Analysis of the ratio expansion watermarking scheme and its improvement

Ratio expansion is a recently proposed, remarkable method for reversible data hiding in 3D meshes. The capacity and distortion are the two most common performance measures for data hiding. This paper analyses the ratio expansion method, and a novel mathematical framework is introduced and described, which proves that the magnitude of the distortion depends on three parameters. By appropriately selecting two of these parameters and a factor called “reference value”, the proposed method achieves reduction of the distortion. Another novel coding and decoding phrase is also proposed to get higher capacity. The experimental results show that this method improves the embedding capacity significantly without affecting the visual quality of 3D mesh model.


Introduction
Due to the recent advances in computing power and transmission bandwidth, three-dimensional (3D) digital content becomes more and more popular in many application domains such as video games, computer animation, virtual reality. As a result, the need for copy protection, authentication and integrity checking of the 3D content emerges. Watermarking is an appropriate approach to address this need. 2D image watermarking is a well-established and studied field for recent years. Difference expansion [10], [25] and histogram shifting [17] methods are nowadays combined [21], [23], [24], [32], [33] creating a mainstream in the 2D reversible watermarking domain. However, the methods that have been proposed so far cannot be directly applied to 3D models due to their numerous alternative representation structures (point clouds, polygonal meshes, parametric surfaces, etc.) and presentation models (texture, reflective characteristics, etc). For example, Luo et al. [16] apply their algorithms on point cloud models, while Garcia and Dugelay [8] work on textured-based 3D objects.
In general, watermarking schemes can be categorized into irreversible and reversible ones. Irreversible watermarking schemes recover only embedded messages, while the reversible ones recover both distorted host signals and embedded messages. Thus, reversible watermarking schemes are also called invertible data hiding, lossless data hiding, or distortion-free data hiding.
The first 3D watermarking algorithm applied on a 3D mesh model was published by Ohbuchi et al. [18]. Since then, numerous algorithms have been developed in the irreversible watermarking domain [1], [4], [5], [6], [13], [19], [27], [30], [31], [34] which aimed at designing robust watermarking algorithms against all possible attacks. Most of these embedding techniques modify, and hence distort, the host signal in order to insert additional information. Often, this embedding distortion is small, yet irreversible. In many applications, the loss of host signal fidelity is not prohibitive as long as original and modified signals (in our case 3D models) are perceptually equivalent. However, in some application domains, such as medical imaging, military imaging and law enforcement, a slight modification can lead to a significant difference in the final decision-making process. Thus, it may affect the right decision and lead to legal problems. The latter implies that reversible (lossless) data embedding techniques are needed where the host signal must be perfectly recovered.
According to different embedding domains, the reversible watermarking schemes can be classified into spatial-domain-based, compressed-domain-based and transform-domain-based methods. In spatial-domain-based methods [7], [28], [29], the task of data embedding is to modify the vertex coordinates, edge connections, face slopes, and so on. These schemes usually have a low computational complexity. The compressed-domain-based methods [15], [22] are used for embedding data with certain compression techniques involved, e.g., vector quantization. In addition, some of these methods are designed for compressed content of 3D models. Their advantage is that they hide data without decompressing the host model. In transform-domain based methods [16], the original model is transformed into a certain transform domain (i.e., wavelet transform domain in their approach) and then data are embedded in the transformed coefficients.
One of the first reversible watermarking methods for 3D mesh models was published in 2006 by Sun et al. [22]. This method utilized a vector quantization compression technique and the space saved due to the compression was used to place the watermark. The performance of this approach was related to the efficiency and the distortion caused by the compression method.
Jhou et al. [9] proposed a method based on histogram shifting for 3D reversible watermarking. They firstly calculated the center of the vertices of the model and all vertices were sorted by their distance to the center, in descending order. The last m bits of the coordinate value of every vertex were used to generate a histogram. Then, the watermark embedding was taken place as a normal histogram shifting method for images.
The prediction error expansion (PEE) method [29] predicts a vertex position by calculating the centroid of its traversed neighbors, i.e., the vertices directly connected and having been previously traversed. By comparing the predicted and real positions, the prediction error can be obtained and expanded to embed one-bit value in each modified coordinate. In order to reduce the distortion, prediction errors with magnitude larger than an adaptively chosen threshold are not being expanded. In another relevant approach called ratio expansion (RE) [12], the distance between an anchor vertex and its adjacent vertices is divided by a reference value to calculate a ratio. Every ratio is used to achieve reversible watermarking by exploiting a modulo arithmetic operation. These two approaches show superior performance compared to the previously mentioned methods and will be used in this paper as a reference for evaluating the proposed scheme. In general, RE outperforms PEE in terms of perceptual quality but has smaller embedding capacity.
In this paper, we propose an improved RE-based method that can increase the embedding capacity and decrease the distortion. The former is achieved by proposing a novel encoding and decoding phrase to get more vertices to embedding. Concerning the latter, mathematical formulae are derived, and parameters that affect the distortion are calculated and experimentally verified. Extensive experimental results performed prove that the proposed scheme can significantly improve the performance of the existing RE [12] methods in terms of both the distortion and the capacity.
The rest of this paper is organized as follows: Section 2 briefly describes the existing ratio expansion algorithms. In Section 3, the improved ratio expansion method is introduced. Then, the expected distortion of the ratio expansion algorithm is estimated, and experiments are conducted to verify it. Experimental results are given in Section 5. Finally, conclusions are drawn in Section 6.

