1 Introduction
Data hiding (DH) technology [1] has been used to embed additional data into the cover media for the purpose of authentication, secret communication and ownership protection etc. The classical DH techniques caused permanent distortions to the cover media and hence the original cover media could not be recovered. However, the exact recovery of the cover media is pertinent to some sensitive application fields such as law enforcement, military, medical imaging and astronomy. Reversible data hiding (RDH) is a type of DH which allows to embed additional data into cover media in a way that the cover media can be losslessly restored after the extraction of embedded data. In recent years, RDH has been extensively studied by the research community [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12].
The existing RDH schemes can be mainly divided into three categories: lossless compression, difference expansion(DE) and histogram shifting(HS). Image lossless compression [2, 3, 4, 5] extracts some features of the original image for lossless compression, and additional data is embedded into the reserved space left by lossless compression. Lossless compression based RDH schemes offer low embedding capacity. To improve the embedding capacity, difference expansion (DE) [6, 7, 8] and histogram shifting (HS) [9, 10, 11, 12] based RDH methods emerged. The DE based methods embeds data into the cover media by expanding the difference of neighboring pixel values resulting in high embedding capacity. In HS based methods, a histogram of the cover image is first generated and then the zero points and peak points of the histogram are found. The pixel values between the peak points and the zero points are shifted, and additional data is embedded by modifying the peak points. The embedding capacity of HS based RDH depends on peak points of image histogram. Since, histograms of most natural images are not sharp enough, the embedding performance of HS based RDH is not satisfactory.
With third party cloud paradigm and privacy preserving applications, the demand for privacy protection is increasing. The combination of encryption and RDH technologies play a crucial role in privacy protection. In order to store or share files securely using third party services, the content owner will use encryption technology to transform the original content into unreadable ciphertext before transmission, and then data hider will embed data into ciphertext for data management, authentication and ownership protection. At the same time, recipient wants to be able to recover the original content losslessly after decryption and data extraction. Such privacy preserving scenarios trigger RDHED for managing ciphertext data.
In recent years, some existing RDHED algorithms for embedding additional data into encryption carriers have been proposed, which can be classified into two categories: vacating room after encryption (VRAE)
[13, 14, 15, 16, 17] and vacating room before encryption (VRBE) [18, 19, 20, 21, 22].Zhang [13]
first proposed the RDHED method, which embedded additional data by modifying the pixels of encrypted images and extracted data by using pixel correlation of original images. The original encrypted image is divided into nonoverlapping blocks and embedded data by flipping the three least significant bits (LSB) of half of the pixels in each block. Due to the spatial correlation of the image, the original image block is much smoother than the modified image block, so the recipient used a smoothness estimation function to estimate the texture complexity of each block for data extraction and image restoration. However, the quality of restored image and the accuracy of data extraction are low. Hong et al.
[14] designed a better blocks smoothness measurement scheme, and further reduced the bit error rate by using the sidematch scheme. However, the extraction accuracy rate is still not very satisfactory. In the methods mentioned above, data extraction and images recovery are carried out at the same time. In order to separate data extraction from image recovery, Zhang [15] proposed a separable RDHED scheme based on LSB compression. This scheme realizes that additional data can be extracted directly from embedding room and images can be restored without data extraction. After that, Qian et al. [17] proposed a scheme that uses stream cipher to encrypt the original images. The datahider compresses a series of selected bits taken from the encrypted images to make room for additional data. If the recipient has the embedding key and the encryption key, then the additional data can be extracted correctly and original images can be perfectly restored using distributed source coding.Compared with VRAE algorithms, VRBE algorithms have better performance in reducing data extraction errors and restoring original images. Ma et al. [18] proposed to reserve room for data embedding before encryption. Ma et al.’s method can guarantee that there are no errors in data extraction and image recovery. Cao et al. [21] further explored the correlation between neighbor pixels. Thanks to the powerful representation of the sparse coding, it reserves a large room for data embedding. Most recently, Puteaux et al. [22] proposed a RDHED scheme in which data hider embeds additional data into the pixels of encrypted images through MSB substitution. The recipient can extract additional data from the MSB plane of the encrypted images. After decryption, the recipient uses correlation between adjacent pixels, and reconstructs the original images through MSB prediction.
According to the above introduction, RDH algorithms for images have been extensively studied for many years, but these algorithms cannot be directly applied to other cover media, such as text, audio, video, and 3D models. Because of the wide range of applications of 3D models (mesh models or point cloud models) and its large intrinsic capacity, 3D models are considered potential covers carriers for RDH. However, research on RDH technology using 3D models as cover media is still in initial stage. According to literature, existing RDH methods of 3D models are mainly divided into four domains: spatial domain, transform domain, compressed domain and encrypted domain.
