Mathematical and experimental validation of the bifocusing method tailored for bistatic measurement

Technical Summary: Mathematical and Experimental Validation of the Bifocusing Method for Bistatic Measurement

Problem Statement
The paper addresses the inverse scattering problem, specifically the rapid identification of small, penetrable dielectric inhomogeneities within a two-dimensional bistatic measurement configuration. While bistatic setups (where the transmitter and receiver are separated by a fixed angle) offer advantages such as reduced signal interference and richer scattering data compared to monostatic systems, efficient imaging algorithms tailored to these setups remain a challenge. Existing quantitative iterative methods (e.g., Newton-type schemes) are computationally expensive and sensitive to initial guesses, while existing non-iterative qualitative methods have not been rigorously analyzed for single-frequency bistatic applications. A critical gap exists in understanding how the bistatic angle ($\alpha$) intrinsically affects imaging resolution and detection capability.

Methodology
The authors propose a Bifocusing Method (BFM) based imaging strategy. The core of the methodology involves:

1. Problem Formulation: Defining a 2D direct scattering problem where small dielectric inclusions are embedded in a homogeneous background. The scattered field is approximated using an integral equation involving the product of Green's functions for the transmitter and receiver.
2. Indicator Function Construction: A bifocusing-based indicator function, $F_{BFM}(x, \alpha)$, is constructed by correlating the measured scattered field data with a test function derived from the product of Green's functions at a test point $x$.
3. Theoretical Derivation: The authors derive a closed-form mathematical structure for the indicator function. By utilizing the asymptotic expansion of Hankel functions and the Jacobi-Anger expansion, they express the indicator function as an infinite series of Bessel functions ($J_q$). This derivation explicitly links the function's behavior to the material properties of the inhomogeneities and the bistatic angle $\alpha$.
4. Numerical Validation: The theoretical findings are validated using the Fresnel experimental dataset. Simulations include single and multiple dielectric circular inhomogeneities, as well as a metallic rectangular object. The data includes 20 dB white Gaussian noise, and imaging results are generated for various bistatic angles ($\alpha$) and operating frequencies.

Key Contributions and Theoretical Findings
The primary contribution of the paper is the rigorous mathematical characterization of the BFM indicator function in a bistatic setting. The derived structure reveals the following properties:

• Dependence on Bistatic Angle: The imaging resolution is intrinsically dependent on $\alpha$. The term governing the primary peak, involving $J_0(k(1+\cos\alpha)|x-z|)J_0(k\sin\alpha|x-z|)$, dictates that resolution is maximized when $\alpha$ approaches $0^\circ$ (quasi-monostatic).
• Degradation at Large Angles: As $\alpha$ approaches $180^\circ$, the resolution degrades significantly.
• Singularity at $180^\circ$: The authors prove that when $\alpha = 180^\circ$, the indicator function becomes independent of the existence of the inhomogeneities ($1+\cos(180^\circ)=0$ and $\sin(180^\circ)=0$), rendering target identification impossible.
• Artifact Analysis: The study identifies a secondary term in the series expansion involving even-order Bessel functions ($J_{2q}$). While this term does not contribute to the primary peak at the target location, it generates artifacts that can disturb recognition, particularly in multi-target scenarios.

Numerical Results
Simulations using the Fresnel dataset confirm the theoretical predictions:

• Single Targets: For a single dielectric inclusion, high-resolution imaging is achieved at $\alpha = 60^\circ$ and $90^\circ$. Resolution visibly degrades as $\alpha$ increases to $120^\circ$ and $135^\circ$, and the target becomes unrecognizable at $\alpha = 180^\circ$.
• Multiple Targets: The method faces challenges with multiple targets. While single targets are well-resolved, distinguishing two closely spaced inhomogeneities becomes ambiguous at certain angles (e.g., $90^\circ$ and $120^\circ$) due to artifact interference, though clear identification was possible at $\alpha = 135^\circ$ in one specific test case.
• Metallic Objects: The indicator function successfully located a metallic rectangle, demonstrating applicability beyond the theoretical assumption of dielectric inhomogeneities, though the exact shape of the non-smooth boundary could not be fully retrieved.
• Frequency Sensitivity: Results indicate that imaging performance is also dependent on the operating frequency, with different angles yielding optimal results at different frequencies (e.g., $2$ GHz vs. $6$ GHz).

Significance and Limitations
The paper establishes that the Bifocusing Method is a viable non-iterative approach for bistatic imaging, provided the bistatic angle is carefully selected. The work provides a theoretical justification for avoiding $\alpha = 180^\circ$ and suggests that angles closer to $0^\circ$ yield superior resolution.

However, the authors modestly acknowledge limitations. The method's performance degrades in the presence of multiple targets compared to single targets, often resulting in ambiguous distinctions between objects. The paper concludes that while the proposed indicator function is effective for rapid identification, further improvements are necessary to handle multi-target scenarios robustly. Additionally, the extension of this work to three-dimensional problems is identified as a promising direction for future research.