A zero-watermarking algorithm for vector geographic data based on feature invariants

Previous studies on zero-watermarking algorithms for vector geographic data focus on improving the robustness against geometrical attacks, compression attacks and object attacks. However, there are limited zero-watermarking algorithms against projection transformation. In this study, we proposed a zero-watermarking algorithm for vector geographic data based on feature invariants. After any projection transformation of vector geographic data, the number of vertices and relative storage order of objects keep consistent. Therefore, the number of vertices and relative storage order of objects can be considered as the feature invariants. The proposed algorithm consists of three steps. Firstly, according to the relative storage order of objects, the watermark bit is determined by comparing the number of vertices between any two objects. Secondly, the watermark index is calculated by the number of vertices of two objects. Then, a feature matrix is constructed by combining the watermark bit and the watermark index. Finally, the XOR operation is performed between the feature matrix and the scrambled watermark image to generate the zero-watermark image. The experimental results show that the watermark information with NC of 1.0 can be detected from the vector geographic data after any projection transformation, which shows that the algorithm can resist any projection transformation. At the same time, the useful watermark information with minimum NC values of 1.0, 0.911, 0.84, and 1.0 can be detected from the vector geographic data after the geometrical attack, object addition attack, object deletion attack and precision reduction attack. Therefore, the proposed algorithm can effectively against geometrical attacks, object addition attacks, object deletion attacks and precision reduction attacks, showing superior performance compared with previous algorithms.


Introduction
With the rapid development of geographic information science, vector geographic data plays an important role in the national economy, urban national security, and environmental protection Xi et al. 2020;Yang et al. 2020). The production cost of vector geographic data is high, the production cycle is long, and the potential value is high. Therefore, the security of vector geographic data has always been concerned (Hou et al. 2018;Yan et al. 2017). However, with the growing demand for vector geographic data sharing, the illegal use and infringement of vector geographic data frequently occur. Thus, effective technologies are urgently needed to protect vector geographic data. Currently, digital watermarking technology is widely used for copyright protection and traceability of vector geographic data (Lopez 2002;Zhu 2017;Abubahia and Cocea 2017; Communicated by: H. Babaie Peng et al. 2019). Traditional digital watermarking techniques embeds watermark information by modifying coordinate values in the spatial domain (Voigt and Busch 2003;Yan et al. 2011;Wang et al. 2018;Ren et al. 2014) or frequency domain Shen et al. 2009;Xu et al. 2011;Xu and Wang 2010), which will reduce the accuracy and is not suitable for high-fidelity vector geographic data (Wang et al. 2007;Cao et al. 2015;Ren et al. 2020). The emergence of zero-watermarking technology has solved this problem very well. Zero-watermarking technology constructs watermarking information by extracting important features of the data itself without any modification on the original data (Wen et al. 2003;Wang et al. 2012a;Peng and Yue 2015). Compared with traditional watermarking technology, zero-watermarking technology is more suitable for high-precision vector geographic data. At the same time, the zero-watermarking technology also balances the contradiction between watermark invisibility and robustness (Zhou et al. 2021;Wang et al. 2012b).
Currently, most zero-watermarking algorithms of vector geographic data are targeted at the robustness to geometrical attacks, compression attacks, and object attacks. For example, the zero-watermarking image was constructed based on the distance ratio of feature vertices. This algorithm has good robustness against geometrical attacks, compression attacks, and object attacks (Peng and Yue 2015). Li et al. constructed the zero-watermarking image by modulating the slopes of adjacent feature points. This algorithm can effectively against geometrical attacks and vertex attacks (Li et al. 2016). Xi et al. proposed a multiple zero-watermarking algorithm. In this algorithm, feature points and non-feature points are divided firstly, then two zero-watermarking images are constructed respectively according to feature points and non-feature points. This algorithm has a good watermark capacity and overall robustness (Xi et al. 2019). Zhu et al. proposed a zero-watermarking algorithm based on the minimum bounding rectangle (MBR) and singular value decomposition. This algorithm has good robustness against translation attacks, compression attacks, and object deletion attacks ). Ren et al. proposed a zerowatermarking algorithm based on concentric circles . This algorithm has good robustness against geometrical attacks, compression attacks, and object attacks. In summary, these zero-watermarking algorithms mainly focus on the robustness to geometrical attacks, object attacks, and vertex attacks, few algorithms consider the robustness of projection transformation. At the same time, the existing zero-watermarking algorithms for vector geographic data improve robustness and efficiency by block operation, and feature matrix is constructed by features such as distance, distance ratio and angle, which can maintain good performance under common attacks. However, the vector geographic data after the projection transformation attacks will have shape distortions, which will lead to great changes in the mentioned features. Therefore, the previous algorithms cannot extract effective copyright information from vector geographic data after any projection transformation attack.
Projection transformation is often used in the processing and application of vector geographic data, which plays an important role in the operation of vector geographic data (Zhou et al. 2021;Yan et al. 2016). Therefore, under the background that the existing algorithms are difficult to resist the projection transformation attacks and the importance of the projection transformation operation, a zerowatermarking algorithm based on the feature invariants of vector geographic data is proposed. The algorithm has a good robustness to common attacks as well as any projection transformation attack, which provides a new scheme for copyright protection of vector geographic data.
The remainder of this paper is organized as follows. "Preliminary" section is about how to use feature invariants. "Methodology" section is the basic ideas and details of the algorithm. "Experiments" section presents the experimental design. Then "Robustness and Analyses" section describes the experimental analyses and results. Finally, "Discussions" section presents a discussion and "Conclusions" section draws the conclusions.

