Deep Bayesian Optimization of Sparse Aperture for Compressed Sensing 3D ISAR Imaging

Highlights

What are the main findings?

• A compressed-sensing 3D ISAR framework couples an adaptive-sparsity OMP reconstructor with a physically realizable, single-direction “I-shaped” sparse aperture found by deep Bayesian optimization to minimize sensing-matrix coherence and suppress sidelobes.
• In chamber tests, the optimized aperture (174 angle samples) achieved low mutual coherence and high correlation to full-aperture references (μ ≈ 0.98), closely matching random sampling while preserving continuous, turntable-feasible motion.

What is the implication of the main finding?

• High-fidelity 3D ISAR imaging can be obtained with substantially fewer measurements, enabling shorter scan times and higher throughput in compact-range metrology without sacrificing dynamic range.
• The optimization principle generalizes to other aperture families and modalities (e.g., polarimetric or multistatic), offering a template for sparsity-aware, mechanically executable sampling policies in practical radar measurement systems.

Abstract

High-resolution three-dimensional (3D) Inverse Synthetic Aperture Radar (ISAR) imaging is essential for the characterization of target scattering in various environments. The practical application of this technique is frequently impeded by the lengthy measurement time necessary for comprehensive data acquisition with turntable-based systems. Sub-sampling the aperture can decrease acquisition time; however, traditional reconstruction algorithms that utilize matched filtering exhibit significantly impaired imaging performance, often characterized by a high peak side-lobe ratio. A methodology is proposed that integrates compressed sensing(CS) theory with sparse-aperture optimization to achieve high-fidelity 3D imaging from sparsely sampled data. An optimized sparse sampling aperture is introduced to systematically balance the engineering requirement for efficient, continuous turntable motion with the low mutual coherence desired for the CS matrix. A deep Bayesian optimization framework was developed to automatically identify physically realizable optimal sampling trajectories, ensuring that the sensing matrix retains the necessary properties for accurate signal recovery. This method effectively addresses the high-sidelobe problem associated with traditional sparse techniques, significantly decreasing measurement duration while maintaining image quality. Quantitative experimental results indicate the method’s efficacy: the optimized sparse aperture decreases the number of angular sampling points by roughly 84% compared to a full acquisition, while reconstructing images with a high correlation coefficient of 0.98 to the fully sampled reference. The methodology provides an effective solution for rapid, high-performance 3D ISAR imaging, achieving an optimal balance between data acquisition efficiency and reconstruction fidelity.

1. Introduction

High-resolution three-dimensional (3D) radar imaging serves as a crucial instrument for the non-invasive characterization and diagnostic evaluation of a target’s scattering properties [1,2]. In critical domains such as aerospace and defense, these techniques are vital for quantitatively assessing the Radar Cross Section (RCS), validating the in situ performance of antennas, and diagnosing intricate electromagnetic scattering phenomena on high-value targets, including aircraft, missiles, and satellites [3,4]. Measurements are typically conducted within the electromagnetically controlled environment of a Compact Antenna Test Range (CATR) to ensure high precision and repeatability. A CATR simulates far-field plane-wave illumination, facilitating precise characterization in laboratory settings [5]. A common approach for producing 3D images in this context entails positioning the target on a high-precision multi-axis turntable and collecting coherent backscattered data across a two-dimensional grid of azimuth and elevation angles, thus creating a synthetic aperture [6,7].

Although accurate, a major operational challenge in turntable-based 3D Inverse Synthetic Aperture Radar (ISAR) imaging is the lengthy measurement time necessary for a complete full-aperture scan. The cross-range resolution of the reconstructed image is directly proportional to the angular extent of the synthetic aperture. Thus, attaining high resolution in both azimuth and elevation dimensions requires precise angular sampling across a broad solid angle [8,9]. The mechanical rotation of the turntable is inherently slow, resulting in the acquisition of this dense dataset being a time-consuming bottleneck. This limitation significantly restricts measurement throughput, impedes the swift iterative design and testing of contemporary radar targets, and elevates operational costs [10].

Various strategies have been investigated to reduce data acquisition time, with the most straightforward method being the implementation of a sparse or sub-sampled aperture. Numerous sparse sampling schemes have been documented, including early Interferometric ISAR (IF-ISAR) configurations [11] and more organized patterns like “1-loop” and “2-loop” apertures [12,13]. Although these methods effectively decrease the number of measurement points, they present a significant new issue when analyzed using traditional imaging algorithms, which are generally founded on Matched Filtering (MF). The MF algorithm is predicated on the assumption of a fully and densely sampled aperture to ensure optimal imaging performance. Violation of this condition due to sparse sampling leads to degradation of the algorithm’s point spread function, resulting in reconstructed images characterized by severe artifacts, elevated side-lobes, and a notably diminished imaging dynamic range. The low quality renders it impossible to differentiate between weak yet significant scattering centers and algorithm-induced clutter, thus compromising the reliability of the measurement for any meaningful diagnostic application.

The theory of Compressed Sensing (CS) has developed into a significant framework that addresses the fundamental trade-off between measurement time and image quality [14,15]. The fundamental principle of compressed sensing is that a signal, which is sparse in a particular domain, can be accurately reconstructed from a limited number of incoherent, sub-Nyquist measurements through the application of convex optimization techniques [16]. This framework is particularly effective for ISAR imaging, as the radar signatures of most man-made targets exhibit natural sparsity in the spatial domain, characterized by a limited number of strong, localized scattering centers [17,18]. Reformulating the imaging task as a sparse recovery problem allows CS-based methods to reconstruct a high-fidelity, artifact-free image from a highly incomplete dataset, thereby fundamentally circumventing the side-lobe issues inherent in traditional MF approaches [19,20].

The effective implementation of Compressed Sensing (CS) relies heavily on the characteristics of the measurement matrix, which must fulfill specific criteria, including the Restricted Isometry Property (RIP) [16]. This involves minimizing the mutual coherence among the columns of the sensing matrix, thereby ensuring the unique and stable recovery of the sparse signal [17]. Research indicates that purely random sampling schemes are theoretically optimal for generating a sensing matrix characterized by low mutual coherence [21,22]. A random sampling path is not operationally feasible for a turntable-based system. The required discontinuous and random movements would compel the turntable to cover the full angular range, resulting in no actual decrease in total measurement time. This highlights a distinct research gap: the necessity for a structured, continuous, and time-efficient sampling path that can be physically implemented using a turntable system, while also producing a sensing matrix with the low-coherence characteristics essential for high-quality compressed sensing reconstruction. This paper presents a comprehensive methodology for achieving efficient and high-fidelity 3D ISAR imaging. This work presents the following primary contributions:

• We present and implement a compressed sensing imaging framework, employing an adaptive-sparsity Orthogonal Matching Pursuit (ASOMP) method, to reconstruct high-dynamic-range 3D pictures from sparsely sampled turntable data. This approach significantly mitigates the side-lobe artifacts that afflict conventional approaches, facilitating unambiguous visualization of target scattering centers.
• We introduce a new aperture optimization method that creates a continuous, efficient sampling path in a methodical way. To ensure strong CS reconstruction and go beyond heuristic-based designs, our technique relies on a Deep Bayesian Optimization framework that finds an aperture structure with a matching sensing matrix that shows low mutual coherence.
• We validate the proposed methodology through a practical experiment involving a physical target. The results indicate that our optimized aperture effectively reduces measurement time while preserving high imaging fidelity, as validated by quantitative correlation analysis with truth data, and it demonstrates the capacity to approximate the performance of theoretically optimal yet physically unattainable sampling schemes.

This paper is organized as follows. Section 2 details the methodology, including the signal model for 3D ISAR, the principles of both MF and CS-based imaging, and the proposed sparse aperture optimization scheme. Section 3 describes the experimental setup, including the test facility, instrumentation, and test target. Section 4 presents the simulation and experimental results, providing a comparative analysis of different aperture schemes. Section 5 provides an in-depth discussion of the results and the method’s performance. Finally, Section 6 concludes the paper.

2. Methods

2.1. Signal Model for Turntable 3D ISAR Imaging

A typical 3D ISAR imaging system is presented in Figure 1. The object under test (OUT) is positioned at the center of the rotary stage. The coordinate systems of the target and the radar is denoted as $x-y-z$ and $u-v-n$ respectively. As the target rotates, coordinate system of the target aligns with the movement in $x-y-z$ and the radar coordinate system $u-v-n$ remains static. The direction of antenna observation is aligned with the $v$-axis. In an elevation-over-azimuth positioning system, the positive elevation angle $\phi$ is defined as counterclockwise around the $n$-axis, and the positive azimuth $\theta$ is counterclockwise around the $x$ axis.