The ratio expansion method
In 2010, Ji et al. [12] proposed a RE scheme for reversible watermarking. In their scheme, one "anchor" (or "center") vertex v 0 and its K adjacent vertices v i can be considered as an embedding unit. One of ICAACE 2020 Journal of Physics: Conference Series 1570 (2020) 012059 IOP Publishing doi:10.1088/1742-6596/1570/1/012059 3 the anchor's adjacent vertices is selected to be the reference vertex v 1 , if it has the smallest index among all neighbouring vertices in an embedding unit. A ratio R of the L 2 -distance D from the center vertex to an adjacent vertex to the distance from the center vertex to the reference vertex is then computed as: Every ratio can be expanded to embed n bits of data (in general, n is at least 1). For each embedding unit with M vertices, the center vertex and a reference vertex should not be modified, thus M−2 vertices can be expanded using the RE method. Therefore, the maximum embedding capacity is • 2 bits. However, every embedding unit should not overlap each other, leading to a not so large embedding capacity of the RE method. Figure 1 shows two valid embedding units. The anchor vertex v 1 is connected to six neighboring vertices: v 2 , v 3 , v 4 , v 5 , v 9 , and v 11 . In this embedding unit, M equals 7 since there are six vertices around the anchor vertex. Thus, we can hide 5n bits of data. The vertex v 2 becomes v 0 as an reference as it has the smallest index. Figure 1. Possible anchor vertices of the RE method. The vertex v 2 cannot be an anchor to form embedding unit since it is a member of the existing unit. For the same reason, v 3 , v 4 and v 5 . The vertex v 6 is not a member of any existing unit, however, its adjacent v 4 is the member of the existing unit, so it should be skipped too. For the same reason, we skip vertices from v 7 to v 13 . The vertex v 14 can form an embedding unit with three neighbouring vertices: v 6 , v 12 , and v 13 . In this embedding unit, M is 3. Even though there are fourteen vertices in figure 1, the embedding capacity is 6n (i.e., (5 + 1)n) bits. Each embedding unit is separated by dashed lines depicted in figure 1.
Two parameters are used in RE: n that determines the number of bits to be hidden and m that specifies the embedding position. By assuming that the secret data (watermark) should have k units W1,W2, ...,Wk, then Wi can be one of a 2 n -ary numbers, i.e., W i ∈ {0, 1, 2, ..., 2 n − 1}. Before expanding R (see equation (1)), we need to know m so as to specify the embedding position in R.
Two distinct steps are followed in the RE method: In the first step, the ratio R is divided by 2 n , and then the quotient is split into two parts according to the position m. The first part is the embedding part R e , and the second is the recording part R r , which are computed as follows: where T m (ꞏ) is a function that truncates the part right to the position m. Embedding is achieved by: , , (7) where is the ratio from the watermarked 3D meshes. The hidden message (watermark) W i can be extracted using: where M(x, y) is the remainder of x divided by y, and is the embedding part of . The function M(x, y) is similar to the modulus operation. The model can be recovered through:

