Parameter Estimation of 3D-GTD Model Based on a Multi-Neural Network

MingruiQi1

LihuaWu1✉Emailwulihua@alumni.nudt.edu.cn

RuiCao1

MinghaoPeng1

1Northwest Institute of Nuclear Technology710038Xi’China

Mingrui Qi1,+, Lihua Wu1,*,+, Rui Cao1, and Minghao Peng1

1 Northwest Institute of Nuclear Technology, 710038 Xi’ an, China

*corresponding.author: wulihua@alumni.nudt.edu.cn

+these authors contributed equally to this work

ABSTRACT

In this article, a novel parameter estimation method of three-dimensional geometrical theory of diffraction(3D-GTD) model based on a multi-neural network is proposed. By leveraging multi-view one-dimensional geometrical theory of diffraction(1D-GTD) parameters as input, the three-branch neural network simultaneously estimates 3D scattering center parameters—position, intensity, and type. Position reconstruction branch employs geometric encoding and gated fusion for joint feature extraction, combined with hierarchical attention and coordinate regression. Scattering intensity estimation branch achieves efficient feature aggregation via multi-level convolutional layers and parameter sharing mechanism, enabling high-precision intensity parameter estimation while ensuring model lightweight. Type identification branch establishes a mapping from numerical features to categorical indices, which converts the regression task into a classification task that conforms to discrete value constraints. It employs a sub-channel processing architecture and a shared multi-layer perceptron (MLP) to effectively aggregate multi-view predictive information, thereby achieving accurate identification of scattering center types. To address the challenge of difficult acquisition of training data, we construct a self-generated dataset with random characteristics, thereby enhancing the generalization performance of the network. Experimental results show the proposed method outperforms traditional approaches in estimation accuracy across different signal-to-noise ratio (SNR) levels and computational efficiency.

Index Terms

Three-dimensional geometrical theory of diffraction(3D-GTD) model

multi-neural network

scattering center parameters

self-generated dataset.

I Introduction

The geometrical theory of diffraction (GTD) model [1] serves as a classical high-frequency regime radar target electromagnetic scattering center model, which characterizes the overall target scattering echo by decomposing it into a superposition of multiple isolated contributions, including point scattering, diffraction, reflection, and other scattering types [2]. Scattering center models can be categorized into three classes based on their spatial descriptive capabilities, including one-dimensional scattering center model (describing single-view scattering characteristics) [3], [4], two-dimensional scattering center models (describing scattering characteristics within a single imaging plane) [5], [6], and three-dimensional models (describing multi-azimuth and multi-elevation scattering characteristics) [7]. Among these, three-dimensional geometrical theory of diffraction(3D-GTD) model can achieve comprehensive characterization of target scattering properties across full attitude angles while preserving physical interpretability. The parameter estimation of the 3D-GTD model has been widely applied in radar target recognition [8], [9], radar echo simulation [10], [11], and high-resolution radar imaging [12], [13].

The parameter estimation methods for target scattering center models can be broadly categorized into two classes. The one is forward deduction method, which can calculate the scattering center model parameters based on the geometric structure and material properties of the radar target [14], [15]. This kind of method depend on accurate modeling of the physical attributes of the target, which often appear significant errors when model mismatch occurs. The other is reverse estimation method, which can extract target scattering center model parameters based on frequency response or radar image [16], [17], [18], [19], [20], [21]. Compared to image-domain processing approaches, frequency-domain processing method can effectively utilize phase information, while avoiding artifacts caused by scattering center segmentation, and achieve higher resolution and estimation accuracy. Therefore, this article is devoted to the parameter estimation method based on electromagnetic scattering frequency response data. Representative algorithms include the multiple signal classification (MUSIC) algorithm [21], [22], the matrix enhancement and matrix pencil (MEMP) algorithm [18], [23], [24], and the estimating signal parameter via rotational invariance techniques (ESPRIT) algorithm [19], [20]. The MUSIC algorithm leverages the orthogonality between the noise subspace and the steering vector to estimate scattering center parameters via spatial spectrum peak search. However, it suffers from large amount of calculation. The MEMP algorithm enhances the separability between the signal subspace and the noise subspace by constructing an augmented matrix pencil. However, the estimation accuracy can be affected by the aperture closure mismatch problem. The ESPRIT algorithm avoids parameter pairing and spectrum peak search by exploiting the rotational invariance property of the signal subspace, yet its performance will be limited by noise. The aforementioned algorithms involve the construction and operation of complex feature matrices, resulting in relatively low computational efficiency. Furthermore, coupled parameter estimation and high-dimensional matrix operations often lead to error propagation and unstable estimation accuracy.

In the past few years, deep learning technology have been widely applied in the field of electromagnetic scattering analysis, such as scattering calculation [25], [26], inverse scattering [27], [28], and scattering characteristic analysis [29], [30]. Neural networks have demonstrated remarkable performance in establishing end-to-end mappings between high-dimensional observational data and underlying physical parameters [31], thus offering new perspectives for improving the performance of scattering center parameter estimation algorithms. Huo et al. employed a sparse autoencoder network to learn one-dimensional high-resolution range profiles (HRRPs) of typical targets and explored the relationship between network weights and scattering center parameters [32], though their work did not utilize neural networks for direct parameter estimation. Xing et al. proposed a convolutional neural network (CNN)-based method for parameter estimation under the bistatic GTD (Bi-GTD) model [33], which effectively compensated for estimation errors caused by phase inaccuracies and achieved superior performance compared to traditional methods. Nevertheless, the research on three-dimensional scattering center parameter estimation based on neural network remains relatively scarce.