The transform between the two coordinate systems is as follows:

$$\left[\begin{array}{c}u\\ v\\ n\end{array}\right]=\left[\begin{array}{ccc}\mathit{cos}\phi & \mathit{sin}\phi & 0\\ -\mathit{sin}\phi \mathit{cos}\theta & \mathit{cos}\phi \mathit{cos}\theta & \mathit{sin}\theta \\ \mathit{sin}\phi \mathit{sin}\theta & -\mathit{cos}\phi \mathit{sin}\theta & \mathit{cos}\theta \end{array}\right]\left[\begin{array}{c}x\\ y\\ z\end{array}\right]$$

(1)

According to the transform described above, the corresponding $v$-coordinate of a point located at ($x,y,z$) is as follows:

$$v(x,y,z,\theta ,\phi )=x\mathrm{s}\mathrm{i}\mathrm{n} \theta +y\mathrm{c}\mathrm{o}\mathrm{s} \theta \mathrm{c}\mathrm{o}\mathrm{s} \phi -z\mathrm{c}\mathrm{o}\mathrm{s} \theta \mathrm{s}\mathrm{i}\mathrm{n} \phi$$

(2)

The network analyzer transmits a stepped-frequency wave given by $e^{j2\pi ft}$. The return from a scattering point at range $R$ is expressed as $s=g\cdot e^{j2\pi f(t-2R/c)}$. Here, $s$ represents the return signal and $g$ is the RCS density. According to the definition of RCS, $\sigma \propto g^2$. When $t=0$, the return signal simplifies to $s=g\cdot e^{-j2\pi f\cdot 2R/c}$.

For any point located at $\left(x_0,y_0,z_0\right)$ in the target field. When the frequency equals $f$, the azimuth equals $\theta$ and the elevation equals $\phi$, the echo $s_0(f,\phi ,\theta )$ is given by

$$s_D(f,\phi ,\theta )=\sum _{k=1}^{N_z}\sum _{j=1}^{N_y}\sum _{i=1}^{N_x}g\left(x_i,y_j,z_k\right)\cdot e^{-j4\pi fR\left(x_i,y_j,z_k,\phi ,\theta \right)/c}$$

(3)

where $R(\cdot )=v(\cdot )+R_0,R_0$ denotes the distance between antenna and the origin of the coordinate. The volume integral in Equation (3) from the entire target field.

$$s(f,\phi ,\theta )={\iiint }_{\mathsf{\Omega }}g(x,y,z)\cdot e^{-j4\pi fR(x,y,z,\phi ,\theta )/c}dxdydz$$

(4)

By dividing the target field into $N_x\times N_y\times N_z$ grids, Equation (4) can expressed in discrete form as

$$\left[\begin{array}{c}s(f_1,{\phi }_1,{\theta }_1)\\ s(f_2,{\phi }_1,{\theta }_1)\\ ⋮\\ s(f_{N_f},{\phi }_1,{\theta }_1)\\ ⋮\\ s(f_1,{\phi }_2,{\theta }_1)\\ ⋮\\ s(f_{N_f},{\phi }_{N_{\phi }},{\theta }_1)\\ ⋮\\ s(f_{N_f},{\phi }_{N_{\phi }},{\theta }_{N_{\theta }})\end{array}\right]=\left[\begin{array}{ccc}{e}^{-j4\pi f_{1}R(x_1,y_1,z_1,{\phi }_1,{\theta }_1)/c}& \dots & e^{-j4\pi f_{1}R(x_{N_x},y_{N_y},z_{N_z},{\phi }_1,{\theta }_1)/c}\\ e^{-j4\pi f_{2}R(x_1,y_1,z_1,{\phi }_1,{\theta }_1)/c}& \dots & e^{-j4\pi f_{2}R(x_{N_x},y_{N_y},z_{N_z},{\phi }_1,{\theta }_1)/c}\\ ⋮& \ddots & ⋮\\ e^{-j4\pi f_{N_f}R(x_1,y_1,z_1,{\phi }_1,{\theta }_1)/c}& \dots & e^{-j4\pi f_{N_f}R(x_{N_x},y_{N_y},z_{N_z},{\phi }_1,{\theta }_1)/c}\\ ⋮& \ddots & ⋮\\ e^{-j4\pi f_{1}R(x_1,y_1,z_1,{\phi }_2,{\theta }_1)/c}& \dots & e^{-j4\pi f_{1}R(x_{N_x},y_{N_y},z_{N_z},{\phi }_2,{\theta }_1)/c}\\ ⋮& \ddots & ⋮\\ e^{-j4\pi f_{N_f}R(x_1,y_1,z_1,{\phi }_{N_{\phi }},{\theta }_1)/c}& \dots & e^{-j4\pi f_{N_f}R(x_{N_x},y_{N_y},z_{N_z},{\phi }_{N_{\phi }},{\theta }_1)/c}\\ ⋮& \ddots & ⋮\\ e^{-j4\pi f_{N_f}R(x_1,y_1,z_1,{\phi }_{N_{\phi }},{\theta }_{N_{\theta }})/c}& \dots & e^{-j4\pi f_{N_f}R(x_{N_x},y_{N_y},z_{N_z},{\phi }_{N_{\phi }},{\theta }_{N_{\theta }})/c}\end{array}\right]\left[\begin{array}{c}g(x_1,y_1,z_1)\\ g(x_2,y_1,z_1)\\ ⋮\\ g(x_{N_x},y_1,z_1)\\ ⋮\\ g(x_1,y_2,z_1)\\ ⋮\\ g(x_{N_x},y_{N_y},z_1)\\ ⋮\\ g(x_{N_x},y_{N_y},z_{N_z})\end{array}\right]$$

(5)

When sampling at several discrete frequencies, azimuths, and elevations, Equation (5) can be expressed in matrix form as follows:

$${\mathbf{s}}_{M\times 1}={\mathsf{\Phi }}_{M\times N}\cdot {\mathbf{g}}_{N\times 1}$$

(6)

where ${\mathbf{s}}_{M\times 1}$ represents the echo from target field, $M$ is the product of frequency sampling number and total number of sampling angles in angular space $(\phi ,\theta )$. $\mathsf{\Phi }$ is the sampling matrix. ${\mathbf{g}}_{N\times 1}$ represents the target field, where $N$ = $N_x\times N_y\times N_z$.

Considering the noise ${\mathbf{n}}_{M\times 1}$, Equation (6) then can be written as

$${\mathbf{s}}_{M\times 1}={\mathsf{\Phi }}_{M\times N}\cdot {\mathbf{g}}_{N\times 1}+{\mathbf{n}}_{M\times 1}$$

(7)

2.2. Matched Filtering Reconstruction

The reconstruction of the target field $\mathbf{g}$, utilizes traditional matched filtering methods that rely on uniform minimum variance unbiased estimation (UMVUE). The solution for UMVUE is given by:

$$\stackrel{\mathbf{ˆ}}{\mathbf{g}}={\left({\mathsf{\Phi }}^H{\mathbf{C}}^{-1}\mathsf{\Phi }\right)}^{-1}{\mathsf{\Phi }}^H{\mathbf{C}}^{-1}\mathbf{s}$$

(8)

where $\stackrel{\mathbf{ˆ}}{\mathbf{g}}$ represent the estimation of $\mathbf{g},\mathbf{C}$ is the covariance matrix of noise $\mathbf{n}$. When $\mathbf{n}$ is white noise, $\mathbf{C}=E\left[{\mathbf{n}\mathbf{n}}^H\right]=$ ${\sigma }^2\mathbf{I}$. Where $\sigma$ is the noise variance and $\mathbf{I}$ represents identity matrix. Then Equation (8) can be transformed to

$$\stackrel{\mathbf{ˆ}}{\mathbf{g}}={\left({\mathsf{\Phi }}^H{\mathbf{C}}^{-1}\mathsf{\Phi }\right)}^{-1}{\mathsf{\Phi }}^H{\mathbf{C}}^{-1}\mathbf{s}$$

(9)

In most radar imaging scenarios, the term ${\left({\mathsf{\Phi }}^H\mathsf{\Phi }\right)}^{-1}$ remains unsolved, resulting in the direct computation of Equation (9) being resource-intensive. For the purpose of simplifying the computation, it is common to assume that ${\mathsf{\Phi }}^H\mathsf{\Phi }$ is approximately equal to $\mathbf{I}$ [23]. This mathematical condition possesses a distinct physical interpretation: it is valid when the system’s point spread function (PSF) exhibits sharpness and impulsiveness, defined by a narrow mainlobe and minimal sidelobes. A desirable point spread function (PSF) is generally attained through a fully-sampled aperture that meets the Nyquist sampling criterion. The preservation of orthogonality in a sparse aperture is crucial, as it directly influences the quality of matched filter reconstruction. A loss of orthogonality results in elevated sidelobes and various imaging artifacts. Consequently, the optimization problem discussed in this paper can be interpreted as a quest for the sparse aperture configuration that most effectively preserves this orthogonality property. Based on this assumption, Equation (9) can be approximated as

$${\stackrel{\mathbf{ˆ}}{\mathbf{g}}}_{mf}=\left({\mathsf{\Phi }}^H\mathsf{\Phi }\right)\stackrel{\mathbf{ˆ}}{\mathbf{g}}={\mathsf{\Phi }}^H\mathbf{s}$$

(10)

Here, $\stackrel{\mathbf{ˆ}}{\mathbf{g}}$ represents best estimation result of $\mathbf{g}$ and ${\stackrel{\mathbf{ˆ}}{\mathbf{g}}}_{mf}$ is an approximation, where $mf$ means matched filtering. It is equivalent to multiply a matrix ${\mathsf{\Phi }}^H\mathsf{\Phi }$ on the left of $\stackrel{\mathbf{ˆ}}{\mathbf{g}}$. Indeed, ${\mathsf{\Phi }}^H\mathsf{\Phi }$ is a matrix exhibiting a sinc response [23]. The computational outcome of the matched filtering method is analogous to the convolution of the system response with $\stackrel{\mathrm{ˆ}}{\mathrm{g}}$. This explains the presence of major lobes and side lobes in matched filtering imaging. The elaboration of Equation (10) constitutes the back projection (BP) algorithm [24].