Proposed method
In this section, we firstly introduce the mathematical framework of the distortion caused by ratio expansion embedding. The framework shows that the distortion relates to three parameters, that is, "reference value", embedding position m and n. Then, the proposed method selects the shortest edge among the edges connected to the center vertex as the reference value to decease the distortion. Moreover, a new encoding and decoding phase are proposed to improve the capacity.

3.1Mathematical Expectation of the Distortion for the RE
In [12], one pixel and its all neighbors are considered as an embedding unit, and the neighbor with the smallest index among all neighbors will be selected as the reference. The selection of reference is not so reasonable. In this section, we try to deduce PDF (probability density function) of distortion caused by one ration expansion embedding, and then obtain the mathematical expectation of the distortion. Based on the expectation, we can select the references and config the other parameters. We firstly review the symbols. R is the ratio before data hiding and is the one after data hiding. There are two parameters, m and n. m is the position in R where the data is embedded. The secret unit is embedded to m right to decimal point of R. n determines the ability for one expansion embedding. For example, if m is 4 and R is 1.12345678, then secret number is embedded to the place of number 4; if n is 2, then we can embed one number of "0, 1, 2, 3" to R for one embedding. W i is the secret number we want to embed, and it can be one of "0, 1, 2, … , 2 n−1 ". R e and R r is the embedded part and recording part of R, respectively. The definitions are in equation 3.
is the embedded part of , please refer to equation 4. Visualization of these symbols can be found in figure 2.
We still have to define three more symbols. Let d denote the distortion measure caused by an expansion and r denote the remainder after dividing by 2 n , that is: where mod is the modulus after division, and R [m+1,∝) is defined as follows: ,∝ • 10 .
An example is made. Assume R is 1.12345678, m is 4, n is 2 and W i is 3. In the ratio expansion, R is firstly divided by 22, and the result is 0.280864195. We can obtain that R e is 0.2808 and R r is 0.000064195, then R e will be multiplied by 2 2 and added by W i at "position 4" to form , that is, 0.2808 × 4 + 3 × 0.0001, so is 1.1235. Rr is appended to to form , that is 1.123564195. In this embedding, r and R [m+1,∝) can be obtained from R. r is 1.12345678 • 10 , 4 , which is 2, and R [m+1,∝) is 0.5678.  r, W i and R [m+1,∝) are mutually independent. In the following, we firstly deduce function , , 1, ∝ , which is in equation (15), and then compute p d .
The distortion caused by one expansion embedding is expressed as follows: where v f is the reference value. Substituting equations (3) and (4) with equation (12), we obtain Then, according to equation (3), R r can be computed as Finally, we can draw Let we define ∇ as follows: For the sake of brevity, we replace a 1 , a 2 , and b with W i , r, and R [m+1, ∝ ) , respectively. The probability density functions p a1 and p a2 of the random variables a 1 , a 2 can be expressed as P(x) in the following based on the above distributions: and b as Then, the probability density function of a 2 + b can be easily derived based on Lemma 1.
Lemma 1. Let X and Y be independent random variables having the respective probability density functions f X (x) and f Y (y). Then the probability distribution function f Z (z) of the random variable Z = X + Y can be given as follows: Proof: see [20].
According to Lemma 2, we can get the probability density function of • as Lemma 2. If X ∼ F X , then F X (x) = Prob{X < x} for x < X < ̅ . Let Y = aX for ax < Y < a ̅ ; then F Y (y) = Prob{Y < y} = Prob{aX < y} = Prob{X < y=a} = F X (y=a). Therefore, F Y (y) = F X (y=a) for ax < Y < a ̅ . Proof: See [20]. Using Lemma 1 again, we can derive the probability density function of |∇| as follows: The probability density function figures of a 1 , a 2 , b, • and ∇ are shown in figure 8.  Figure 3. Probability density function of the ratio expansion scheme. Then, the mathematic expectation of |∇| can be computed according to its definition, and the result is given as follows: Equation (15) shows 10 • |∇| • . According to the linearity of the expectation operator, the expectation of d is given as follows: This equation implies that:  The distortion d is proportional to 10 −m and v f .  d is an increasing function of n.