In this article, we proposed a multi-neural network for estimating parameters of the 3D-GTD model. The branch networks independently process the multi-observation one-dimensional scattering center parameters of the target, separately accomplishing the estimation of three-dimensional scattering center location, type, and intensity parameters. This structure decouples the multi-parameter estimation task, thereby avoiding error amplification issues. Furthermore, we constructed a self-generated dataset based on random parameters to solve the problem of obtaining training data. This approach additionally enables the network to adaptively learn and compensate for errors arising from intermediate processes and noise interference, thereby enhancing the model's robustness and generalization capability. To evaluate the performance of the proposed method, we conducted comprehensive experiments, including analyses of parameter estimation accuracy, radar cross section (RCS) extrapolation, and computational efficiency. The experimental results demonstrate that the proposed method achieves higher parameter estimation accuracy, superior noise robustness, and greater computational efficiency compared to traditional approaches.

The remainder of this article is organized as follows. In Section II, the concept of 3D-GTD model is introduced, and the technological process of the proposed method along with the structures of the neural networks are presented. In Section III, the self-generated method of dataset and the configuration of network training are expounded. In Section IV, the experimental results and the performance analysis of the proposed method are provided. In Section V, the conclusion and further discussion are presented.

II Materials and Methods

A༎ The concept of 3D-GTD model

The GTD model incorporates diffraction coefficients to describe the phenomenon of diffraction occurring when incident rays encounter edges or tips, making it a scattering center model that closely aligns with the actual physical scattering mechanisms. According to the classical GTD scattering theory, in the high-frequency area, the total scattering field of the target can be equivalently regarded as the coherent superposition of scattering fields generated by discrete, localized scattering centers. Each scattering center is associated with local geometric structures of the target, such as edges, ridges, tips, flat plates, curved surfaces, of which the scattering characteristics can be parametrically modeled by a set of parameters with clear physical significance. Under the far-field conditions, a typical GTD scattering center model can be expressed as

$E\left( {{f_m}} \right)={\sum\limits_{{i=1}}^{I} {{A_i}\left( {j\frac{{{f_m}}}{{{f_0}}}} \right)} ^{{\alpha _i}}}\exp \left( {\frac{{ - j4\pi {f_m}{r_i}}}{c}} \right)$

where

is the number of scattering centers.

${r_i}$

${A_i}$

${\alpha _i}$

denote the range, scattering intensity, and scattering type of the ith scattering center.

${f_m}$

is the current frequency of the incident electromagnetic wave

${f_m}={f_0}+m\Delta f,m=1,2, \ldots M$

The values of

${\alpha _i}$

corresponding to typical scattering structures are listed in Table 1.

Table 1
Values of scattering type parameter corresponding to typical scattering structures.
${\alpha _i}$	Scattering structure	Scattering mechanism
-1	Tip, corner edge	Corner diffraction
-0.5	Curved edge	Edge diffraction
0	Doubly curved surface, straight edge	Point scattering, specular scattering
0.5	Singly curved surface, top-hat	Corner diffraction
1	Flat plate, dihedral	Corner diffraction

where

${f_0}$

$\Delta f$

are the initial frequency, frequency step. M is the total number of frequency points.

Furthermore, to characterize the full-aspect scattering properties of the target, the GTD model can be extended into three-dimensional space, incorporating the effects of noise, which can be expressed as

$\begin{gathered} E\left( {{f_m},{\theta _n},{\varphi _k}} \right)={\sum\limits_{{i=1}}^{I} {{{\hat {A}}_i}\left( {j\frac{{{f_m}}}{{{f_0}}}} \right)} ^{{{\hat {\alpha }}_i}}} \cdot \exp [ - j4\pi {f_m}({x_i}cos{\theta _n}\cos {\varphi _k} \hfill \\ +{y_i}cos{\theta _n}\sin {\varphi _k}+{z_i}\sin {\theta _n})/c]+\omega \left( {{f_m},{\theta _n},{\varphi _k}} \right) \hfill \\ \end{gathered}$

where

${x_i}$

${y_i}$

${z_i}$

are the transversal position, longitudinal position, and vertical position of the ith scattering center.

$\omega$

denotes the gaussian white noise.

${\theta _n}$

${\varphi _k}$

are the current azimuth angle and pitching angle of the incident electromagnetic wave direction vector, which can be expressed as

${\theta _n}={\theta _0}+n\Delta \theta ,n=1,2, \ldots N$

${\varphi _k}={\varphi _0}+k\Delta \varphi ,k=1,2, \ldots K$

Where

${\theta _0}$

$\Delta \theta$

are the initial azimuth angle, azimuth angle step.

${\varphi _0}$

$\Delta \varphi$

are the initial pitching angle, pitching angle step. N, K are the total number of azimuth angles and pitching angles.

B༎ The proposed method

1) Methodology and process

The process of the proposed method is illustrated in Fig. 1. Firstly, a computation is performed using Eq. (3) to determine the electromagnetic scattering frequency response data of the target when illuminated by electromagnetic waves at various frequencies and incident angles. The extracted 1D-GTD model parameters from various incident angles are then processed by a multi-neural network, which leverages a dedicated three-branch architecture to estimate the comprehensive 3D-GTD model parameters.

Fig. 1

Process of the proposed method.