Preliminary
Before introducing the algorithm in this paper, there are three problems to be explored: (1) what feature invariants do vector geographic data have after projection transformation; (2) how to keep the watermark synchronization by using the feature invariants; (3) how to use the feature invariants to construct the feature matrix. These three questions are the key points of the proposed algorithm in this paper.

Feature invariants of projection transformation
Projection transformation is an important part of map projection and map compilation. It mainly studies the theory and method of transforming one map projection into another map projection. The projection can be divided into four types: the equal area projection, the conformal projection, the equidistant projections, and the compromise projection. All different map projections change the relative position and geometry of the vector geographic data (Li and Stefanakis 2020). How to choose suitable feature invariants for projection transformation of vector geographic data is the key point for the zero-watermarking algorithm. (Banesh et al. 2021). In summary, we find that the number of vertices is a feature invariant for the projection transformation, regardless of any projection transformation. Thus, the number of object vertices is a suitable feature invariant.
At the same time, the relative storge order of objects is also a feature invariant for vector geographic data . In general, this feature invariant is rarely noticed, so this feature invariant is rarely attacked. Furthermore, the relative storage order of objects remains unchanged after geometrical attacks, compression attacks, and projection transformation attacks. Therefore, the relative storage order of the objects can also be used as a feature invariant. Figure 1 shows that the relative storage order of objects keep consistent after object addition and object deletion. Figure 1a shows object addition, where A, B, C and D are the storage order of the original data. When an X is added, the object storage order becomes A, X, B, C, and D. Figure 1b shows object deletion, where when B is deleted from the original data, the object storage order becomes A, C, D, and E. Figure 1 shows that the relative storage order keep consistent in any way after object addition and object deletion. Likewise, the relative storage order of objects is also a stable characteristic under other types of attack. Therefore, we choose the number of object vertices and the relative storage order as the feature invariants. The following is how we use these two feature invariants to design the algorithm.

Rules for generating watermark index based on the number of object vertices
In order to ensure the synchronization of the watermark, the numbers of object vertices of any two objects are operated to calculate the watermark bits in this paper. Firstly, this paper uses a multiplication operation to expand the range of values. Secondly, the square of the previous object's vertices number is multiplied by the next object's vertices number. Finally, a remainder operation is performed between the obtained result and the original watermark length to obtain the watermark index of these two objects. This can be calculated by the Eq. (1).
where index ij ∈ [0, L − 1] , N i and N j mean the number of vertices of any two objects in the relative storage order, and L means the length of original watermark information.

Rules for generating watermark bit based on feature invariants
Since the relative storage order and the number of object vertices are two feature invariants to the projection transformation, it is crucial to construct the watermark by using these two feature invariants. In general, the watermark is a binary sequence generated by 0 and 1. The watermark bit is determined by judging the number of vertices of any two objects. If the number of vertices of the former object is larger than the number of vertices of the latter object, the watermark bit is 1, otherwise the watermark bit is 0. This way, we generate a watermark sequence containing only 1 and 0. The length of the watermark information depends on the number of objects. It can be expressed as Eq. (2). where n indicates the number of objects and m means the length of the watermark information. The sum needs to be calculated with a lower limit i is 2, upper limit n and the sum part is i − 1 . For example, when n is 4, the length of watermark information is 6.