3.2Verification
In order to further verify the conclusions of the previous subsection, we use the Bunny as a test model, choosing different values for the parameters m and n, and D r . A standard metric, the signal-to-noise ratio (SNR), is used as a performance measure.
The first experiment is conducted to identify the effect of the reference values. In this experiment, the shortest edge among the edges connected with the center will be chosen as the reference for every embedding step. Figure 4 verifies that the method with new reference produces less distortion. when other parameters are fixed. In table 1, we report how parameters n and m affect distortion. In this experiment, we embed 3,000 units of secret data into Dragon, Happy, and Bunny with different parameter values of n and m, respectively. Table 1 shows that smaller n and larger m produce smaller distortion. As a result, we select the shortest adjacent edge among the three sides of a triangle as the reference value, and set m = 2 and n = 3 in all experiments that follow. should be sent to decoder as the header information. The embedding steps are as follows.
Step 1. Set the flags of all vertices to 0 and the set V to null.
Step 2. Traverse each vertex of the model. For each point v i , set the set V i to null as initial value. Traverse each adjacent vertex of v i , and if the flag of the adjacent vertex is 0, add this vertex to V i . If the flag of v i is 1 or the cardinality of V i is less than 2, set V i to null; else the flags of v i and the vertices in V i are all set to 1.
Step 3. For every vertex v i in V, if V i is not null set, each distance from v i to the vertex in V i is computed. The reference value is the minimum distance, which will respectively divide other distances, and the ratios will be saved. Step 4. For the results obtained from step 3, we embed watermarks by ratio expansion, and then the ratios are changed.
Step 5. According to the changed distance ratios, relevant points are shifted along certain direction, finally forming a watermarked model.

3.3.2Decoding
Phase. In The restoration of original model and extraction of watermarks is just reverse to the procedure of data embedding. Steps are as follows.
Step 1. Follow the same way as steps 1 and 2 did in embedding. Not only the flag values of all vertices are obtained, but also the set queue V i is attained, where ∈ 1,2, ⋯ , , and lv is the index of the last embedded vertex.
Step 2. The decoding uses the opposite direction of the encoding, from the last embedded vertex to the first vertex. For every vertex V i , if V i is not a null set, the ratios are obtained as step 3 does in embedding, and then the flags of v i and the vertices in V i are all set to 0; else go to previous vertex, and do step 2 again.
Step 3. For the ratios obtained in step 2, we extract the watermarks, and recover the model by ratio expansion method.
Step 4. Go to previous vertex and then do step 2 again until finishing processing the first vertex.

Experiments
In this section, we firstly demonstrate the visual quality of the proposed method, and then we compare our method with the work in [10] in terms of embedding capacity versus distortion. The test models are shown in figure 5. In the experiments, the parameter values m = 3 and n = 2 are used.  Figure 6 depicts the achieved embedding capacity versus distortion of the proposed method compared against the RE method. It is obvious that the proposed method clearly outperforms the RE method.  Table 2 shows the embedding capacities of the RE and the proposed method. It is evident that the proposed method allows for much greater embedding capacity in all four test models.

Conclusion
In this paper, the ratio expansion based watermarking method is analysed and improved. In order to find the minimum distortion for ratio expansion embedding, an expected distortion measure was modelled, and several experiments were conducted to verify the model. Based on this model, the minimum distance to the adjacent vertices is selected as a reference value, but not the minimum vertex index in [12]. Moreover, a new encoding and decoding phase is also proposed, so that more vertices are used to embed data. As a result, the embedding capacity increases. The experiments show that the proposed scheme can improve the performance on both the capacity and the distortion, and the distortion model for ratio expansion is current.