The frequency response data are obtained using a linear array configuration, during which the incident azimuth and elevation angles (just as Fig. 2) are adjusted in a prescribed co-stepping manner, which can be expressed as

$O=\left[ {\begin{array}{*{20}{c}} {{\theta _1}}&{{\varphi _1}} \\ {{\theta _2}}&{{\varphi _2}} \\ \vdots & \vdots \\ {{\theta _N}}&{{\varphi _N}} \end{array}} \right]$

Fig. 2

Line array observation with co-stepping incident azimuth and elevation angles.

Compared to planar array observations, the use of linear arrays enhances computational efficiency and reduces data requirements. The parameters of the 1D-GTD model for the target at various incident angles can then be estimated using established algorithms, such as the ESPRIT algorithm [19], [33], the generalized likelihood ratio test (GLRT)-based method [34], and the MUSIC algorithm [35], [36]. Since this article does not focus on parameter estimation for the 1D-GTD model, the specific methodologies are not elaborated here. Set the estimator for the parameters of the 1D-GTD under the nth incident angle as

$\Lambda \left( n \right)=\left\{ {{{\hat {r}}_i}\left( n \right),{{\hat {A}}_i}\left( n \right),{{\hat {\alpha }}_i}\left( n \right)} \right\}_{{i=1}}^{I}$

Where

${\hat {r}_i}\left( n \right)$

${\hat {A}_i}\left( n \right)$

, and

${\hat {\alpha }_i}\left( n \right)$

are the range, scattering intensity, and scattering type parameter estimations of the ith scattering center under the nth incident angle, respectively.

Subsequently, association matching of the 1D-GTD parameters is conducted to ensure that each parameter group corresponding to the same spatial position originates from an identical physical scattering center. In this work, the association of range parameters is accomplished through a method integrating the Kalman filter (KF) and the nearest-neighbor standard filter (NNSF) [37]. This approach capitalizes on the gradual displacement of scattering centers under small-angle stepping observation conditions by treating their one-dimensional range history as target trajectories, thereby reformulating the scattering center association task as a multi-target track formation problem. Based on the association results of the range parameters, the 1D-GTD parameters across different incident angles are reorganized accordingly.

It should be noted that the aforementioned method may introduce non-negligible errors to a certain degree, which subsequently require adaptive learning and compensation by the neural network.

2) Branch 1: The neural network for position parameter reconstruction

The neural network for position parameter reconstruction employs a multi-stage feature fusion strategy and a hierarchical cross-view attention mechanism to achieve end-to-end implicit reconstruction of scattering centers from multi-view one-dimensional ranges to three-dimensional spatial positions. The architecture of the network is illustrated in Fig. 3.

Fig. 3

The architecture of the neural network for position parameter reconstruction.

The network consists four key modules, including an enhanced feature fusion module, a hierarchical cross-view attention module, a dynamic feature aggregation module, and a coordinate regression module. The input of the network consists of multi-view one-dimensional scattering center ranges and projection vectors, which can be denoted as

${x_r} \in {{\text{R}}^{N \times I \times 1}}$

and

${x_v} \in {{\text{R}}^{N \times 3}}$

within each batch sample.

${x_r}=\left[ {\begin{array}{*{20}{c}} {r_{1}^{1}}&{r_{1}^{2}}& \cdots &{r_{1}^{I}} \\ {r_{2}^{1}}&{r_{2}^{2}}& \cdots &{r_{2}^{I}} \\ \vdots & \vdots & \ddots & \vdots \\ {r_{N}^{1}}&{r_{N}^{2}}& \cdots &{r_{N}^{I}} \end{array}} \right]$

${x_v}=\left[ {\begin{array}{*{20}{c}} {\cos {\theta _1}\cos {\varphi _1}}&{\cos {\theta _1}\sin {\varphi _1}}&{\sin {\theta _1}} \\ {\cos {\theta _2}\cos {\varphi _2}}&{\cos {\theta _2}\sin {\varphi _2}}&{\sin {\theta _2}} \\ \vdots & \vdots & \vdots \\ {\cos {\theta _N}\cos {\varphi _N}}&{\cos {\theta _N}\sin {\varphi _N}}&{\sin {\theta _N}} \end{array}} \right]$

To facilitate fusion processing, the projection vectors can be expanded along the second dimension, which is

${x_v} \in {{\text{R}}^{N \times I \times 3}}$

. Input

${x_r}$

and

$x{}_{v}$

through single-channel and three-channel pathways, respectively, followed by feature extraction via two fully-connected layers. One branch contains fully-connected layers with hidden dimensions of 32 to 128, while another contains fully-connected layers with hidden dimensions of 64 to 128, both activated by the gaussian error linear unit (GELU) activation function. The range distribution features

${F_r}$

and projection features

${F_v}$

can be expressed as

${F_r}={{\text{W}}_g}\left[ {{f_{GELU}}\left( {{{\text{W}}_g}\left[ {{x_r}} \right]} \right)} \right]$

${F_v}={{\text{W}}_g}\left[ {{f_{GELU}}\left( {{{\text{W}}_g}\left[ {{x_v}} \right]} \right)} \right]$

where

${{\text{W}}_g}\left[ \cdot \right]$

denotes the processing by the fully-connected layers,

${f_{GELU}}\left( \cdot \right)$

denotes the GELU activation function. The GELU activation function facilitates more effective utilization of neurons under negative input conditions due to its non-zero gradients, while its smooth gradient profile contributes to enhanced training stability, which can be approximately expressed as

${f_{GELU}}\left( x \right) \approx 0.5x\left( {1+\tanh \left[ {\sqrt {\frac{2}{\pi }} \left( {x+0.044715{x^3}} \right)} \right]} \right)$

${F_r}$

and

${F_v}$

are concatenated and processed by a fully connected layer (with 128 hidden dimensions) and a sigmoid activation function to generate gating weights, which can be expressed as

${\beta _g}=\sigma \left( {{{\text{W}}_g}\left[ {{F_r}\left\| {{F_v}} \right.} \right]} \right)$

$\sigma \left( \cdot \right)$

denotes the sigmoid activation function. Then the features from the two branches can be dynamically weighted, which can be expressed as

${F_{fused}}=\beta \odot {F_r}+\left( {1 - \beta } \right) \odot {F_v}$

where

$\odot$

denotes element-wise multiplication,

${F_{fused}}$

is the gated fused feature. The adaptive gating fusion mechanism helps to avoid feature redundancy commonly caused simple concatenation, while promoting fusion efficiency and enabling the network to dynamically adjust the importance of features across different pathways.

