## 4 Experiments and Analysis
### Robust Experiments and Analysis  
Robustness is the most significant difference between the perceptual hash algorithm and cryptographic hash algorithm. For HRRS images, robustness means that the perceptual hash values of the images with the same content should be the same or very similar. The important goal of our algorithm is to improve the robustness of the algorithm while maintaining the tamper sensitivity. In this study, we use the modified U-net model to learn the previously constructed training samples based on deep learning theory. If the corrected HRRS images (including geometrically corrected and radiation corrected) have undergone content-preserving operations, such as digital watermark embedding or format conversion, the perceptual hash values of the original image \(l_{o}\) and the processed one \(l_{p}\) should satisfy:  
\[Dis(PH(l_{o}),PH(l_{p}))<T \tag{4}\]  
where \(T\) is the pre-determined threshold.  
Firstly, we take two content-preserving operations, watermark embedding and data format conversion, as examples to test the robustness of our algorithm. For data format conversion operations, we take the conversion between TIFF and BMP as an example. For digital watermark embedding, the least significant bit (LSB) is taken as an example. Here, the pre-determined threshold is set as 0. For the six selected HRRS images, we take the percentage of the divided images whose perceptual hash value has not changed to describe the robustness of the algorithm. As shown in Table 1, all of the perceptual hash values of the divided images have not been changed. In other words, our algorithm can keep well robust to data format conversion of TIFF to BMP and LSB watermark embedding. In contrast, cryptographic hash algorithms treat the above manipulation as illegal operations and cannot perform well for the authentication of HRRS images.  
Figure 10: Comparison between the robust features extracted by U-net and the features extracted by Canny operator: (**a**) Examples of the original HRRS images, (**b**) edge detected by Canny operator, (**c**) robust edge detected by the modified U-net model.  
Next, we test the robustness of the algorithm to data compression operations, which is compared with the algorithm in [10]. Here, the lossless compression (PNG compression), 95% JPEG compression, and 99% JPEG compression are taken as examples. In [10], the threshold \(T\) is set to 0.20. Here, to make it easier to compare the results, both of the thresholds of this algorithm and the algorithm in [10] are set to 0.10. The proportion of images that the _Dis_ is greater than \(T\) are shown in Table 2.  
It can be seen from Table 2 that the robustness of the proposed algorithm to data compression operations is improved compared to the algorithm in [10], with an average increase of 10%. The robustness of the algorithm can be adjusted by changing the algorithm parameters such as \(T\). It should be pointed out that robustness only makes sense if the algorithm can effectively detect the malicious tampering of HRRS images. If robustness is excessively emphasized, sensitivity may be directly affected and some tampering of HRRS image may be impossible to identify, which affects the authentication accuracy of HRRS image. Since the ability of conventional image perceptual hash algorithms to detect tampering is insufficient to meet the authentication requirements of HRRS images, the robustness of the proposed algorithm is not compared with them.  
In addition, the robustness of the proposed algorithm depends largely on the production of the training sample. If the training sample deletes too many edges, the robustness will be enhanced, but it will affect the tamper sensitivity of the algorithm.  
It should be noted that the authentication object of our algorithm is the corrected (including geometric correction and radiation correction) HRRS image, the rotation and scaling robustness are out of our consideration.