Basic idea
In order to solve the three problems mentioned in "Preliminary" section, the following three aspects are proposed to design the zero-watermarking algorithm: (1) The number of object vertices and the relative storage order are used as two feature invariants for the projection transformation.
(2) The numbers of two objects vertices are used to determine the watermark index. (3) The numbers of two objects vertices in the two relative storage orders are used to generate the watermark bit.
The scheme mainly includes three parts: the generation of original watermark information, the construction of the zerowatermark image and the detected of the zero-watermark image. The detailed descriptions of each part are shown below.

The scrambled of original watermark information
In order to improve the security of the original watermark information in practical applications, the watermark information is preprocessed by scrambling transformation before the construction of zero-watermark image. In this paper, we choose the Arnold transform for watermark scrambling Saadi et al. 2019). The calculate of the Arnold transform can be mathematically expressed as Eq. (3).
In Eq. (3), (x, y) is the coordinate of each pixel in the original watermark image; (x � , y � ) is the coordinate of the corresponding pixel after the Arnold transform; N is the image size, when the length and width of the image are equal, then N takes this value, when the length and width are not equal, N 1 and N 2 take the length and width respectively, as shown in Eq. (4).
Take a watermark image of size 32 × 64 as an example. Figure 2 shows original watermark and the watermarks with 1, 12, 24 and 32 times of transform, respectively. As can be seen from Fig. 2e, when the image is transformed 32 times, the scrambled image is restored to the original image. Thus, the period of the Arnold transform of this image is 32 times.

The construction of zero-watermark image
The process of constructing a zero-watermark image is shown below. Firstly, the numbers of any two objects' vertices in the relative storage order are counted to determine the watermark bit. Secondly, the watermark index is determined by performing several mathematical operations on numbers of any two objects vertices. Thirdly, the feature matrix is constructed by combining the watermark bit, watermark index and voting principle. Finally, the XOR operation is performed on the scrambled watermark image and the feature matrix to generate a zero-watermark image. Figure 3 shows the flowchart of the proposed method.

The detection of zero-watermark image and copyright information
The detection of zero-watermarked image and copyright information is roughly the same as the construction of zerowatermarked image.

Dataset
In order to evaluate the performance of the algorithm, we performed experiments by Python3.7 in Windows10. The proposed algorithm in this paper is suitable for polylines and polygons. Four vector geographic datasets with the same scale of 1:400,000 are used. The format of the four experimental data is the ESRI Shapefile. These four datasets are different types of vector geographic data from different regions, including rivers, highways, administrative divisions of prefectural boundary, and administrative divisions of county boundary. Figure 4 shows the four pieces of experimental data. Table 1 shows the basic information of experimental data, including data name, data type, number of objects, number of vertices, and file size.

Experiment design and implementation
This section is to perform attack experiments with different types and intensities to verify the robustness of the proposed algorithm. The possible attacks in the actual use of vector geographic data are listed, including geometrical attack, object attack, precision reduction attacks, and projection transformation. The proposed algorithm focuses on these attacks as an important aspect of the robustness evaluation. The geometrical attack includes translation, scaling, and rotation. The object attack includes object addition and object deletion. There are four types of map projection on projection transformation attack, including the equal area projection, the conformal projection, the equidistant Algorithm 1 Construction of Zero-watermark Image projection, and the compromise projection. A qualified zerowatermarking algorithm can against the above attacks. The detailed attack settings for every type of attack will be given later. Table 2 shows the corresponding sub-types of different attack types.
Three algorithms are selected for comparative experiments, and the details of the three algorithms are referred to as (Wang et al. 2012b;Zhu et al. 2021;Ren et al. 2021). The algorithm proposed by Wang constructs concentric rings of all vertices and then constructs zero-watermark image by quantifying the number of vertices in each ring. The algorithm proposed by Zhu divides the vector geographic data into average blocks and calculates the singular values in the X and Y directions of each block respectively to construct zero-watermark image. The algorithm proposed by Ren draws several concentric rings according to the size of the watermark image and converts the angle of feature points into several zero-watermark images.
In order to demonstrate the robustness of projection transformation attacks, we select four different data for experiment and display the results in the projection transformation attack. In view of the length limitation of paper, other attack experiments only show the experimental results of data (D).

Geometrical attacks
Geometrical attacks generally include translation, scaling, and rotation. In translation attack experiments, the experimental data is simultaneously translated in both X and Y directions. The translation distance ranges from 0 to 3000 m with interval of 500 m. Scaling attacks are divided into uniform scaling and non-uniform scaling. Uniform scaling means that the scale factor is the same in both X and Y directions, denoted as Sx,Sy . Non-uniform scaling means that the scale factor is not the same in the X and Y directions, which also leads to shape distortion. In the rotation attack experiments, the experimental data is rotated clockwise from the data center, the rotation angle from 0° to 360° with a step of 60°. The partial attack results are shown in Fig. 5.