2.3. Compressed Sensing Reconstruction and Incoherent Sampling

Equation (6) delineates a linear sampling model. Where $\mathbf{g}$ is an unspecified natural signal. Typically, the majority of natural signals exhibit sparse representation when subjected to a particular transformation basis, hence

$$\mathbf{g}={\mathsf{\Psi }}_{\mathbf{x}}$$

(11)

where $\mathbf{x}$ is the sparse representation of $\mathbf{g}$ under the base $\mathsf{\Psi }$.

Substitute Equation (11) into Equation (7)

$$\mathbf{s}=\mathsf{\Phi }\mathsf{\Psi }\mathbf{x}+\mathbf{n}=\mathbf{A}\mathbf{x}+\mathbf{n}$$

(12)

where $\mathbf{A}=\mathsf{\Phi }\mathsf{\Psi }$ is the sensing matrix.

For ISAR images, the target generally comprises a finite number of scattering centers, signifying pronounced sparsity. In this instance, signal ${\mathbf{g}}_{N\times 1}$ no longer requires sparse representation, the sparse representation matrix $\mathsf{\Psi }$ transforms into the identity matrix, therefore $\mathbf{g}=\mathsf{\Psi }\mathbf{x}=\mathbf{I}\mathbf{x}=\mathbf{x}$ and $\mathbf{A}=\mathsf{\Phi }$. Equation (12) is expressed as:

$$\mathbf{s}=\mathbf{A}\mathbf{g}+\mathbf{n}$$

(13)

Assume that there are $K$nonzero values in ${\mathbf{g}}_{N\times 1}$, where $K\ll N$, making g a K-sparse signal.

The theory of compressed sensing posits that if $\mathbf{g}$ is sparse and the sensing matrix $\mathbf{A}$ meets the low mutual coherence criterion, the estimation of $\mathbf{g}$ can be formulated as a ${\mathcal{l}}_1$-norm optimization problem [25].

$${\stackrel{\mathbf{ˆ}}{\mathbf{g}}}_{cs}=\underset{\mathbf{g}}{\mathrm{m}\mathrm{i}\mathrm{n}}\Vert \mathbf{g}{\Vert }_1, \mathrm{s}.\mathrm{t}. \Vert \mathbf{s}-\mathbf{A}\mathbf{g}{\Vert }_2^2<\epsilon$$

(14)

where ${\stackrel{\mathbf{ˆ}}{\mathbf{g}}}_{cs}$ is the estimate of $\mathbf{g}$, and $\epsilon$ depends on noise level.

A sensing matrix $\mathbf{A}$ with low mutual coherence guarantees precise reconstruction. The definition of mutual coherence is as follows:

$$\rho =\underset{i\ne j}{\mathrm{m}\mathrm{a}\mathrm{x}}{\displaystyle \frac{|⟨\mathbf{A}(i),\mathbf{A}(j)⟩|}{\Vert \mathbf{A}\left(i\right){\Vert }_2\Vert \mathbf{A}\left(j\right){\Vert }_2}}$$

(15)

where $⟨\mathbf{A}\left(i\right),\mathbf{A}\left(j\right)⟩$ refers to the inner product of the $i$ column and the $j$ column of sensing matrix A.

In turntable 3D ISAR imaging, as described by Equations (7) and (15), the mutual coherence $\rho$ is a function of the variables $(x,y,z,f,\phi ,\theta )$. Let it be assumed that the target field ($x,y,z$) and wide frequency band signal $f$ have been previously defined. Thus, $\rho$ is solely determined by parameters ($\phi ,\theta$), indicating that $\rho$ is exclusively influenced by the sampling aperture.

To solve the ${\mathcal{l}}_1$-norm optimization problem in Equation (13), this paper employs an adaptive-sparsity orthogonal matching pursuit (ASOMP) algorithm [26].

In conventional OMP, the sparsity level $K$ must be predetermined; however, in practical imaging scenarios, $K$ is seldom known and heuristic selections (e.g., $K\approx 1\%\times N$) are susceptible to over-or under-fitting based on aperture coverage and SNR. The implementation maintains the standard OMP selection and least-squares update, while incorporating an adaptive, energy-based termination rule that deduces effective sparsity from the evolution of the residuals. The reconstruction persists in focusing on Equation (14), while the sensing design is guided by the mutual-coherence aspect outlined in Equation (15).

We first normalize the columns of the sensing matrix to equalize atom magnitudes. Let $s\in {\mathbb{C}}^M$ denote the measurement vector and $g\in {\mathbb{C}}^N$ the unknown sparse reflectivity. At iteration $k$, OMP maintains a support set ${\mathcal{S}}_k\subseteq \{1,\dots ,N\}$ and computes the restricted least-squares estimate

$${\hat{g}}_k=\mathrm{a}\mathrm{r}\mathrm{g}\underset{\theta \in {\mathbb{C}}^{\left|{\mathcal{S}}_k\right|}}{\mathrm{m}\mathrm{i}\mathrm{n}} \Vert s-A_{{\mathcal{S}}_k}\theta {\Vert }_2$$

(16)

where $A_{{\mathcal{S}}_k}\in {\mathbb{C}}^{M\times \left|{\mathcal{S}}_k\right|}$ is the submatrix formed by the columns of $A$ indexed by ${\mathcal{S}}_k$. The residual is updated as

$$r_k\triangleq s-A{\hat{g}}_k$$

(17)

where $r_k\in {\mathbb{C}}^M$ is the residual at iteration $k$; $s$ and $A$ are as defined above; and ${\hat{g}}_k$ is the least-squares estimate at iteration $k$.

The adaptive termination evaluates both the normalized residual energy and its reduction per iteration. Reconstruction ceases when the residual energy is less than a specified fraction of the signal energy.

$${\displaystyle \frac{\Vert r_k{\Vert }_2^2}{\Vert s{\Vert }_2^2}}<{\eta }_{\mathrm{energy}}$$

(18)

where ${\eta }_{\mathrm{energy}}\in (\mathrm{0,1})$ is a dimensionless threshold; $\Vert r_k{\Vert }_2^2$ is the squared residual energy at iteration $k$; and $\Vert s{\Vert }_2^2$ is the total signal energy, or if the relative decrease in residual energy becomes negligible,

$${\displaystyle \frac{\Vert r_{k-1}{\Vert }_2^2-\Vert r_k{\Vert }_2^2}{\Vert r_0{\Vert }_2^2}}<{\eta }_{\mathrm{decrease}}$$

(19)

where $r_0\equiv s$ is the initial residual; ${\eta }_{\mathrm{decrease}}\in (\mathrm{0,1})$ is a dimensionless threshold that limits insignificant improvements between iterations.

