# A Zero-Watermarking Algorithm Based on Scale-Invariant Feature Reconstruction Transform
## 2 Related Work
### Multi-Radius Local Binary Pattern Descriptors  
Generally, constructing a zero-watermark feature matrix requires multiple sources of feature data and diverse computational approaches. If only the reduced redundancy keypoint descriptor matrix is used to construct the target matrix, both the computational form and the source of the feature data would appear to be overly singular, resulting in poor discriminability of the generated zero-watermark images. To enhance the discriminability and robustness of the feature matrix, this paper introduces texture feature descriptors beyond those used for the aforementioned feature data. These descriptors set the keypoint coordinates as the origin and compute appropriate distance neighborhood texture features. Figure 1: The construction process of SIFT feature matrix. It is worth noting that both types of feature descriptor need to satisfy similar properties. When generating texture features, the algorithm needs to ensure that the computed texture feature descriptors possess rotational invariance and compatibility. We improve the traditional LBP algorithm and propose a novel MrLBP algorithm for extracting texture feature information from multiple circular neighborhoods around feature keypoints. Initially, we compute three keypoints (x\({}_{\text{n}a}\), y\({}_{\text{n}a}\)), (x\({}_{\text{nb}}\), y\({}_{\text{nb}b}\)), and (x\({}_{\text{n}c}\), y\({}_{\text{n}c}\)), which are geometrically close to the feature keypoint (x\({}_{\text{n}}\), y\({}_{\text{n}}\)), with distance R\({}_{\text{i}}\), where i \(\in\) {a, b, c}. The initial angle \(\theta_{\text{i}}\), where i \(\in\) {a, b, c}, is determined by the point with the maximum pixel value within a single circular neighborhood. Twelve directional angles are set clockwise, with Ri as the radius, and the pixel values of the sampling points on these 12 angles are compared with the pixel value of the central keypoint to generate 36-dimensional MrLBP descriptors. For a given keypoint (x\({}_{\text{n}}\), y\({}_{\text{n}}\)), its neighborhood pixel positions are calculated using Formulas (1) and (2):  
\[x_{nri}= x_{n}+Ri\cos\left(\frac{2\pi p}{P}+\theta i\right) \tag{1}\] \[y_{nri}= y_{n}-Ri\sin\left(\frac{2\pi p}{P}+\theta i\right) \tag{2}\]  
where P represents the total number of sampling points in a single-radius neighborhood; thus, P = 12. The number of sampling points can be flexibly adjusted according to the actual dimensional requirements, where p denotes the (p + 1)-th sampling point in the single-radius neighborhood, with p \(\in\) [0, 11] and p being an integer. Due to the discrete nature of digital images, the sampling point coordinates in circular neighborhoods are typically non-integer, and their coordinates do not correspond to pixel values in the discrete image. To address this issue, this paper introduces bilinear interpolation to approximate the pixel values of these virtual points. Bilinear interpolation reduces the loss of feature information during information transmission compared to the nearest neighbor interpolation method, preserving the grayscale continuity of the image and better matching the analog gradient relationships in the real world. This facilitates the generation of more accurate texture feature descriptors, as illustrated in Figure 2.  
The pixel values of the four points in Figure 2 are represented by different colors. According to the principle diagram, it can be observed that the pixel value of the virtual point (x, y) is derived from the pixel values of its four neighboring points, weighted based on their geometric distances. The closer the distance, the larger the weighting coefficient, and the larger the corresponding colored area in the image. Suppose the coordinates of the sampling point (x, y) do not correspond to actual points on the 2D image.