Furthermore, just as expressed in Eq. (15), a geometric prior encoding branch is incorporated. Specifically,

${x_v}$

is normalized and subsequently processed by two fully connected layers (with hidden dimensions of 32 to 128) and a GELU activation function, aiming at extracting geometric prior features, which are added to the gated fusion features.

${F_{geo}}={{\text{W}}_g}\left[ {{f_{GELU}}\left( {{{\text{W}}_g}\left[ {{{\left\| {{x_v}} \right\|}_2}} \right]} \right)} \right],{F_{comb}}={F_{geo}}+{F_{fuse}}$

where

${F_{comb}}$

is the combined feature, which is the input of the hierarchical cross-view attention module.

The learnable view-specific positional encodings can be incorporated to embed positional awareness.

$V=Embed\left( {{v_{idx}}} \right)$

where

${v_{idx}}$

is the index of the projection vector, which helps to implement the geometric constraints for tree-dimensional reconstruction. Subsequently, the features are fed into two separate multi-head attention modules (8 heads in this work). The global attention module processes the complete feature sequence to model dependencies across all views, and the output can be denoted as

${A_{global}}$

. The local attention module employs masked gating parameters to restrict attention within a window (the window length is 5 in this work), capturing relationships among neighboring views, and the output can be denoted as

${A_{local}}$

. Finally, the outputs from both branches are processed by a GELU activation function and then combined via weighted summation to achieve hierarchical fusion.

${F_{attn}}={\beta _l} \cdot {f_{GELU}}\left( {{A_{local}}} \right)+\left( {1 - {\beta _l}} \right) \cdot {f_{GELU}}\left( {{A_{local}}} \right)$

where

${F_{attn}}$

is the cross-view fusion feature,

${\beta _l}$

is the local attention weight (set as 0.6 in this work). This hierarchical attention mechanism incorporates both local adjacent views and global view relationships, thereby enhancing cross-view consistency modeling and offering greater physical plausibility compared to the single attention mechanism.

${F_{attn}}$

is the input of the dynamic feature aggregation module, which is processed by two fully connected layers (with hidden dimensions of 64 to 1) and a GELU activation function. The output is subsequently normalized via a softmax function to generate importance weights for each view, achieving attention-based scoring.

${\beta _a}={\text{Softmax}}\left( {{{\text{W}}_g}\left[ {{f_{GELU}}\left( {{{\text{W}}_g}\left[ {{F_{attn}}} \right]} \right)} \right]} \right)$

where

${\beta _a}$

is the view weight coefficient. After that, the weighted summation of the hierarchical perspective fusion features is performed.

${F_{agg}}=\sum\nolimits_{{n=1}}^{N} {\beta _{a}^{n}} \cdot F_{{attn}}^{n}$

where

${F_{agg}}$

is the view-weighted fusion feature, which is subsequently processed by two fully connected layers (with hidden dimensions of 64 to 3) and a gated linear unit (GLU) activation function.

${F_{out}}={f_{GLU}}\left( {{{\text{W}}_g}\left[ {{F_{agg}}} \right]} \right)$

where

${F_{out}}$

is the final output features. The GLU activation function splits the input tensor into two equal parts along a specified dimension. One half is transformed by a sigmoid function to generate a gating signal, which is then multiplied element-wise with the other half.

${f_{GLU}}\left( x \right)=\left( {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{x} W+b} \right) \odot \sigma \left( {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{x} V+c} \right)$

where

and

are learnable weight matrixes,

and

are bias vectors. The gating mechanism of the GLU activation function enables the network to dynamically enhance important features while suppressing less relevant ones. In summary, the dynamic feature aggregation module achieves feature compression while adaptively enhancing the contributions of informative views and reducing errors caused by occlusion and erroneous feature matching across partial views.

Finally, the output features are fed into the coordinate regression module, which are processed by two fully connected layers (with hidden dimensions of 64 to 3) and a GELU to obtain the three-dimension spatial coordinates of target scattering.

$\hat {P}={{\text{W}}_g}\left[ {{f_{GELU}}\left( {{{\text{W}}_g}\left[ {{F_{out}}} \right]} \right)} \right]=\left[ {\begin{array}{*{20}{c}} {{{\hat {x}}_1}}&{{{\hat {y}}_1}}&{{{\hat {z}}_1}} \\ {{{\hat {x}}_2}}&{{{\hat {y}}_2}}&{{{\hat {z}}_2}} \\ \vdots & \vdots & \vdots \\ {{{\hat {x}}_I}}&{{{\hat {y}}_I}}&{{{\hat {z}}_I}} \end{array}} \right]$

3) Branch 2: The neural network for scattering intensity parameter estimation

The neural network for scattering intensity parameter estimation adopts the position-independent processing architecture. Specifically, the multi-view 1D-GTD intensity parameters of each scattering center are processed independently in separate channels to estimate the 3D-GTD intensity parameters. Feature extraction and parameter estimation are accomplished through multi-level convolutional layers and multi-layer perceptron (MLP). The architecture of the network is illustrated in Fig. 4.

