Inversion of limited-aperture Fresnel experimental data using orthogonality sampling method with single and multiple sources

TL;DR

This work analyzes the Orthogonality Sampling Method (OSM) for limited-aperture inverse scattering of small objects, first deriving a precise structure for the single-source indicator in terms of $J_0(k_b|r'-r|)$ plus an infinite Bessel-series and revealing strong dependence on emitter location and frequency. To address these limitations, the authors design a multi-source indicator (MOSM) whose imaging functional involves squared Bessel terms, yielding reduced artifacts and (under suitable conditions) unique detection. They provide theoretical expressions and asymptotic justifications, and validate them with 2D Fresnel experimental data, showing improved performance over the single-source approach across frequencies and emitter configurations. The results offer a fast, non-iterative imaging framework for small objects in practical limited-aperture settings and establish guidance for deploying multi-source configurations in applications such as biomedical imaging and sensing.

Abstract

In this study, we consider the application of orthogonality sampling method (OSM) with single and multiple sources for a fast identification of small objects in limited-aperture inverse scattering problem. We first apply the OSM with single source and show that the indicator function with single source can be expressed by the Bessel function of order zero of the first kind, infinite series of Bessel function of nonzero integer order of the first kind, range of signal receiver, and the location of emitter. Based on this result, we explain that the objects can be identified through the OSM with single source but the identification is significantly influenced by the location of source and applied frequency. For a successful improvement, we then consider the OSM with multiple sources. Based on the identified structure of the OSM with single source, we design an indicator function of the OSM with multiple sources and show that it can be expressed by the square of the Bessel function of order zero of the first kind an infinite series of the square of Bessel function of nonzero integer order of the first kind. Based on the theoretical results, we explain that the objects can be identified uniquely through the designed OSM. Several numerical experiments with experimental data provided by the Institute Fresnel demonstrate the pros and cons of the OSM with single source and how the designed OSM with multiple sources behave.

Figures (11)

• Figure 1: Illustration of measurement configuration corresponding to the location of emitter.
• Figure 2: Plots of $|J_0(k_{\mathrm{b}} x)|$ and $\mathcal{D}_1(x)$ for $-1\leq x\leq1$ and $f=2GHz$.
• Figure 3: Illustration of antenna arrangement with transmitters $\mathcal{A}_1$, $\mathcal{A}_{10}$, and $\mathcal{A}_{25}$.
• Figure 4: Maps of $\mathfrak{F}_{\mathop{\mathrm{OSM}}\limits}(\mathbf{r}',\mathbf{a}_1)$. White colored circles describes the boundary of objects.
• Figure 5: Maps of $\mathfrak{F}_{\mathop{\mathrm{OSM}}\limits}(\mathbf{r}',\mathbf{a}_{10})$. White colored circles describes the boundary of objects.

Theorems & Definitions (12)

• Lemma 2.1: Asymptotic expansion formula AK2CV
• Theorem 3.1
• proof
• Remark 3.1: Availability and limitation of object detection
• Remark 3.2: Detection of multiple objects
• Remark 3.3: Effect of the factor $\mathcal{E}(\mathbf{r}',\mathbf{r},\mathbf{a}_m)$
• Theorem 5.1
• proof
• Remark 5.1: Availability of object detection
• Remark 5.2: Effect of the factor $\mathcal{M}(\mathbf{r}',\mathbf{r})$