Object attacks
Object attacks include object addition and object deletion. Object attacks change the number of objects, without effects on the basic shape of objects. The addition rate and deletion rate are equal from 0 to 50% of the original object number, with the step of 10%. Figure 6 shows object addition and Fig. 7 shows object deletion.

Precision reduction attacks
Precision reduction attack refers to an attack by reducing the accuracy of data without affecting the use of data. The effective decimal digit of vector geographic data used in this paper is 11 digits. Each attack is reduced by one digit until there are no decimal digits.

Projection transformation attacks
A good zero-watermarking algorithm should be against any projection transformations. To evaluate the robustness of Algorithm 2 Detection of Zero-watermark Image and Copyright Information the proposed algorithm to projection transformations, four different projection types are employed. Three subtypes are selected for each type of projection, for a total of 12 projection transformations. This is also to verify the feasibility of the proposed method. Table 3 gives the specific information of projection transformation types and subtypes. Figures 8,  9, 10 and 11 are visualizations of four projection transformations, taking the data (D) as an example.

Evaluation
It is necessary to compare the detected watermark image with the original watermark image. Normalized correlation (NC) is often used to evaluate watermark quality. In this paper, we set the threshold value of NC as 0.80. If the value   of NC is higher than 0.80, the copyright information will be detected successfully, otherwise, it will not be detected. The mathematical equation of NC is as follows: where W(i, j) and W � (i, j) denote the original watermark image and the detected watermark image respectively; M × N is the size of the watermark image.

Robustness and analyses
The robustness of geometrical attacks Figure 12 shows the results of geometrical attack. The algorithm proposed by Wang, Zhu, and Ren have the same effect as the proposed algorithm in this paper in terms of translation attacks. In addition, both the comparison algorithm M × N and the proposed algorithm have good robustness against uniform scaling attacks. However, at the non-uniform scaling attacks, the robustness of the comparison algorithm decreases at the X direction both decreases and increases, which cannot meet the requirements. The reason is that nonuniform scaling leads to changes in the feature information in the grid division and concentric rings. In terms of rotation attacks, the algorithm proposed by Zhu is not robust to rotation attacks, because the grid division cannot against rotation attacks. The algorithms proposed by Wang and Ren have good robustness against rotation attacks because of the advantages of concentric rings. As can be seen from Fig. 12, the NC detected from the geometrical attack is always kept as 1, because the two feature invariants used in this paper do not change. Therefore, the proposed algorithm has good robustness against geometric attacks.

The robustness of object attacks
The results of object attack are shown in Fig. 13. The NC values of object addition and object deletion vary with the ratio of addition and deletion. In the object addition attacks, the NC values detected by Wang, Zhu and the algorithm proposed decreases with the increase of the addition ratio. However, Ren's algorithm is robust to object addition because it generates several zero-watermark images. Object addition attack has certain randomness, and different addition produces different results. In this object addition attacks, the minimum NC values of the proposed algorithm and the algorithm proposed by Wang, Zhu and Ren are 0.965, 0.815, 0.597 and 1.0, respectively. It can be seen from Fig. 13a that the fluctuation range of the algorithm proposed by Wang is large, and the algorithm proposed, Wang's and Ren's algorithm have good robustness in the object addition attacks. For object deletion attacks, it can be seen from Fig. 13b that the NC values of the algorithm in this paper and the other three algorithms will decrease with the increase of the deletion ratio. However, the algorithm proposed and Zhu's algorithm can still detect NC values greater than 0.8 even if 50% of the objects are deleted, which is more advantageous than Wang's and Ren's algorithm. In summary, the algorithm proposed in this paper has good robustness in object addition attacks and object deletion attacks, and is better than the other three algorithms.

