## 3 Materials and Method
### Proposed Method
#### 3.3.2 Probability Density Extraction Module  
The proposed probability density extraction module aims to capture the global distribution characteristics of raw data, providing precise probabilistic features to the subsequent information entropy fusion module, as shown in Figure 4. Unlike attention mechanisms that focus on feature weighting, this module emphasizes the consistency of data distribution on a global scale. While attention mechanisms typically assign different weights to specific regions of features to enhance task-specific performance, the probability density extraction module models the data's probability distribution to ensure that the obfuscated data retain statistical characteristics similar to the original data. In its design, the probability density extraction module integrates kernel density estimation (KDE) and convolutional neural networks (CNNs) to transform raw data into high-dimensional distribution features. The input data have dimensions \(W\times H\times C\), where \(W=128\), \(H=128\), and \(C=3\), which are chosen to accommodate common image inputs. First, KDE calculates the probability distribution of the data. The formula for KDE is as follows:  
\[\hat{f}(x)=\frac{1}{nh}\sum_{i=1}^{n}K\left(\frac{x-x_{i}}{h}\right), \tag{18}\]  
where \(K(x)\) is the Gaussian kernel function defined as  
\[K(x)=\frac{1}{\sqrt{2\pi}}e^{-\frac{x^{2}}{2}}. \tag{19}\]  
This process generates the initial probability density distribution of the data, capturing its global characteristics. Subsequently, the input data undergo feature extraction through a three-layer convolutional neural network. The first convolutional layer uses kernels of size \(3\times 3\), with 64 channels, a stride of 1, and ReLU as the activation function. The second convolutional layer increases the number of channels to 128, with the same kernel size but a stride of 2. The third convolutional layer uses 256 channels, with a kernel size of \(3\times 3\) and a stride of 2. The computation for each convolutional layer is defined as  
\[F_{l}=\sigma(W_{l}*F_{l-1}+b_{l}), \tag{20}\]  
Figure 4: Architecture of the probability density extraction module. where \(*\) denotes the convolution operation, \(W_{l}\) and \(b_{l}\) represent the weights and biases of the \(l\)-th layer, and \(\sigma\) is the ReLU activation function. After the convolution operations, the feature maps are flattened and passed through the fully connected layers to generate high-dimensional distribution feature vectors. This process is represented as  
\[Z=\phi(WZ_{l-1}+b), \tag{21}\]  
where \(\phi\) is a nonlinear activation function. The final output is a distribution feature vector with a dimension of \(1\times 64\). To validate the accuracy and fidelity of the estimated probability density, several additional validation methods are employed beyond the L_density loss term. In practice, the performance of the KDE-based density estimation is evaluated through its consistency with the true data distribution using the Kullback-Leibler (KL) divergence, which measures the discrepancy between the true and estimated probability distributions. The KL divergence is defined as  
\[D_{\text{KL}}(P||Q)=\sum_{x}P(x)\log\frac{P(x)}{Q(x)}, \tag{22}\]  
where \(P(x)\) represents the true probability density and \(Q(x)\) is the estimated probability density generated by the module. Minimizing the KL divergence ensures that the estimated density closely matches the true data distribution, providing a more robust validation of the global distribution characteristics captured by the module. Additionally, the ability of the module to retain global distribution characteristics is further validated through a data reconstruction process. In this process, new data samples are generated using the estimated probability density and compared to the original data to ensure that the generated samples preserve key statistical features, such as mean and variance.