## 3 Proposed Scheme
### Authentication Stage
#### 3.2.1 Perceptual Hash Value Generation Based on Modified U-Net Model  
Figure 6 shows the process framework of the perceptual hash value generation for HRRS images, which consists of two main stages. (1) A modified U-net-based robust edge feature extraction process, which extracts the robust edge feature of HRRS image. (2) A principal component analysis (PCA)-based encoder F, which reduces the dimensionality of robust features and then generates perceptual hash value PH for HRRS image. Detailed steps are listed as follows:  
(1) Taking the characteristics of multi-bands of remote sensing images into account, bands of HRRS image are fused together to generate the input data for our modified U-net model. For an image with \(N\) bands, the fusion model used in this paper is as follows:  
\[F(x,y)=\sum_{k=1}^{N}\alpha_{k}R^{k}(x,y) \tag{1}\]  
where \(R^{k}(x,y)\) and \(F(x,y)\) respectively denote the pixel values of the \(K\)th band \(R^{k}\) and the fused image \(F\) coordinates (\(x,y\)), \(\alpha_{k}\) denotes weighting factor. In different applications, the importance of different bands is often different, so the determination of the weighting factor \(\alpha_{k}\) should be based on the specific application. For example, in our previous study [42], we used the information entropy of the band to determine the weighting factor for TM images. For HRRS images with three bands, we set the weighting coefficients as 0.30, 0.59, and 0.11 respectively according to the human eye's visual characteristics. For single-band HRRS image, this band fusion operation is not required. (2) Robust edge feature extraction. The trained U-net models can be directly used to extract the edge feature of the pre-processed HRRS images. However, due to the limitation of the graphics memory, the fused HRRS image is normalized by interpolation algorithm (in this paper, we take bilinear interpolation as an example) into an image of \(M\times M\) pixel size (in this paper, \(M\) = 256). Then we take the normalized image as input data for the trained U-net model, the prediction of the image resulted in a \(256\times 256\) labeled image, which is the robust edge image and is denoted as \(E\). Unlike the result of the Canny operator, the labeled image \(E\) is a grayscale image rather than a binary image. (3) PCA-based encoder. Compactness is one of the most important characteristics that perceptual hash algorithms should satisfy, which means the perceptual hash value should be as compact as possible to make it convenient to transfer, storage, and use. In the encoder phase, we did not use a machine-learning-based approach like some other studies. This is because the machine-learning-based encoding method requires a large number of training samples, and it is difficult to build a training sample set that meets the requirements due to the handcrafted robust edge images. To achieve feature compression, we use a PCA-based encoder on the feature matrix. PCA is a common transform that is often used in the numerical analysis of matrices [43; 44; 45]. By performing PCA on the edge feature matrix, we can remove the linear correlation of the matrix element to achieve data compression and noise elimination. Before we perform PCA on the edge feature matrix, the size of the labeled image \(E\) is resampled to \(m_{1}\times m_{1}\) by bilinear interpolation to achieve data compression (in this algorithm, \(m_{1}\) = 64). Figure 6: Framework of the perceptual hash value generation. Then we serialize the resampled labeled image \(\mathbf{E}\) to construct the edge feature matrix which is denoted as \(\mathbf{ME}\). As the labeled image here is a grayscale image rather than a binary image, the serialization process needs to record the decimal value of each pixel. The principal components of the matrix \(\mathbf{ME}\) are then normalized to get fixed-length string, in which the principal components are binarized to a 0-1 sequence, denoted as \(\mathit{PSString}\).