\infty,+\infty\right\}\backslash\left\{0\right\}

Figures (12)

  • Figure 1: Illustration of the background (left), single (center) and multiple (right) small objects.
  • Figure 2: (Example \ref{['ex1']}) Distribution of the singular values of $\mathbb{K}(C)$ at $f=1GHz$ with various $C$.
  • Figure 3: (Example \ref{['ex1']}) Maps of $\mathfrak{F}(\mathbf{z},C)$ at $f=1GHz$ with various $C$. Black-colored dashed line describes the $\partial D_1$.
  • Figure 4: (Example \ref{['ex2']}) Distribution of the singular values of $\mathbb{K}(C)$ at $f=1GHz$ with various $C$.
  • Figure 5: (Example \ref{['ex2']}) Maps of $\mathfrak{F}(\mathbf{z},C)$ at $f=1GHz$ with various $C$. Black-colored dashed lines describe the $\partial D_1\cup\partial D_2$.
  • ...and 7 more figures

Theorems & Definitions (12)

  • Theorem 3.1
  • proof
  • Remark 3.1: Detectability of the object
  • Remark 3.2: Impact of the constant $C$ and its best selection
  • Remark 3.3: Ideal conditions to obtain good results
  • Remark 3.4: Practical condition to obtain good results
  • Corollary 3.1: Unique identification of object
  • Example 4.1: Imaging of single object
  • Example 4.2: Imaging of multiple objects
  • Example 4.3: Imaging of large object
  • ...and 2 more