Fig. 4

The structure of the neural network for scattering intensity parameter estimation.

The multi-view 1D-GTD intensity parameters can be denoted as

${x_A} \in {{\text{R}}^{N \times I}}$

, specifically expressed as

${x_A}=\left[ {{{\text{A}}_1},{{\text{A}}_2}, \ldots {{\text{A}}_I}} \right]=\left[ {\begin{array}{*{20}{c}} {A_{1}^{1}}&{A_{1}^{2}}& \cdots &{A_{1}^{I}} \\ {A_{2}^{1}}&{A_{2}^{2}}& \cdots &{A_{2}^{I}} \\ \vdots & \vdots & \ddots & \vdots \\ {A_{N}^{1}}&{A_{N}^{2}}& \cdots &{A_{N}^{I}} \end{array}} \right]$

${{\text{A}}_1}$

${{\text{A}}_I}$

are fed channel-wise into the feature extraction module, which consists of five sequentially connected convolutional blocks, each comprising a one-dimensional convolutional layer, a linear rectification unit (ReLU) activation function, and a pooling layer. The convolution kernels slide with the change of view angle. The depths of the five convolutional layers are 32, 64, 128, 256, and 512. The kernel size of the first convolutional layer is set as 5, while the other convolutional layers are set as 3. The ReLU activation function can be expressed as

${f_{ReLU}}\left( x \right)=\hbox{max} (0,x)$

It helps to facilitate effective gradient propagation and accelerates network convergence. The pooling layers employ the one-dimensional average pooling operation. For the first four blocks, the kernel size is set as 3, with the stride of 2 and padding of 1. After the pooling operation, the length of output feature for each block can be expressed as

${L_{out}}=\frac{{{L_{in}} - 1}}{2}+1$

The last pooling layer employs adaptive average pooling to control the output feature length to 1. The feature extraction module hierarchically expands the feature dimensions through successive convolutional layers, thereby strengthening the capacity for representing higher-order abstractions. Concurrently, pooling operations reduce the temporal dimensionality, preserving critical patterns while significantly boosting computational efficiency.

The output features are passed into the parameter estimation module, where they are initially flattened. Following this, both feature aggregation and parameter estimation are implemented via a sequence of four fully-connected layers, between which ReLU activation functions are applied. The dimensionality of these layers progressively decreases in the sequence of 512, 256, 64, 32, and finally I, ultimately yielding the estimated 3D-GTD intensity parameters

$\left[ {{{\hat {A}}_1},{{\hat {A}}_2} \ldots ,{{\hat {A}}_I}} \right]$

4) Branch 3: The neural network for scattering type parameter estimation

The neural network for scattering type parameter estimation establishes the mapping relationship between type parameters and their corresponding category labels, while employing a position-independent processing architecture. It transforms the estimation of 3D-GTD model type parameters into a set of independent channel-wise classification tasks. Such a design effectively incorporates the discrete value constraints inherent to type parameters and allows parallel prediction across channels. The architecture of the network is illustrated in Fig. 5.

Fig. 5

The structure of the neural network for scattering type parameter estimation.

The multi-view 1D-GTD type parameters can be denoted as

${x_\alpha } \in {{\text{R}}^{N \times I}}$

, specifically expressed as

${x_\alpha }=\left[ {{{\text{\varvec{\upalpha}}}_1},{{\text{\varvec{\upalpha}}}_2}, \ldots {{\text{\varvec{\upalpha}}}_I}} \right]=\left[ {\begin{array}{*{20}{c}} {\alpha _{1}^{1}}&{\alpha _{1}^{2}}& \cdots &{\alpha _{1}^{I}} \\ {\alpha _{2}^{1}}&{\alpha _{2}^{2}}& \cdots &{\alpha _{2}^{I}} \\ \vdots & \vdots & \ddots & \vdots \\ {\alpha _{N}^{1}}&{\alpha _{N}^{2}}& \cdots &{\alpha _{N}^{I}} \end{array}} \right]$

After mapping the type parameter values [-1, -0.5, 0, 0.5, 1] to the corresponding class labels [0, 1, 2, 3, 4], the result can be denoted as

${x_l}=\left[ {{{\text{L}}_1},{{\text{L}}_2}, \ldots {{\text{L}}_I}} \right]=\left[ {\begin{array}{*{20}{c}} {l_{1}^{1}}&{l_{1}^{2}}& \cdots &{l_{1}^{I}} \\ {l_{2}^{1}}&{l_{2}^{2}}& \cdots &{l_{2}^{I}} \\ \vdots & \vdots & \ddots & \vdots \\ {l_{N}^{1}}&{l_{N}^{2}}& \cdots &{l_{N}^{I}} \end{array}} \right]$

III Results

A༎ Self-generated dataset construction

Considering the scarcity of open-source electromagnetic scattering datasets and the substantial computational burden and prolonged construction cycles associated with data acquisition through electromagnetic calculations, a self-generating dataset construction method is proposed in this article. The specific procedures are outlined as follows.

Step 1

Generate multiple randomly distributed scattering centers within a certain three-dimensional space. Specifically, the number of scattering centers varies randomly within a predetermined range, and their spatial dispersion is constrained to a certain random interval under the condition of ensuring a minimum distinguishable distance between adjacent centers.

