## 2 QIM and Multiple QIM  
Watermark algorithms generally involve three steps: watermark generation, watermark embedding, and watermark detection. The robustness of the algorithm is usually used to evaluate the quality of watermark algorithms. Robustness pertains to the capacity of the embedded watermarks to retain their attributes in the cover data following data processing. In order to ensure the robustness of a watermarking algorithm, as discussed in Section 1, scholars often focus on designing special mechanisms during the embedding stage, while the proposed algorithm employs a novel embedding approach during this stage. Quantization index modulation (QIM), achieved by modulating the host data with the embedded information, has become a popular watermark-embedding algorithm because it is a method that rejects host interference and has provably good performance in terms of rate distortion robustness [39]. The vector geographic data comprise the fundamental units of points, polylines, and polygons, represented by coordinate vertices. These vertices are utilized to embed watermark bits for implementing the watermarking algorithm. The QIM technique modulates an index with the watermarks and subsequently quantizes the coordinate vertices of the vector geographic data into different quantization intervals based on the associated quantizer. The watermarks are divided into meaningful and meaningless watermarks. In the proposed algorithm, the meaningless watermarks, pseudorandom binary sequences comprising \(1\) and \(-1\), which distribute randomly, were used to embed more watermarks. The random \(1\) or \(-1\) is called the watermark bit. The vertices coordinates of the cover vector geographic data are divided into several intervals according to the change in the watermark bits, which is referred to as QIM. Figure 1 shows the rule of QIM. The \(X\) values of the vertices are quantized into \(-1\) and \(1\) intervals, where \(s\) represents the quantization step size. The vertices coordinates of the cover data are denoted as \(vc\), in accordance with the watermark embedding rule:  
\[\left\{ \begin{array}{l}vc=vc+s\;if\;\left(\left(\left(vc\%(2*s)\right)<s\right) \&\&\&(wb==1)\right)\\ vc=vc-s\;if\;\left(\left(\left(vc\%(2*s)\right)\geq s\right)\&\&(wb==-1)\right), \end{array} \right. \tag{1}\]  
where \(wb\) is the embedding watermark bit. Similarly, the watermark extraction rule is followed:  
\[\left\{ \begin{array}{l}wb==-1\;if\;\left(vc\%(2*s)\right)<s\right)\\ wb==1\;if\;\left(vc\%(2*s)\right)\geq s)\,.\end{array} \right. \tag{2}\]  
The above represents the QIM technique for a single watermark algorithm. By adhering to the rules of the single quantization index, it becomes possible to repeatedly embed one watermark within the cover data, thereby enhancing its robustness. Considering the characteristics of the single quantization index, the intervals can be quantized many times based on the original quantization index rule to embed multiple watermarks. The multiple quantization index includes the up-down and down-up quantization indexes, according to different quantization orders. In the up-down quantization, half of the quantization step size is used as the next one. In Figure 2a, 0\(\sim\)2s is the interval to be quantized, and the first step size is \(s\). The second quantization step size is \(s/2\), as shown in Figure 2b. The third quantization step size is \(s/4\), as shown in Figure 2c. In this way, this process goes on until multiple watermarks are embedded. The rules of watermark extraction are shown in Table 1, corresponding to the up-down multiple quantization mechanism. The next quantization step size is twice the original quantization step size in the down-up quantization mechanism, as shown in Figure 3. In Figure 3a, 0\(\sim\)8s is the interval to be quantized, and the first step size is \(s\). The second quantization step size is \(2s\), as shown in Figure 3b. The third quantization step size is \(4s\), as shown in Figure 3c. The process continues in this way until multiple watermarks are embedded.