To promote identifiability and numerical stability of the least-squares stage, we bound the admissible sparsity,

$$K_{\mathrm{m}\mathrm{i}\mathrm{n}}=10,K_{\mathrm{m}\mathrm{a}\mathrm{x}}=\mathrm{m}\mathrm{i}\mathrm{n}\{M, \lfloor N/3\rfloor \}$$

(20)

where $K_{\mathrm{m}\mathrm{i}\mathrm{n}}$ and $K_{\mathrm{m}\mathrm{a}\mathrm{x}}$ are the minimum and maximum allowed support sizes; $M$ is the number of measurements, i.e., the number of rows of $A$; $N$ is the dictionary size, i.e., the number of columns of $A$; and $\lfloor \cdot \rfloor$ is the floor operator. The stopping rules in Equations (18) and (19) are evaluated only after $K_{\mathrm{m}\mathrm{i}\mathrm{n}}$ iterations and then at each subsequent iteration until one rule triggers; the triggering iteration defines $K_{\mathrm{opt}}=k$, and the algorithm outputs

$${\hat{g}}_{\mathrm{cs}}={\hat{g}}_{K_{\mathrm{opt}}}$$

(21)

where ${\hat{g}}_{\mathrm{cs}}\in {\mathbb{C}}^N$ is the final reconstruction and $K_{\mathrm{opt}}$ is the selected sparsity

When neither criterion triggers before the upper bound is reached, we select the model size by a light complexity-penalized residual score,

$$K_{\mathrm{opt}}=\mathrm{a}\mathrm{r}\mathrm{g}\underset{1\le k\le K_{\mathrm{m}\mathrm{a}\mathrm{x}}}{\mathrm{m}\mathrm{i}\mathrm{n}}(\Vert r_k{\Vert }_2^2+\lambda k)$$

(22)

where $\lambda >0$ is a regularization weight; $\Vert r_k{\Vert }_2^2$ is the residual energy at support size $k$; and $k$ plays the role of model order. A robust default is

$$\lambda ={\eta }_{\mathrm{energy}} {\displaystyle \frac{\Vert s{\Vert }_2^2}{K_{\mathrm{m}\mathrm{a}\mathrm{x}}}}$$

(23)

where $\Vert s{\Vert }_2^2$ is the total signal energy.

For practical operation, fixed thresholds can be replaced by data-driven ones using an estimate of the measurement noise variance. A robust median-absolute-deviation (MAD) estimator on the initial residual yields

$${\hat{\sigma }}^2={\displaystyle \frac{\mathrm{m}\mathrm{e}\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{n}\left(\right|r_0-\mathrm{m}\mathrm{e}\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{n}\left(r_0\right)|{)}^2}{(0.6745{)}^2}}$$

(24)

where ${\hat{\sigma }}^2$ is the variance estimate of the complex residual amplitude and $0.6745$ is the Gaussian MAD consistency constant, which suggests setting

$${\eta }_{\mathrm{energy}}=\mathrm{m}\mathrm{i}\mathrm{n}\{0.02, {\displaystyle \frac{{\hat{\sigma }}^2}{\Vert s{\Vert }_2^2/M}}\},{\eta }_{\mathrm{decrease}}=\mathrm{m}\mathrm{a}\mathrm{x}\{0.003, 0.5{\eta }_{\mathrm{energy}}\}$$

(25)

where the cap $0.02$ prevents overly loose stopping; $\Vert s{\Vert }_2^2/M$ normalizes the noise power per measurement; and the floor $0.003$ avoids premature termination in low-SNR scenes.

This mechanism maintains the simplicity and computational efficiency of OMP while eliminating its most fragile hyperparameter. Termination is linked to the residual-energy trajectory, leading the solver to balance fidelity and parsimony. In scenarios where the scene is simple or the aperture is sparse, the residual decays rapidly, resulting in a small $K_{\mathrm{opt}}$. Conversely, in richer scenes or those with higher SNR, additional atoms are incorporated until the residual approaches the estimated noise floor or its decline halts. The upper bound $K_{\mathrm{m}\mathrm{a}\mathrm{x}}$ enhances the regularization of the least-squares stage by maintaining a model size that is proportional to the number of measurements. The default (or MAD-adapted) thresholds demonstrate reliable performance without the need for per-dataset retuning, ensuring consistent quality–efficiency trade-offs across various aperture configurations. The adaptive OMP stage is entirely compatible with subsequent processing; for thoroughness, any post hoc consistency projection utilized for comparison is applied downstream to ${\hat{g}}_{\mathrm{cs}}$ without alteration.

2.4. Optimized Sparse Aperture Design for Low Mutual Coherence

Nevertheless, the conventional sparse aperture configurations, including “IF-ISAR,” “1-loop,” “loop-cross,” and “2-loop,” are derived from practical engineering experience and do not possess a theoretical foundation, as illustrated in Figure 2. Traditional sparse sampling apertures in the spatial frequency domain (k-space), represented by coordinates ($k_u,k_v,k_n$). The red and blue lines depict the sampling trajectories, while the arrows indicate the sequential direction of data acquisition along these paths. Mathematical calculations must be utilized in the design of the sampling aperture to improve its effectiveness.

The mathematical model for aperture design is articulated as follows: under predetermined sampling frequency and target space conditions, the objective is to minimize $\rho$ by through the selection of an appropriate sequence of $(\mathsf{\phi },\mathsf{\theta })$ combinations.

$$\left\{\left(\phi ,\theta \right)\right\}=\underset{\left\{\left(\phi ,\theta \right)\right\}\stackrel{cond}{\subset }MA}{argmin}\rho \left(\mathit{A}\right)$$

(26)

where $\left\{\left(\phi ,\theta \right)\right\}$denotes the set of combinations of sampling azimuth and elevation angles, representing the specific sampling positions. MA refers to the maximum aperture under full sampling conditions. $\left\{\left(\phi ,\theta \right)\right\}\stackrel{cond}{\subset }MA$ indicates that, under certain constraints (cond), the set $\left\{\left(\phi ,\theta \right)\right\}$ is a subset of the maximum aperture MA.

This section presents a sparse aperture optimization scheme that improves the efficacy of sparse apertures within the framework of the compressed sensing radar imaging algorithm. This study will evaluate the effectiveness of the optimized aperture in comparison to loop apertures. We compare the optimized aperture with a random sampling aperture, which is suggested by compressed sensing theory as the optimal sampling method.

This research aims to reduce measurement duration in 3D turntable ISAR imaging, while preserving an adequate imaging dynamic range. Two critical factors must be taken into account to attain that objective. The optimized sparse aperture must include an efficient sampling path to substantially decrease measurement time. The sensing matrix of the optimized sparse aperture must demonstrate low mutual coherence to ensure the reconstruction stability of the CSOMP algorithm. Based on the initial factor and application habits, we formulated the aperture optimization scheme depicted in Figure 3. Two recessed structures, designated as “CDE” and “IJK,” are constructed inwardly in accordance with the “1-loop” aperture. The design demonstrates ingenuity through the inward formation of two recessed sampling structures, which enhance aperture complexity while minimally prolonging the rotation stage’s motion duration. The optimized aperture structure excludes any repeated paths. In contrast, sampling with the “loop-cross” or “2-loop” apertures leads to repeated stage movements and greater time expenditure. The travel distance in the elevation direction for this optimized aperture is the same as that of the “1-loop” aperture; only the travel distance in the azimuth direction is larger.

Figure 3a illustrates that the sampling arrows in the spatial frequency domain form a unidirectional path, where arrows delineate a continuous, unidirectional sampling trajectory. This method demonstrates greater efficiency compared to loop-cross aperture and 2-loop aperture configurations. All arrows are oriented either horizontally or vertically, aligning with established engineering practices. The dual-indented aperture structure enhances geometric complexity, thereby potentially reducing the mutual coherence $\rho$ of the corresponding sensing matrix. As depicted in Figure 3b, the exact shape of the optimized aperture is exclusively determined by the lengths of six primary arc segments, labeled A--F. These six arcs collectively define an I-shaped trajectory comprising twelve discrete segments labeled A--L. Given that the mutual coherence $\rho$ is directly derived from the sensing matrix associated with the aperture geometry, a quantitative relationship can thus be established linking the mutual coherence $\rho$ to the specific lengths of the aforementioned six arcs.

2.5. Deep Bayesian Optimization of a Structured I-Shaped Sparse Aperture

Section 2.4 established that the reconstruction fidelity of CS-based 3D ISAR is governed by the mutual coherence $\rho \left(\mathbf{A}\right)$ of the sensing matrix $\mathbf{A}$ and that, under fixed target voxels and bandwidth, $\rho$ is determined by the angular sampling aperture $\left\{\right(\phi ,\theta \left)\right\}$. In our optimization framework, we consider the target characteristics, including spatial extent and scattering properties, along with the operational frequency band, as predetermined and fixed conditions. This assumption is practical in numerous measurement scenarios where the target class is predetermined and the equipment bandwidth is established. Therefore, with the frequency grid and target volume fixed, we design a continuous and turntable-feasible aperture by minimizing $\rho \left(\mathbf{A}\right)$ through a Bayesian deep-learning surrogate [27] that guides the exploration of a structured, I-shaped family of paths.

The selection of a deep Bayesian optimization framework is driven by its remarkable sample efficiency, which is essential for our application. The forward imaging process, which functions as our objective criterion, is computationally demanding. This renders solutions necessitating several evaluations, such as genetic algorithms or simulated annealing, unworkable for this issue. Bayesian optimization excels in these situations by constructing a probabilistic surrogate model of the goal function, enabling the intelligent selection of the most informative evaluation points, thereby significantly decreasing the necessary number of simulations.