Step 2

For each scattering center, the scattering intensity parameter is assigned a random value within a specified range, while the scattering type parameter is randomly assigned within the bounds outlined in Table I. This process yields the ground truth parameters of the three-dimensional scattering centers. Based on that, the target dataset can be constructed.

Step 3

According to the observation array shown in Fig. 2, the frequency response data of the target under different incident frequencies, azimuth angles, and elevation angles can be calculated by using (3), with random noise introduced.

Step 4

Estimating the parameters of the 1D-GTD model across different azimuth and elevation angles, followed by association and matching. Subsequently, the matching parameters can be used for constructing the training dataset.

It can be seen that various kinds of random variables are introduced in the process of constructing the dataset, aiming to mitigate the influence of fixed parameters on network training and thereby facilitating the development of models with stronger generalization capabilities. Following the aforementioned steps, a dataset comprising 5,000 samples was generated.

B༎ Loss functions and training environment

The Mean Squared Error (MSE) loss function is employed for both the branch 1 and branch 2, which can be expressed as

${L^{MSE}}=\frac{1}{n}\sum\limits_{{i=1}}^{n} {{{\left( {{y_i} - {{\hat {y}}_i}} \right)}^2}}$

where

is the sample size,

${y_i}$

${\hat {y}_i}$

denote the predicted value truth value and the of the ith sample. MSE loss function is widely applied in regression analysis, owing to its computational straightforwardness and effective optimization capabilities. The principal merits of MSE comprise its convex nature and continuous differentiability across the domain, enabling robust convergence in gradient-based optimization methods.

Considering the estimation of branch 3 is fundamentally a classification task, the cross-entropy loss function is employed, which can be expressed as

${L^{CrossEntropy}}= - \frac{1}{n}\sum\limits_{{i=1}}^{n} {\log \left( {{p_{i,yi}}} \right)}$

where

${p_{i,yi}}$

denotes the prediction probability of the ith sample for the real category. The cross-entropy loss function directly quantifies the discrepancy between the predicted probability distribution and the ground-truth distribution, making it a prevalent choice for classification tasks. It provides larger gradients when errors are significant, facilitating rapid model correction, while reducing gradients as predictions approach accuracy, thereby minimizing oscillations and promoting stable convergence.

The training process is finished on an NVIDIA Quadro P6000 GPU (24 GB VRAM) based on PyTorch. The training procedure comprised a pre-training stage and a subsequent fine-tuning stage. In the pre-training stage, the Adam optimizer was applied with a learning rate set to 10⁻⁴ over the course of 10 epochs. For the fine-tuning stage, a lower learning rate of 10⁻⁵ was adopted, and the training spanned 50 epochs. The complete process required approximately 8 hours to finish. The training loss curves of the three-branch neural network, presented in Fig. 6, demonstrate that the loss for each branch has been successfully reduced to below 1. This signifies that the multi-branch architecture has been adequately trained, with the loss function effectively minimized, thereby endowing the model with competent predictive capabilities.

Fig. 6

The training loss of three branch neural network.

IV Contrast Experiment and Discussion

To validate the performance of the proposed method, several comparative experiments were conducted with three well-established algorithms, including 3D-ESPRIT, 3D-MUSIC, and 3D-MEMP.

A༎ Electromagnetic scattering data acquisition

In order to obtain the electromagnetic scattering data required for the experiment, we select a radar target consists of ten scattering centers and the values of parameters are provided in Table 2. Set the incident frequency as

$f=\left[ {10.00,10.02, \ldots 11.00} \right]{\text{GHz}}$

Table 2
Parameters of the ten scattering centers.
NO	${{{x_i}} \mathord{\left/ {\vphantom {{{x_i}} m}} \right. \kern-0pt} m}$	${{{y_i}} \mathord{\left/ {\vphantom {{{y_i}} m}} \right. \kern-0pt} m}$	${{{z_i}} \mathord{\left/ {\vphantom {{{z_i}} m}} \right. \kern-0pt} m}$	${\alpha _i}$	${A_i}$
Scattering center 1	4.41	5.68	-5.22	-1	4.5302
Scattering center 2	5.79	1.85	-5.63	-1	2.3846
Scattering center 3	-3.10	0.66	6.41	1	1.4857
Scattering center 4	6.51	-4.79	6.59	-0.5	4.4741
Scattering center 5	6.40	-0.20	4.20	-1	5.7511
Scattering center 6	-5.01	-1.10	5.82	-0.5	3.1937
Scattering center 7	4.09	6.43	2.18	0.5	4.8276
Scattering center 8	-6.50	4.89	6.08	0	1.9344
Scattering center 9	2.50	3.61	3.40	0.5	3.2279
Scattering center 10	-1.51	2.18	-4.60	0.5	4.5468

It should be noted that, the three conventional algorithms require frequency response data obtained from a small-angle planar array observation, which means that the azimuth and elevation angles of the incident electromagnetic wave should be stepped incrementally. Specifically, set the initial pitching angle

${\theta _0}$

as 90°, the pitching angle step

$\Delta \theta$

is 0.02°, the number of pitching angles N is 20, the initial azimuth angle

${\varphi _0}$

is 90°, the azimuth angle step

$\Delta \varphi$

is 0.02°, the number of azimuth angles K is 20. The total number of incident angles is

$N \times K=400$

. As for the proposed method, set the incident angle as

$\left[ {\varphi \theta } \right]=\left[ {\begin{array}{*{20}{c}} {{0^ \circ }}&{ - {{45}^ \circ }} \\ {{4^ \circ }}&{ - {{44}^ \circ }} \\ \vdots & \vdots \\ {{{356}^ \circ }}&{{{44}^ \circ }} \end{array}} \right]$

the total number of incident angles is 90°.

