# Fine Recognition of MEO SAR Ship Targets Based on a Multi-Level Focusing-Classification Strategy
## 4 Fine-Level Focusing and Classification of MEO SAR Ship Targets
### Ship Target Fine-Focusing Based on Local Region Slice  
The fine-refocusing process compensates for defocusing caused by ship rotations induced by ocean waves. The phase errors induced by target rotation exhibit space-varying characteristics. Although the motions of individual scattering points may differ, it is reasonable to assume that scattering points within a small localized region of the target share similar motion errors. Accordingly, the coarsely focused SAR image can be divided into multiple sub-blocks along the range direction based on the structural characteristics of the target. For each sub-image block, high-order phase errors estimation and compensation are performed to achieve local fine-refocusing. The finely focused sub-image slices are then fed into the subsequent fine classification network.  
To estimate phase errors in arbitrary forms without parametric modeling, the phase errors in every frequency bin are directly treated as independent variables, while the image entropy of sub-blocks serves as the objective function for the optimization model in the fine refocusing. The optimization model can be constructed as  
\[\Delta\boldsymbol{\varphi}=\underset{\Delta\boldsymbol{\varphi}}{\text{ argmin}}\;IE\left(\sum_{k=0}^{L-1}G(k,n)e^{j(2\pi\frac{kl}{L}+\Delta\varphi_{k})}\right) \tag{26}\]  
in which  
\[\Delta\boldsymbol{\varphi}=\left[\Delta\varphi_{1}\quad\Delta\varphi_{k}\quad \ldots\quad\Delta\varphi_{L}\right]^{T} \tag{27}\]  
is the azimuth frequency phase error vector to be estimated. Compared to the coarse-focusing process, the fine-focusing procedure requires solving for an \(L\)-dimensional parameter space instead of a single dimension, leading to a dramatic increase in computational load for parameter estimation. The high-dimensional optimization problem is solved using the quasi-Newton iteration method. The gradient of the objective function is given by the following equation:  
\[\frac{\partial IE}{\partial\Delta\varphi_{k}}=-2\text{Im}\left(G^{*}(k,n)e^{-j (2\pi\frac{kl}{L}+\Delta\varphi_{k})}I(l,n)\right)\sum_{l=0}^{L-1}\sum_{n=0}^{N- 1}\left(\text{ln}|I(l,n)|^{2}+1\right) \tag{28}\]  
Frequency-domain phase errors are then estimated via iterative calculation. The DFP (Davidon-Fletcher-Powell) algorithm is utilized to calculate the optimal solution [34; 35]. The DFP algorithm does not require computing the second-order derivatives of the objective function, which exhibits super linear convergence and is widely applied to large-scale optimization problems. Let \(\Delta\boldsymbol{\varphi}_{0}\) be the initial zero vector and \(\Delta\boldsymbol{\varphi}_{p}\) be the vector of the \(p\)-th iteration. \(\mathbf{D}_{0}\) is an identity matrix of size \(L\times L\). Then the searching direction of the \(p\)-th iteration is  
\[\mathbf{d}_{P}=\mathbf  
where \(\mathbf{s}_{p}=\lambda_{p}\cdot\mathbf{d}_{p}\), \(\mathbf{y}_{p}=\
abla\mathbf{J}_{p+1}-\
abla\mathbf{J}_{p}\). The iteration terminates when \(\left\|\
abla\mathbf{J}_{p+1}\right\|\leq\varepsilon\). The obtained final phase vector is utilized to obtain the fine-refocused SAR image slice, which is then input to the second-level classification network.