## 2 Methods  
### Extraction of Multidimensional Features for Underwater Targets  
#### 2.1.1 Pseudo-Three-Dimensional Spatial Features  
Acquiring sonar images of high-speed moving targets is difficult; however, low-resolution highlight images of these targets can be obtained using high-resolution DOA estimation techniques [9]. These images clearly reflect the key spatial features of the target, including its contour, scattering strength distribution, and relative attitude, thereby facilitating the classification of the target from interference. After the active sonar emits a narrowband pulse signal, the target can be modeled as a coherent sound source consisting of scattering bright spots. Subsequently, the observation data vector for the active sonar two-dimensional array is  
\[\mathbf{x}(t)=\sum_{q=1}^{Q}x_{q}(t)\mathbf{b}(\theta_{q},\psi_{q},\rho_{q})+ \mathbf{n}(t) \tag{1}\]  
Here, \(x_{q}(t)\) denotes the complex amplitude of the \(q\)th narrowband distributed source at time \(t\). The term \(\mathbf{n}(t)\) represents an additive random noise vector that is independent of the signal. Additionally, \(\mathbf{b}(\theta_{q},\psi_{q},\rho_{q})\) signifies the generalized steering vector of the \(q\)th distributed source, where \(\theta_{q}\), \(\psi_{q}\), and \(\rho_{q}\) correspond to the relative horizontal orientation angle, relative pitch angle, and spatial angular distribution parameters of the \(q\)th narrowband distributed source, respectively. The covariance matrix of the observation data is expressed as \(\mathbf{R}=E\{\mathbf{x}_{t}\mathbf{x}_{t}^{H}\}\). Its singular value decomposition (SVD) can be written as follows:  
\[\mathbf{R}=\mathbf{U}_{S}\Sigma_{S}\mathbf{U}_{S}^{H}+\mathbf{U}_{N}\Sigma_{N }\mathbf{U}_{N}^{H} \tag{2}\]  
where signal subspace \(\mathbf{U}_{S}\) consists of the singular vectors corresponding to the \(Q\) largest singular values, and noise subspace \(\mathbf{U}_{N}\) is composed of the singular vectors corresponding to the \(M-Q\) smallest singular values. \(\Sigma_{S}\) and \(\Sigma_{N}\) are diagonal matrices containing the singular values associated with the signal and noise subspaces, respectively. Assuming that the mathematical forms of the spatial angular density functions for all distributed sources are identical and known and that a closed-form solution for the generalized steering vector \(\mathbf{b}(\theta_{q},\psi_{q},\rho_{q})\) can be derived, the generalized MUSIC spectrum for distributed signal sources can then be obtained by exploiting the orthogonality between the noise subspace and steering vector, as shown in Equation (3):  
\[f(\theta_{q},\psi_{q},\rho_{q})=\frac{1}{\mathbf{b}^{H}(\theta_{q},\psi_{q}, \rho_{q})\hat{\mathbf{U}}_{N}\hat{\mathbf{U}}_{N}^{H}\mathbf{b}(\theta_{q}, \psi_{q},\rho_{q})} \tag{3}\]  
where \(\hat{\mathbf{U}}_{N}\) represents the estimated noise subspace. Active sonar systems encounter various types of interference and target signals. Accurately determining the angular density function and parameters of the distribution source, particularly obtaining a precise \(\mathbf{b}(\theta_{q},\psi_{q},\rho_{q})\), presents a significant challenge. To avoid the substantial computational burden associated with searching for these parameters, it is assumed that the angular density function of the source space satisfies a unimodal symmetry constraint. Consequently, the estimation of the center wave arrival direction of the distribution source corresponds to solving the minimization problem presented in Equation (4) [23]:  
\[P(\theta_{q},\psi_{q})=\arg\min_{(\theta,\psi)}\lambda_{\min}(\mathrm{Re}[ \mathbf{\Phi}^{H}(\theta,\psi)\hat{\mathbf{U}}_{N}\hat{\mathbf{U}}_{N}^{H} \mathbf{\Phi}(\theta,\psi)]) \tag{4}\]  
where \(\lambda_{\min}(\cdot)\) denotes the minimum eigenvalue of the matrix, and \(\mathbf{\Phi}(\theta,\psi)\) is a diagonal matrix constructed using the steering vector \(\mathbf{a}(\theta,\psi)\) of the sonar array. The aforementioned assumptions may inevitably result in parameter estimation errors for multi-source targets.