B༎ Performance of position parameter reconstruction

First, we compared the spatial positions of reconstructed scattering centers with the true positions under the SNR of 0dB, the results are shown in Fig. 7.

Fig. 7

The true three-dimensional spatial position of scattering center and the positions reconstructed by (a) 3D-ESPRIT, (b) 3D-MEMP, (c) 3D-MUSIC, and (d) the proposed method under the SNR of 0dB.

A comparison between the reconstructed scattering centers and the ground truth reveals a high degree of positional agreement. In terms of reconstruction performance, the 3D-ESPRIT and 3D-MEMP algorithms are comparable, and both are generally superior to the 3D-MUSIC algorithm. The proposed method demonstrates the highest positional accuracy, providing estimates that are consistently closest to the actual positions. Subsequently, we evaluate the reconstruction accuracy under SNRs of -10dB to 20dB, at interval is 1dB, using the root mean square error (RMSE) between the estimated values of transversal position x, longitudinal position y, vertical position z and their true values in Table 2 as the quantitative metric. RMSE can be expressed as

$RMSE\left( \varsigma \right)=\sqrt {\frac{{\sum\limits_{{i=1}}^{I} {\left( {{{\hat {\varsigma }}_i} - {\varsigma _i}} \right)} }}{I}}$

where

${\hat {\varsigma }_i}$

${\varsigma _i}$

denote the estimated value and the true value of the parameters for the ith scattering center, respectively. The results are shown in Fig. 8.

Fig. 8

RMSE of the estimated under different SNRs. (a) transversal position x; (b) longitudinal position y; (c) vertical position z.

It can be observed that the reconstruction accuracy of all methods begins to improve when SNR>-10dB and tends to stabilize after SNR>-5dB. The proposed method demonstrates greater stability in reconstructing the parameter z. Overall, the reconstruction errors of the proposed method for the parameters x, y, and z remain at lower levels under different SNRs, indicating superior performance of position parameter reconstruction.

C༎ Performance of scattering intensity and scattering type parameters estimation

The estimation accuracy of the scattering intensity parameters under different SNRs is evaluated quantitatively using the RMSE, the results are shown in Fig. 9.

Fig. 9

RMSE of the estimated intensity parameters under different SNRs.

It can be observed that as the SNR increases, the estimation errors of all methods decrease to varying degrees. Notably, the 3D-ESPRIT algorithm exhibits a more significant reduction in RMSE compared to the 3D-MEMP and 3D-MUSIC algorithms. Overall, the RMSE of the intensity parameter estimated by conventional algorithms generally exceeds 1. The proposed method consistently sustains an RMSE below 0.5 across all tested SNRs, demonstrating its superior estimation accuracy.

Subsequently, we compared the scattering type parameter estimation performance under different SNRs, using the estimation accuracy rate as the quantitative metric, which can be expressed as

$Accuracy rate=\frac{{\sum\limits_{{i=1}}^{I} {C\left( {{{\hat {\alpha }}_i}={\alpha _i}} \right)} }}{I} \times 100\%$

where

$C\left( \cdot \right)$

denotes the indicator function, which equals 1 if the condition

${\hat {\alpha }_i}={\alpha _i}$

is true and 0 otherwise. The results are shown in Fig. 10.

Fig. 10

Accuracy rate of the estimated type parameters under different SNRs.

A clear trend is observed wherein the estimation accuracy of the three conventional algorithms begins to increase significantly once the SNR surpasses 5 dB, ultimately plateauing around a maximum of 80%. The proposed method exhibits superior robustness, with its estimation accuracy starting to improve at an SNR as low as − 10 dB. The accuracy rate reaches 90% when the SNR is ≥ 2 dB and attains complete reliability (100%) at an SNR of 16 dB. These results, obtained under comprehensive testing, confirm that the proposed method consistently outperforms the conventional algorithms in type parameter estimation across SNRs.

D༎ RCS extrapolation accuracy

To further validate the performance of the proposed method, we analyze the extrapolation accuracy of the RCS. Based on the estimated parameters, the far-field RCS can be calculated as

$\sigma =\mathop {\lim }\limits_{{R \to \infty }} 4\pi {R^2}\frac{{{{\left| {{E^s}} \right|}^2}}}{{{{\left| {{E^i}} \right|}^2}}}$

where

${E^s}$

and

${E^i}$

denote the scattering electric field and the incident electric field,

is the far field distance. We extrapolate two sets of RCS data under the SNR of 0dB. The first set is the narrowband RCS of the target with the incident angle

$\varphi =\theta ={45^ \circ }$

, while the incident frequency f sweeps from 10GHz to 12GHz, the results are shown in Fig. 11(a). The second set is the narrowband RCS with the incident pitching angle

$\theta ={45^ \circ }$

and the fixed frequency f = 10GHz, while the azimuth angle

$\varphi$

varies from 100° to 260°, the results are shown in Fig. 11(b). It can be observed that under both conditions, the RCS extrapolation results obtained by the proposed method exhibit the closest agreement with the theoretical values, indicating the superior performance in parameter estimation.

Fig. 11

Accuracy rate of the estimated type parameters under different SNRs. (a)

$\varphi =\theta ={45^ \circ }$

, f sweeps from 10GHz to 12GHz; (b) Fixed frequency f = 10GHz, the azimuth angle

$\varphi$

varies from 100° to 260°..