We parameterize the I-shaped aperture by a six-dimensional integer vector

$$\mathbf{s}=(a,b,c,d,e,f)\in {\mathbb{Z}}_{+}^6$$

(27)

where $a,b,c,d,e,f$ are nonnegative integers (arc lengths on the discrete angular grid) that determine twelve connected horizontal/vertical arc segments composing the I-shaped path, and ${\mathbb{Z}}_{+}^6$ denotes the six-dimensional nonnegative integer lattice.

To guarantee feasibility on the angular grids $\{{\phi }_k{\}}_{k=0}^{N_{\phi }-1}$ (azimuth) and $\{{\theta }_{\mathcal{l}}{\}}_{\mathcal{l}=0}^{N_{\theta }-1}$ (elevation), we enforce

$$c<N_{\phi }-b,f<N_{\theta }-e$$

(28)

where $N_{\phi }$ and $N_{\theta }$ are the total numbers of azimuth and elevation samples, respectively; these constraints prevent the horizontal and vertical strokes from exceeding grid limits.

Consistent with the forward model in Section 2.1, Section 2.2 and Section 2.3, the phase for a measurement at frequency $f$ (in hertz) and angles $(\phi ,\theta )$ (in radians) for a voxel at $\mathbf{r}=(x,y,z)\in {\mathbb{R}}^3$ is

$$\psi (\phi ,\theta ,f;\mathbf{r})=-{\displaystyle \frac{4\pi f}{c}}\stackrel{\mathbf{ˆ}}{\mathbf{k}}(\phi ,\theta )\cdot \mathbf{r}$$

(29)

where $c$ is the propagation speed (m/s), $\stackrel{\mathbf{ˆ}}{\mathbf{k}}(\phi ,\theta )=[\mathrm{s}\mathrm{i}\mathrm{n} \phi , \mathrm{c}\mathrm{o}\mathrm{s} \phi \mathrm{c}\mathrm{o}\mathrm{s} \theta , -\mathrm{c}\mathrm{o}\mathrm{s} \phi \mathrm{s}\mathrm{i}\mathrm{n} \theta {]}^{\mathrm{\top }}$ is the unit look-direction vector, and “$\cdot$” denotes the Euclidean inner product.

The $(m,n)$-th entry of the sensing matrix $\mathbf{A}$ induced by $\mathbf{s}$ is then

$$A_{mn}=\mathrm{e}\mathrm{x}\mathrm{p}\left(\mathrm{i}\psi \right({\phi }_m,{\theta }_m,f_m;{\mathbf{r}}_n\left)\right)$$

(30)

where $({\phi }_m,{\theta }_m,f_m)$ is the $m$-th angle–frequency triple on the I-shaped path generated by $\mathbf{s}$, and ${\mathbf{r}}_n$ is the $n$-th voxel location.

The black-box objective associated with $\mathbf{s}$ is defined by

$$f\left(\mathbf{s}\right)=\rho \left(\mathbf{A}\right(\mathbf{s}\left)\right)$$

(31)

where $\mathbf{A}\left(\mathbf{s}\right)$ denotes the sensing matrix constructed from the angles determined by $\mathbf{s}$ together with the fixed frequency grid and voxel set, and $\rho (\cdot )$ is the mutual coherence in Equation (15).

Direct evaluation of $f\left(\mathbf{s}\right)$ is expensive because it requires assembling $\mathbf{A}\left(\mathbf{s}\right)$ and computing column-wise correlations over the full voxel and bandwidth sets. To amortize this cost while preserving global search, we adopt a deep-ensemble surrogate within a Bayesian optimization loop. This method functions as a robust and scalable substitute for conventional Gaussian Process Regression (GPR). The selection is driven by deep ensembles’ capacity to accurately represent the intricate, non-linear correlation between high-dimensional aperture parameters and the resultant mutual coherence, thereby addressing the computational difficulties encountered by conventional Gaussian Process Regressions as the number of evaluations increases. At iteration $t$, the evaluated dataset is

$${\mathcal{D}}_t=\left\{\right({\mathbf{s}}_i,y_i){\}}_{i=1}^{N_t},y_i=f({\mathbf{s}}_i)+{\epsilon }_i$$

(32)

where $N_t$ is the number of evaluated designs, $y_i$ is the measured maximum coherence, and ${\epsilon }_i\sim \mathcal{N}(0,{\sigma }_{\epsilon }^2)$ is zero-mean Gaussian observation noise with variance ${\sigma }_{\epsilon }^2$.

We train an ensemble of $M$ regressors $\left\{g_m\right(\cdot ;{\mathit{\theta }}_m){\}}_{m=1}^M$ on bootstrap resamples of ${\mathcal{D}}_t$. Each regressor maps a handcrafted feature vector $\mathbf{z}(\mathbf{s})\in {\mathbb{R}}^d$ to a scalar prediction, where $\mathbf{z}\left(\mathbf{s}\right)$ concatenates normalized arc lengths $(a,b,c,d,e,f)$, feasibility margins from Equation (18), and simple geometry ratios (dimension $d=14$ in our implementation). The posterior mean and variance surrogates are

$${\mu }_t(\mathbf{s})={\displaystyle \frac{1}{M}}\sum _{m=1}^M{g}_m(\mathbf{z}(\mathbf{s});{\mathit{\theta }}_m)$$

(33)

$${\sigma }_t^2(\mathbf{s})={\displaystyle \frac{1}{M}}\sum _{m=1}^M(g_m(\mathbf{z}(\mathbf{s});{\mathit{\theta }}_m)-{\mu }_t(\mathbf{s}){)}^2+{\sigma }_{\epsilon }^2$$

(34)

where ${\mu }_t\left(\mathbf{s}\right)$ and ${\sigma }_t^2\left(\mathbf{s}\right)$ approximate the posterior mean and variance, respectively, and ${\mathit{\theta }}_m$ denotes the parameters of the $m$-th network.

Each network is trained by minimizing an ${\mathcal{l}}_2$-regularized mean-squared error over its bootstrap subset ${\mathcal{B}}_m$,

$${\mathcal{L}}_m({\mathit{\theta }}_m)={\displaystyle \frac{1}{\left|{\mathcal{B}}_m\right|}}\sum _{({\mathbf{s}}_i,y_i)\in {\mathcal{B}}_m}(g_m(\mathbf{z}({\mathbf{s}}_i);{\mathit{\theta }}_m)-y_i{)}^2+\lambda \Vert {\mathit{\theta }}_m{\Vert }_2^2$$

(35)

where $\left|{\mathcal{B}}_m\right|$ is the subset size and $\lambda >0$ is the regularization coefficient.

Acquisition follows an Upper Confidence Bound (UCB) rule that balances exploitation and exploration:

$${\mathcal{A}}_{\mathrm{UCB}}(\mathbf{s} | {\mathcal{D}}_t)={\mu }_t(\mathbf{s})-\beta {\sigma }_t(\mathbf{s})$$

(36)

where ${\mathcal{A}}_{\mathrm{UCB}}(\cdot )$ is the acquisition value and $\beta >0$ weighs uncertainty exploration (larger $\beta$ favors exploration).

From a feasible candidate set ${\mathcal{S}}_c=\{{\mathbf{s}}^{\left(j\right)}{\}}_{j=1}^{N_c}$, we pick

$${\mathbf{s}}_t^{\star }=\mathrm{a}\mathrm{r}\mathrm{g} \underset{\mathbf{s}\in {\mathcal{S}}_c}{\mathrm{m}\mathrm{i}\mathrm{n}} {\mathcal{A}}_{\mathrm{UCB}}(\mathbf{s} | {\mathcal{D}}_t)$$

(37)

where ${\mathbf{s}}_t^{\star }$ is the next design to evaluate, and $N_c$ is the candidate-set size. The full-fidelity objective is then computed as $y_t^{\star }=f\left({\mathbf{s}}_t^{\star }\right)$, and the dataset is augmented by

$${\mathcal{D}}_{t+1}={\mathcal{D}}_t\cup \left\{\right({\mathbf{s}}_t^{\star },y_t^{\star }\left)\right\}$$

(38)

We alternate acquisition and evaluation until the task-level stopping criterion is met

$$\rho \left(\mathbf{A}\right({\mathbf{s}}_t^{\star }\left)\right)<{\rho }_{\mathrm{tol}}\mathrm{or}t\ge T_{\mathrm{m}\mathrm{a}\mathrm{x}}$$

(39)

where ${\rho }_{\mathrm{tol}}>0$ is the coherence tolerance and $T_{\mathrm{m}\mathrm{a}\mathrm{x}}$ is the maximum number of iterations. In practice, candidate screening uses a low-fidelity evaluator that decimates the angle/frequency/voxel grids, followed by full-grid evaluation for the selected ${\mathbf{s}}_t^{\star }$; this multi-fidelity strategy changes only the cost of obtaining $y$, not the Bayesian updates. The numerical implementation can optionally accelerate matrix assembly and correlation by GPU kernels, but the Algorithm 1 itself is agnostic to hardware.