The robustness of precision reduction attacks
The precision reduction attack results are shown in Fig. 14. Since the effective decimal places of the data are 11, the following 6 decimal places are discarded from the data under the premise of guaranteeing availability. The algorithm proposed by Wang relies on the number of vertices in concentric rings, and discarding a few decimal places Fig. 6 The original data after object addition attacks. (a) Original data, (b) Addition radio = 10%, (c) Addition radio = 20%, (d) Addition radio = 30%, (e) Addition radio = 40%, (f) Addition radio = 50% has almost no effect on the calculation of the number of vertices. Zhu's algorithm is also against precision reduction attacks because singular value decomposition can have a good effect on small perturbations of the matrix.
The algorithm proposed by Ren transforms the angle into the feature matrix, but the precision reduction attack is difficult to affect the integer part of the angle. The algorithm proposed in this paper completely depends on the number

The robustness of projection transformation attacks
The result of the projection transformation attacks is shown in Fig. 15. The four figures are the experimental results of four kinds of data. As can be seen from the Fig. 15, the algorithm proposed by Wang has an overall NC value of 0.80, which can meet the needs of projection transformation attacks in most cases. However, it is not robust in individual projection transformations attacks. The algorithm proposed by Zhu is completely different. Since the projection transformation attacks can cause shape distortions, it has a great impact on the grid division. Therefore, the NC value is basically less than 0.80. Therefore, the algorithm proposed by Zhu is basically not have robust to projection transformations attacks. The algorithm proposed by Ren is not robust to projection transformation attacks because projection transformation attacks cause the vertices and vertex angles of concentric rings to change greatly simultaneously. However, the algorithm proposed in this paper only uses the number of vertices and the relative storage order, which is the feature invariants after projection transformation. Therefore, it has good robustness to projection transformation in theory. At the same time, the experiment also proves that the detected NC value of the algorithm proposed in this paper remains at 1 for 12 different projection types. Therefore, it can be shown that the algorithm proposed has a good effect on projection transformation attacks and is more effective than the other three algorithms.

Discussions
The experimental results show that the proposed algorithm has strong robustness in many types of attacks, especially in projection transformation attacks. The proposed algorithm overwhelms the other algorithms. In order to better illustrate the advantages of the algorithm in this paper, the following three aspects are discussed.

Feature invariants of projective transformation
Vector geographic data are deformed to different degrees after projection transformation. For previous algorithms, most of them use features such as distance and angle to construct feature matrices, and these often cannot against the projection transformation attacks. The proposed algorithm against projection transformation attacks by using the number of vertices and relative storage order as feature invariants. Geometrical attacks and precision reduction attacks change the coordinate values of vector geographic data, but do not affect the number of vertices on the objects. For object attacks, the increase and decrease of object does not affect the number of object and the relative storage order of vector geographic data. In addition, the projection transformation also only distorts the data, but does not affect the number of vertices and the relative storage order, which are feature invariants. The experimental results prove that these feature invariants used in the four types of projection transformation attacks are useful. Thus, the algorithm can effectively against any projection transformation attacks.

The analysis of the watermark index and watermark bit
The number of object vertices is an important feature invariant of projection transformation attacks. The watermark index is determined by the number of any two object vertices. However, adding and deleting vertices will lead to errors in the watermark index, which is also a limitation of the proposed algorithm. A feasible method is that we can compress the vector geographic data in advance, so that adding or deleting non-feature points will not lead to watermark index errors. The watermark bit and watermark index can be obtained through the operation of feature points.
Secondly, since each watermark bit is used multiple times, the voting principle are used multiple times in this algorithm. The voting principle can modify the wrong result induced by object addition and deletion and greatly improve the probability of watermark detection. Therefore, the voting principle plays an  important role in object addition and object deletion, which has also been well proved in the experiment.

Applicable data type
In this algorithm, the number of vertices of polyline data or polygon data is used to establish the watermark index between the original watermark and the watermark information. The whole process of generating the feature matrix is determined by the number of vertices. However, it is not suitable for point data. The reason is that a point object has only one vertex. The watermark index is generated in the relative storage order are single. Therefore, the watermark index between the original watermark information and watermark information cannot be constructed effectively.

Conclusions
Currently, limited zero-watermarking algorithms of vector geographic data can against projection transformation attacks, since it is difficult to find the feature invariant of projection transformation. Two feature invariants are introduced  (D) in this paper, namely the number of vertices and the relative storage order. The former is used to determine the watermark index, and the latter is used to determine the watermark bit, which is the basis of the algorithm in this paper. Experimental results show that the algorithm can against any projection transformation attacks and is robust to geometrical attack, object attack, and precision reduction attacks. The proposed algorithm provides a new exploration for zero-watermarking algorithm of vector geographic data against projection transformation attacks. However, the limitation of this algorithm is not applicable to point data. A feasible solution is to construct Voronoi diagrams so that each point has a Thiessen polygon, but the number of vertices of the Thiessen polygon is limited and cannot be completely applicable. This will also be further explored in `further work.