E༎ Calculation efficiency

To validate the calculation efficiency of the proposed method, we measure the average time required for the four methods to complete 50 operations on the same computational platform, the results are shown in Table 3.

Table 3
Average calculation time for 50 operations.
Method	Average calculation time(s)
3D-MEMP	27.12
3D-ESPRIT	16.45
3D-MUSIC	17.43
The proposed method	1.12

It can be observed that the proposed method exhibits a significant reduction in calculation. On the one hand, the proposed method avoids complex feature calculations. On the other hand, it benefits from the lightweight network architecture. Specifically, the model parameters of network 1, network 2, and network 3 are 0.26M, 0.68M, and 0.07M, respectively.

V Conclusions

In this article, a novel multi-neural network architecture is proposed for estimating the parameters of the 3D-GTD model. A three-branch neural network model with exceptional generalization capability and robustness is trained on a dataset possessing stochastic characteristics generated via a proper self-generating method, which helps to achieve aggregate estimation of the position, type, and intensity parameters of three-dimensional scattering centers. The superiority and effectiveness of the proposed method are validated through multiple experiments. The results demonstrate that, compared to conventional algorithms, the proposed method achieves higher parameter estimation accuracy, exhibits better noise robustness, and significantly improves computational efficiency.

However, the proposed method still has certain limitations. Primarily, the proposed method is based on the 1D-GTD model parameters, which introduces certain errors caused by the processes of parameter estimation and matching. It is necessary to carry out further research on parameter estimation network that operates directly on the original electromagnetic scattering data, avoiding the impact of intermediate processes on network performance.

Author Contribution

Conceptualization, Methodology, Software, Writing Original Draft, Mingrui Qi; Formal analysis, Investigation, Data Curation, Writing Review &amp; Editing, Funding acquisition, Lihua Wu; Visualization, Project administration, Supervision, Rui Cao; Resources, Supervision, Minghao Peng.

Data Availability

All data supporting the findings of this study are available within the paper and its Supplementary Information.

Acknowledgement

The authors thank the National Natural Science Foundation of China [grant numbers 92366301].

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Funding Source

Declaration

This work was supported by the National Natural Science Foundation of China [grant numbers 92366301].

Author contributions statement

Conceptualization, Methodology, Software, Writing Original Draft, Mingrui Qi; Formal analysis, Investigation, Data Curation, Writing Review & Editing, Funding acquisition, Lihua Wu; Visualization, Project administration, Supervision, Rui Cao; Resources, Supervision, Minghao Peng.

Additional information

Accession codes

Data available on request from the authors.

Conflicts of Interest:

No potential conflict of interest was reported by the authors.

Figure 1.

Process of the proposed method.

Figure 2.

Line array observation with co-stepping incident azimuth and elevation angles.

Figure 3.

The architecture of the neural network for position parameter reconstruction.

Figure 4.

The structure of the neural network for scattering intensity parameter estimation.

Figure 5.

The structure of the neural network for scattering type parameter estimation.

Figure 6.

The training loss of three branch neural network.

Figure 7.

The true three-dimensional spatial position of scattering center and the positions reconstructed by (a) 3D-ESPRIT, (b) 3D-MEMP, (c) 3D-MUSIC, and (d) the proposed method under the SNR of 0dB.

Figure 8.

RMSE of the estimated under different SNRs. (a) transversal position x; (b) longitudinal position y; (c) vertical position z.

Figure 9.

RMSE of the estimated intensity parameters under different SNRs.

Figure 10.

Accuracy rate of the estimated type parameters under different SNRs.

Figure 11.

Accuracy rate of the estimated type parameters under different SNRs. (a)

$\varphi =\theta ={45^ \circ }$

, f sweeps from 10GHz to 12GHz; (b) Fixed frequency f = 10GHz, the azimuth angle

$\varphi$

varies from 100° to 260°..

Table 1. Values of scattering type parameter corresponding to typical scattering structures.

${\alpha _i}$	Scattering structure	Scattering mechanism
-1	Tip, corner edge	Corner diffraction
-0.5	Curved edge	Edge diffraction
0	Doubly curved surface, straight edge	Point scattering, specular scattering
0.5	Singly curved surface, top-hat	Corner diffraction
1	Flat plate, dihedral	Corner diffraction

Table 2. Parameters of the ten scattering centers.

NO	${{{x_i}} \mathord{\left/ {\vphantom {{{x_i}} m}} \right. \kern-0pt} m}$	${{{y_i}} \mathord{\left/ {\vphantom {{{y_i}} m}} \right. \kern-0pt} m}$	${{{z_i}} \mathord{\left/ {\vphantom {{{z_i}} m}} \right. \kern-0pt} m}$	${\alpha _i}$	${A_i}$
Scattering center 1	4.41	5.68	-5.22	-1	4.5302
Scattering center 2	5.79	1.85	-5.63	-1	2.3846
Scattering center 3	-3.10	0.66	6.41	1	1.4857
Scattering center 4	6.51	-4.79	6.59	-0.5	4.4741
Scattering center 5	6.40	-0.20	4.20	-1	5.7511
Scattering center 6	-5.01	-1.10	5.82	-0.5	3.1937
Scattering center 7	4.09	6.43	2.18	0.5	4.8276
Scattering center 8	-6.50	4.89	6.08	0	1.9344
Scattering center 9	2.50	3.61	3.40	0.5	3.2279

Table 3. Average calculation time for 50 operations.

Method	Average calculation time(s)
3D-MEMP	27.12
3D-ESPRIT	16.45
3D-MUSIC	17.43
The proposed method	1.12

Yes