Algorithm 1. Deep Bayesian Optimization Algorithm for I-Shaped Sparse Aperture Design.

$\mathrm{Define} \mathrm{the} \mathrm{set} \mathcal{S}=\left\{(a,b,c,d,e,f)\in {\mathbb{Z}}_{+}^6\right\}$ as the parameter space for the I-shaped aperture design. $\mathrm{Obtain} \mathrm{the} \mathrm{initial} \mathrm{sample} \mathrm{set} {\mathcal{D}}_0={\left\{\left({\mathbf{s}}_i,y_i\right)\right\}}_{i=1}^{N_0}$$\mathrm{by} \mathrm{evaluating} N_0$ initial designs using a lowfidelity evaluator. $\mathrm{Initialize} \mathrm{the} \mathrm{surrogate} \mathrm{models} {\left\{g_m(\mathbf{s})\right\}}_{m=1}^M$$\mathrm{with} M$$\mathrm{deep} \mathrm{neural} \mathrm{networks}, \mathrm{trained} \mathrm{on} \mathrm{the} \mathrm{initial} \mathrm{dataset} {\mathcal{D}}_0$. $\mathbf{For} t=1,2,\dots ,T_{\mathrm{m}\mathrm{a}\mathrm{x}}$ $\mathrm{Compute} \mathrm{the} \mathrm{posterior} \mathrm{mean} {\mu }_t(\mathbf{s})$$\mathrm{and} \mathrm{variance} {\sigma }_t^2(\mathbf{s})$ using the ensemble of surrogate models. $\mathrm{Generate} N_c$$\mathrm{feasible} \mathrm{candidate} \mathrm{designs} {\mathcal{S}}_c={\left\{{\mathbf{s}}^{(j)}\right\}}_{j=1}^{N_c}$ from the current model. $\mathrm{Calculate} \mathrm{the} \mathrm{acquisition} \mathrm{function} {\mathcal{A}}_{\mathrm{U}\mathrm{C}\mathrm{B}}(\mathbf{s})={\mu }_t(\mathbf{s})-\beta {\sigma }_t(\mathbf{s})$ for each candidate. $\mathrm{Select} \mathrm{the} \mathrm{top} N_{\mathrm{sel} }$$\mathrm{candidates} \mathrm{from} {\mathcal{S}}_c$ by minimizing the acquisition function. $\mathrm{Evaluate} \mathrm{the} \mathrm{selected} \mathrm{candidates} \mathrm{at} \mathrm{full} \mathrm{fidelity}, \mathrm{obtaining} \mathrm{the} \mathrm{objective} \mathrm{values} y_t^{\star }=f\left({\mathbf{s}}_t^{\star }\right)$$. \mathrm{Augment} \mathrm{the} \mathrm{dataset} {\mathcal{D}}_{t+1}={\mathcal{D}}_t\cup \left\{\left({\mathbf{s}}_t^{\star },y_t^{\star }\right)\right\}$. $\mathrm{If} \mathrm{stopping} \mathrm{criterion} \mathrm{is} \mathrm{met}: \rho \left(\mathbf{A}\left({\mathbf{s}}_t^{\star }\right)\right)<{\rho }_{\mathrm{tol}}$, break. $\mathrm{Periodically} \mathrm{retrain} \mathrm{the} \mathrm{surrogate} \mathrm{models} \mathrm{every} 5 \mathrm{iterations} \mathrm{on} \mathrm{the} \mathrm{augmented} \mathrm{dataset} {\mathcal{D}}_{t+1}$. $\mathrm{Output}: \mathrm{The} \mathrm{optimized} \mathrm{aperture} \mathrm{design} {\mathbf{s}}^{†}={\mathbf{s}}_t^{\star }$$\mathrm{and} \mathrm{its} \mathrm{corresponding} \mathrm{mutual} \mathrm{coherence} \rho \left(\mathbf{A}\left({\mathbf{s}}^{†}\right)\right)$.

3. Experimental Setup

Figure 4 illustrates the schematic diagram of the testing system utilized in this study. A stepped-frequency radar system comprises a two-port vector network analyzer (VNA), a low-noise amplifier (LNA), and two double-ridged horn antennas. A foam pillar measuring 70 cm in height is situated on a high-precision turntable, with the test target positioned at the center of both the foam pillar and the turntable. The VNA is managed via the instrumentation system to facilitate the transmission and reception of electromagnetic waves. Measurements were performed using the S21 parameter of a vector network analyzer. Port 1 was configured as the transmit port and connected directly to the transmitting antenna via a microwave cable. Port 2 served as the receive port; to improve the receiver signal-to-noise ratio, an X-band low-noise amplifier with a gain of 30 dB was inserted between the receiving antenna and Port 2. The measurement system also included a turntable controller and a two-axis rotary table, a plastic foam support column, and low-scattering shields. The foam column and low-scattering shields were used to mitigate coupling between the metal turntable and the target under test.

Background cancellation was implemented during the measurements to reduce the impact of background clutter on the results. Furthermore, each measurement produced full-aperture sampled data; the data needed for specific sparse apertures were derived by extracting the relevant subsets from the full-aperture dataset. This method guarantees that all data used for comparing imaging performance across apertures comes from a single measurement, thus enhancing the credibility of the comparison.

The choice of parameters in the measurement process of 3D imaging significantly affects the quality of the resultant image. This study employs measurement parameters, detailed in

Table 1. Testing parameters for 3D imaging.

Symbol	Parameter	Description
$P$	VV	Polarization mode of the antenna
$R$	4 m	Distance from the Antenna to the Target Center
$B$	4 GHz	Sweep Bandwidth
$f_c$	10 GHz	Center frequency
$\mathsf{\Theta }$	16°	Azimuth Sweep Range
$\mathsf{\Phi }$	16°	Elevation Sweep Range
$\mathsf{\Delta }f$	50 MHz	Frequency Interval
$\mathsf{\Delta }\theta$	0.5°	Azimuth Interval
$\mathsf{\Delta }\phi$	0.5°	Elevation Interval
$N$	401	Number of Frequency Sampling Points
$M$	33	Number of Azimuth Sampling Points
$K$	33	Number of Elevation Sampling Points

, to maintain image integrity and achieve comparable lateral, longitudinal, and vertical resolutions. The angle formed by the line connecting the transmitting and receiving antennas with respect to the target is less than 5°, signifying monostatic radar reception.

In this study, a tank model was selected as the object under test (OUT). Because the model is made of plastic, copper foil was applied to its side skirts and gun barrel to improve surface conductivity, thereby approximating optical-regime scattering characteristics. Photographs of the model and its key dimensions are shown in Figure 5a,b, while images of the actual test configuration in the anechoic chamber are presented in Figure 5c,d.

4. Results

4.1. Optimized Sparse Aperture

Table 2. Angle sampling details of the apertures.

Aperure	Definition	Sampling Angles Number
full	Sampling at each angle. $\begin{array}{ll}& \theta =\left({\theta }_{\mathrm{m}\mathrm{i}\mathrm{n}},{\theta }_{\mathrm{m}\mathrm{a}\mathrm{x}}\right)=\left(-{8.0}^{\circ },{8.0}^{\circ }\right),{\theta }_{\mathrm{step} }={0.5}^{\circ }\\ & \phi =\left({\phi }_{\mathrm{m}\mathrm{i}\mathrm{n}},{\phi }_{\mathrm{m}\mathrm{a}\mathrm{x}}\right)=\left(-{8.0}^{\circ },{8.0}^{\circ }\right),{\phi }_{\mathrm{step} }={0.5}^{\circ }\end{array}$	1089
IF-ISAR	Sampling through 2 arcs. $\begin{array}{ll}& \theta ={\theta }_{\mathrm{m}\mathrm{i}\mathrm{n}}=-{8.0}^{\circ },\phi =\left({\phi }_{\mathrm{m}\mathrm{i}\mathrm{n}},{\phi }_{\mathrm{m}\mathrm{a}\mathrm{x}}\right)=\left(-{8.0}^{\circ },{8.0}^{\circ }\right),{\phi }_{\mathrm{step} }={0.5}^{\circ }\\ & \theta ={\theta }_{\mathrm{m}\mathrm{a}\mathrm{x}}={8.0}^{\circ },\phi =\left({\phi }_{\mathrm{m}\mathrm{a}\mathrm{x}},{\phi }_{\mathrm{m}\mathrm{i}\mathrm{n}}\right)=\left({8.0}^{\circ },-{8.0}^{\circ }\right),{\phi }_{\mathrm{step} }={0.5}^{\circ }\end{array}$	66
1-loop	Sampling through 4 arcs. $\begin{array}{ll}& \theta ={\theta }_{\mathrm{m}\mathrm{i}\mathrm{n}}=-{8.0}^{\circ },\phi =\left({\phi }_{\mathrm{m}\mathrm{i}\mathrm{n}},{\phi }_{\mathrm{m}\mathrm{a}\mathrm{x}}\right)=\left(-{8.0}^{\circ },{8.0}^{\circ }\right),{\phi }_{\mathrm{step} }={0.5}^{\circ }\\ & \phi ={\phi }_{\mathrm{m}\mathrm{a}\mathrm{x}}={8.0}^{\circ },\theta =\left({\theta }_{\mathrm{m}\mathrm{i}\mathrm{n}},{\theta }_{\mathrm{m}\mathrm{a}\mathrm{x}}\right)=\left(-{8.0}^{\circ },{8.0}^{\circ }\right),{\theta }_{\mathrm{step} }={0.5}^{\circ }\\ & \theta ={\theta }_{\mathrm{m}\mathrm{a}\mathrm{x}}={8.0}^{\circ },\phi =\left({\phi }_{\mathrm{m}\mathrm{a}\mathrm{x}},{\phi }_{\mathrm{m}\mathrm{i}\mathrm{n}}\right)=\left({8.0}^{\circ },-{8.0}^{\circ }\right),{\phi }_{\mathrm{step} }={0.5}^{\circ }\\ & \phi ={\phi }_{\mathrm{m}\mathrm{i}\mathrm{n}}=-{8.0}^{\circ },\theta =\left({\theta }_{\mathrm{m}\mathrm{a}\mathrm{x}},{\theta }_{\mathrm{m}\mathrm{i}\mathrm{n}}\right)=\left({8.0}^{\circ },-{8.0}^{\circ }\right),{\theta }_{\mathrm{step} }={0.5}^{\circ }\end{array}$	128
loop-cross	A 1-loop aperture with a cross path composed of 2 arcs. $\begin{array}{ll}& \theta =0^{\circ },\phi =\left({\phi }_{\mathrm{m}\mathrm{i}\mathrm{n}},{\phi }_{\mathrm{m}\mathrm{a}\mathrm{x}}\right)=\left(-{8.0}^{\circ },{8.0}^{\circ }\right),{\phi }_{\mathrm{step} }={0.5}^{\circ }\\ & \phi =0^{\circ },\theta =\left({\theta }_{\mathrm{m}\mathrm{i}\mathrm{n}},{\theta }_{\mathrm{m}\mathrm{a}\mathrm{x}}\right)=\left(-{8.0}^{\circ },{8.0}^{\circ }\right),{\theta }_{\mathrm{step} }={0.5}^{\circ }\end{array}$	189
2-loop	A 1-loop aperture with a similar small-loop in it. $\mathrm{The} \mathrm{maximum} \mathrm{angles} \mathrm{of} \mathrm{the} \mathrm{small}-\mathrm{loop} \mathrm{are} \pm {6.0}^{\circ }$.	224

shows the angle sampling details of the traditional sparse aperture, they are attractive because their trajectories are easy to execute and the scan efficiency is acceptable, yet their angular (and corresponding wavenumber-domain) sampling exhibits repeated directions and localized clustering that raise the mutual correlation of the sensing matrix $A$. this elevated correlation propagates to sparse reconstructions as stronger sidelobes and a compressed dynamic range. To address this limitation, we target the minimization of the maximum mutual correlation ${\rho }_{\mathrm{m}\mathrm{a}\mathrm{x}}=\underset{i\ne j}{\mathrm{m}\mathrm{a}\mathrm{x}}|a_i^{\mathrm{H}}{a}_j|$ under practical feasibility constraints and automatically search a structured single-direction trajectory that remains implementable on a positioner while keeping the number of samples under control.

The search space is a six-parameter piecewise single-direction path $[a,b,c,d,e,f]$ constrained to avoid reversals and to respect mechanical travel limits, the loss is the maximum mutual correlation ${\rho }_{\mathrm{m}\mathrm{a}\mathrm{x}}$ (smaller is better), and deep Gaussian-process modeling with a UCB acquisition guides sampling; the optimizer is initialized with $n_0=100$ designs and runs for $T=50$ evaluations with $\beta =2.0$ and an ensemble size of five, converging within budget at iteration $t^{\mathrm{*}}=50$.

The resulting aperture is I-shaped and strictly single-direction, with two concave vertical legs that diversify sampling directions while preserving mechanical executability; the trajectory covers $M_{\mathrm{sparse}}=174$ unique angular grid points with zero reversals and achieves ${\rho }_{\mathrm{m}\mathrm{a}\mathrm{x}}=0.1947$. Its sampling pattern on the $(\phi ,\theta )$ grid is shown in Figure 6, which highlights the intended directional diversity and the avoidance of redundant revisits; the final parameter vector is $[a,b,c,d,e,f]=[11,19,10,12,11,11]$.

For a fair, metric-only comparison focused on sparsity and correlation,

Table 3. Sampling points and maximum cross-correlation coefficient at different sparse aperture angles.

	IF-ISAR	1-Loop	Loop-Cross	2-Loop	Optimized	Random
$M_{\mathrm{sparse}}$	66	128	189	224	174	174
${\rho }_{\mathrm{m}\mathrm{a}\mathrm{x}}$	0.9689	0.4923	0.4043	0.3280	0.1947	0.1913

lists the angular sample count $M_{\mathrm{sparse}}$ and the maximum mutual correlation ${\rho }_{max}$ for the four baselines and for the proposed aperture; all apertures are subsampled from the same full dataset and the columns of $A$ are ${\mathcal{l}}_2$-normalized before computing $\rho$. These results indicate that, at a comparable or smaller sampling load than the dense multi-loop baselines, the proposed trajectory attains substantially lower mutual correlation, which is the desired precondition for improved sidelobe suppression and dynamic-range performance in the ASOMP reconstructions presented next. Compressed sensing theory holds that random sampling is more likely to ensure sampling incoherence. Accordingly, to compare the aperture-recovery performance of the algorithm proposed herein, we also adopted a random sampling pattern with the same 174 angular sampling points. Such random sampling more effectively enforces the incoherence of the sampling matrix A. Table 3 reports, for different sampling apertures, the maximum mutual coherence ${\rho }_{\mathrm{m}\mathrm{a}\mathrm{x}}$ and the number of angular sampling points $M_{\mathrm{sparse}}$. The results indicate that the optimized sampling aperture computed by our method yields a smaller mutual coherence than the 2-loop scheme and is already close to that of the random-sampling aperture, implying superior theoretical recovery performance relative to 2-loop and potentially approaching that of random sampling.

4.2. Imaging Result

The 3D radar imaging results presented in this study are based on amplitude-normalized images, with amplitude values represented in decibels (dB). This normalization technique ensures that the dynamic range of the imaging data is suitable for comparison and viewing, thereby enhancing the clarity of the reconstructed images. The use of dB as the unit for amplitude enhances the differentiation of signal intensities, especially in the presence of noise and background interference, which is crucial for precise target detection and characterization.

The target imaging space has the following coordinate ranges: X spans from -0.4 m to 0.4 m, Y from -0.3 m to 0.3 m, and Z from -0.2 m to 0.2 m, with an interval of 0.01 m. The total is N = 81 × 61 × 41 = 202581 points in the target space.

Figure 7 displays the complete aperture MF-BP imaging result. The distribution of strong scattering centers on OUT is clearly observable, with a primary concentration on the copper foil coated gun barrel and tracks.

Figure 8 illustrates the ASMOP imaging outcomes across various sparse apertures. Figure 8a–d represent random sampling apertures and provide a more accurate reference baseline. Figure 8 presents a visual comparison indicating that the 1 loop aperture and the IF ISAR aperture result in suboptimal reconstruction, as they do not effectively focus on the dominant scattering centers of the OUT and instead display multiple spurious artifacts. In contrast, the results from the 2 loop aperture, the Loop Cross aperture, and the optimized aperture yield scattering distributions that more closely resemble those of the random sampling case, although significant differences persist in the locations and amplitudes of strong scattering centers.

For each aperture, a top view patch at a randomly selected location is enlarged. The zoomed-in views demonstrate that the optimized sampling aperture derived from our method aligns more closely with the random sampling aperture. The Section 5 will present a quantitative evaluation of imaging accuracy across apertures.

5. Discussion

The MF-BP algorithm functions by executing coherent phase summation to estimate the reflectivity g at each voxel, followed by the mapping of the recovered values to their corresponding spatial locations based on the coordinates $(x, y, z)$. The algorithm provides the most accurate solution without introducing approximations, though it incurs significant computational complexity. The output from conventional MF BP methods serves as a benchmark for quantitatively evaluating the imaging accuracy of the proposed method. Examine Equations (11) and (14), modify certain variable annotations, and present the revised equations below.

$${\stackrel{\mathbf{ˆ}}{\mathbf{g}}}_{mf}=\left({\mathbf{A}}_{fa}^H{\mathbf{A}}_{fa}\right){\stackrel{\mathbf{ˆ}}{\mathbf{g}}}_U={\mathbf{A}}_{fa}^H\mathbf{s}$$

(40)

$${\stackrel{\mathbf{ˆ}}{\mathbf{g}}}_{cs}=\underset{\mathbf{g}}{\mathrm{m}\mathrm{i}\mathrm{n}}\Vert \mathbf{g}{\Vert }_1, \mathrm{s}.\mathrm{t}. {‖\mathbf{s}-{\mathbf{A}}_{sa}\mathbf{g}‖}_2^2<\epsilon$$

(41)

where ${\stackrel{\mathbf{ˆ}}{\mathbf{g}}}_U$ is the UMVUE result. ${\stackrel{\mathbf{ˆ}}{\mathbf{g}}}_{mf}$ is the imaging result of MF-BP method and it is an approximate result of ${\stackrel{\mathbf{ˆ}}{\mathbf{g}}}_U.{\stackrel{\mathbf{ˆ}}{\mathbf{g}}}_{cs}$ is the ${\mathcal{l}}_1$-norm optimization estimation result of compressed sensing method. Matrix ${\mathbf{A}}_{fa}$ means the sampling matrix of the full aperture (fa) and ${\mathbf{A}}_{sa}$ means the sampling matrix of a sparse aperture.

For linear estimation models, regarded the UMVUE result as the best solution. To evaluate the imaging result of CS method, we can compare the similarity between ${\stackrel{\mathbf{ˆ}}{\mathbf{g}}}_{cs}$ and ${\stackrel{\mathbf{ˆ}}{\mathbf{g}}}_U$. But ${\stackrel{\mathbf{ˆ}}{\mathbf{g}}}_U$ can’t be directly calculated. In order to do the comparison, we can times $\left({\mathbf{A}}_{fa}^H{\mathbf{A}}_{fa}\right)$ at the left of ${\stackrel{\mathbf{ˆ}}{\mathbf{g}}}_{cs}$ as

$${\stackrel{\mathbf{ˆ}}{\mathbf{g}}}_{cs}^{\prime }=\left({\mathbf{A}}_{fa}^H{\mathbf{A}}_{fa}\right){\stackrel{\mathbf{ˆ}}{\mathbf{g}}}_{cs}$$

(42)

After this changing, we can compare ${\stackrel{\mathbf{ˆ}}{\mathbf{g}}}_{cs}$ and ${\stackrel{\mathbf{ˆ}}{\mathbf{g}}}_U$ by comparing ${\stackrel{\mathbf{ˆ}}{\mathbf{g}}}_{cs}^{\prime }$ and ${\stackrel{\mathbf{ˆ}}{\mathbf{g}}}_{mf}$. To calculate the similarity between signal ${\stackrel{\mathbf{ˆ}}{\mathbf{g}}}_{cs}^{\prime }$ and ${\stackrel{\mathbf{ˆ}}{\mathbf{g}}}_{mf}$, correlation coefficient can be used as

$$\mu ={\displaystyle \frac{\left|<{\stackrel{\mathbf{ˆ}}{\mathbf{g}}}_{mf},{\stackrel{\mathbf{ˆ}}{\mathbf{g}}}_{cs}^{\prime }>\right|}{{‖{\stackrel{\mathbf{ˆ}}{\mathbf{g}}}_{mf}‖}_2{‖{\stackrel{\mathbf{ˆ}}{\mathbf{g}}}_{cs}^{\prime }‖}_2}}$$

(43)

Table 4. Correlation coefficient between ${\stackrel{\mathbf{ˆ}}{\mathbf{g}}}_{mf}$ and ${\stackrel{\mathbf{ˆ}}{\mathbf{g}}}_{cs}^{\prime }$ of each aperture in practical test.

Aperture	Angle Sampling Points	$\mathit{K}$	$\mathit{\mu }$
IF-ISAR	66	124	0.2319
1-Loop	128	122	0.4721
Loop-cross	189	120	0.9036
2-Loop	224	126	0.8435
optimized	174	114	0.9754
random	174	111	0.9913

presents a theoretically consistent ordering of sparse designs by the correlation coefficient μ between the MF-BP reference and the ASOMP reconstruction.

Table 4 indicates that the random and optimized apertures achieve the highest fidelity ($\mu$ = 0.9913 and 0.9754, respectively) while maintaining the smallest learned model orders ($K$ ≈ 111–114). In contrast, conventional engineering baselines—Loop-cross ($\mu$ = 0.9036, 189 points, $K$ = 120), 2-Loop ($\mu$ = 0.8435, 224 points, $K$ = 126), 1-Loop (μ = 0.4721, 128 points, $K$ = 122), and IF-ISAR ($\mu$ = 0.2319, 66 points, $K$ = 124)—demonstrate significantly lower performance, even when utilizing a greater number of samples in several instances. This pattern suggests that reconstruction fidelity is primarily influenced by the conditioning of the sensing operator rather than by the sampling count alone. Apertures that distribute measurements to minimize directional redundancy and local clustering reduce mutual correlations among sensing atoms, allowing ASOMP to accurately retrieve the correct support with fewer atoms and to mitigate side-lobe artifacts.

The mechanism aligns with compressed sensing theory, indicating the recovery quality enhances as the maximum mutual coherence ${\rho }_{\mathrm{m}\mathrm{a}\mathrm{x}}$ of the measurement matrix diminishes. With the sampling budget set at 174 points, the advantages of the optimized and random apertures in Table 4 correspond with their lower ρ_max values reported in Table 3 (approximately 0.19, compared to 0.33–0.97 for conventional designs), which also accounts for their decreased adaptive sparsity $K$. The values of $\mu$ and the learned $K$ depend on SNR and the termination threshold of ASOMP; however, the ordering observed with equal sampling budgets is consistent and methodologically interpretable. The evidence indicates that a mechanically executable, structured single-direction path can approximate the statistical performance of random sampling within the same sampling budget, resulting in significant reductions in scan time and system complexity while maintaining reconstruction fidelity in three-dimensional radar imaging.

6. Conclusions

This study presents a compressed-sensing 3D ISAR imaging framework which combines an adaptive-sparsity OMP reconstructor with a physically realizable sparse-aperture design, optimized through deep Bayesian search to reduce the mutual coherence of the sensing operator. The proposed method effectively resolves the enduring trade-off between scan efficiency and image fidelity in turntable measurements, facilitating accurate reconstructions from significantly fewer angular samples while maintaining continuous and time-efficient motion of the positioner. The approach is based on a variant of Orthogonal Matching Pursuit (OMP) that eliminates the need for a priori sparsity selection through a data-driven termination rule. It utilizes a structured, single-direction “I-shaped” trajectory, with parameters optimized to minimize column correlations in the measurement matrix, thereby enhancing the theoretical foundations for sparse recovery.

Comprehensive simulations and a controlled chamber experiment, utilizing full-aperture acquisitions followed by identical sub-sampling for equitable comparisons, demonstrated that the optimized path achieves significantly lower maximum mutual coherence than conventional loop-based schemes. Additionally, it yields ASOMP reconstructions that closely align with MF-BP references and approach the random-sampling baseline, all while maintaining a comparable sampling budget and ensuring zero reversals to uphold scan efficiency. The findings indicate that imaging quality in the sparse regime is primarily influenced by the conditioning of the sensing operator, rather than solely by the number of samples. Distributing measurements to minimize directional redundancy and localized clustering effectively suppresses sidelobes and preserves dynamic range.

The proposed design provides an efficient method to reduce measurement time in compact-range 3D imaging while maintaining reconstruction fidelity, utilizing a straightforward trajectory compatible with standard elevation-over-azimuth positioners. The optimization principle can be applied to various aperture families, polarimetric operations, and multi-static or multi-sensor environments. When integrated with online, measurement-in-the-loop optimization and enhanced priors, it can improve robustness against SNR and calibration variations. We expect that these guidelines will enhance applicability across various targets and facilities, facilitating the routine implementation of sparsity-aware sampling in high-value radar metrology. The core optimization methodology introduced herein, while primarily utilized in ISAR imaging, possesses considerable promise for wider application in other scientific and technical fields where data gathering is constrained by physical limitations. For instance, in Magnetic Resonance Imaging (MRI), this approach could be modified to create non-Cartesian k-space trajectories that are both feasible for the gradient system and optimized for compressed sensing reconstruction, potentially decreasing scan durations while minimizing artifacts. In sparse-angle Computed Tomography (CT), the framework can identify an appropriate, non-uniform sequence of projection angles for a rotating gantry to reduce radiation exposure while maintaining diagnostic picture quality. This study addresses a specific difficulty in radar metrology and provides a versatile computational framework for efficient, physics-informed sparse sensing across many image modalities. Currently, compressed sensing algorithms fail to satisfy the demands of real-time imaging. Future research will concentrate on enhancing the computational efficiency of these techniques.