Spatial domain RDH methods [23, 24, 25, 26, 27, 28, 29, 30] embedded additional data into 3D models by slightly modifying vertex coordinates. Transform domain RDH methods [31, 32] embedded additional data into the transformation coefficients of the models. Compressed domain RDH methods [33, 34, 35]
use vector quantization(VQ) for compressing the vertices of 3D models and then embedded data into compressed meodels stream. At present, there are only two papers on 3D mesh models based RDHED
[36, 37]. Jiang et al.’s [37] method mapped vertex coordinates of 3D mesh to integers using scaling and quantization and then encrypted the integer coordinates to obtain the encrypted mesh. Additional data is embedded by flipping several LSBs of the encrypted coordinates. At the recipient side, the marked encrypted mesh is firstly decrypted and then realizes data extraction and mesh recovery by using the designed smoothing measure function. Jiang et al.’s [37] method uses symmetric encryption algorithm and is an inseparable scheme, i.e., mesh decryption and additional data extraction are carried out at the same time. Method [37] has the disadvantages of low embedding capacity, poor recovery mesh quality and larger errors in data extraction. In [36], a twotier RDHED method for 3D mesh models using homomorphic Paillier cryptosystem is proposed, which is more suitable for cloud data management. Due to the large ciphertext expansion and high computational complexity of the Paillier cryptosystem, the method in [36] is not efficient in practice. The separable method proposed in this paper mainly analyzes and compares the performance with [37] based on symmetric encryption algorithm. Experimental results show that compared with the [37] method, our method improves the embedding ability and quality of the recovered mesh, and realizes errorfree extraction of the data.Our method first maps signed floating vertex coordinates values to integer values, analyzes and finds out the vertex index information with prediction error in the original 3D meshes, and then uses stream cipher to encrypt the original 3D meshes. Finally, data is embedded by replacing the MSB of the encrypted vertex coordinates. Recipient can correctly extract the additional data and recover the meshes perfectly from the encryption meshes.
The contributions of our method are summarized as follows:
(1) A separable RDHED method based on MSB prediction for 3D meshes is proposed.
(2) Our method guarantees higher quality of recovered 3D meshes, while achieving higher embedding capacity and freeerror data extraction.
The rest of this paper is organized as follows: The proposed method is presented in Section 2. The experimental results and some discussion are demonstrated in Section 3. And finally, we summarize our work in Section 4.
2 Proposed Method
Our goal is to hide additional data in 3D mesh models by slightly modifying the mesh vertices without changing the mesh topology, while allowing the recipient to correctly extract additional data and perfectly recover the original mesh models. Our method consists of preprocessing, encryption, data embedding, data extraction and mesh recovery. Fig. 1 illustrates the proposed method.
First, the sender analyzes the original 3D mesh to find the vertex index number information with prediction error and records it. Then, the sender uses the encryption key Ke to encrypt the original mesh M to get the encrypted 3D mesh E(M). In the data embedding stage, additional data is embedded into the encrypted mesh E(M) to obtain the marked encrypted mesh E(M)w. After the mesh encryption and data embedding stage, the sender sends the encrypted auxiliary information along with E(M)w as supplementary data.
There are three scenarios for the recipient. For the recipient has only the data hiding key Kw, the recipient can extract additional data from E(M)w. For the recipient has only the encryption key Ke, a highquality recovered mesh can be obtained. For the recipient with encryption Ke and data hiding keys Kw, the recipient can extract the data and recover the mesh.
2.1 Preprocessing
3D mesh models are represented in various file formats such as OFF, PLY, OBJ, etc. The 3D mesh is composed of vertices data and faces data. Vertices data include coordinates data of vertices represented as V= {v 1iN} , where the vertex is represented as v=(v,v,v), and N is the number of vertices. Note that each coordinate v 1 and jx,y,z. F=(f,f…f) represent faces sequence when we traverse the faces data, where f=(v,v,v), M is the number of faces.
The uncompressed vertex coordinates of a 3D mesh are typically represented by 32bit floating point numbers with precision of 6 digits. In most application scenarios, such high precision is not required, and we can perform lossy compression of vertex coordinates according to the recommendation of [38]. According to the different precision m, the corresponding integer value is between 0 and , where m[133]. Normalizing floating point coordinates v to integer coordinates as
(1) 
Where i is the ith vertex, jx,y,z, v is the original set of floating point vertices and is the set of integer vertices.
In the mesh recovery stage, recipient can convert the processed integer coordinates to floating point coordinates by Eq. (2).
(2) 
The value of m corresponds to the bit length (bitlen) of integer coordinates as
(3) 
Different m values correspond to different bitlen, which means that m values affect the quality of the recovery mesh and the time cost of each stage, including encryption, data embedding, data extraction and mesh recovery.
2.2 Prediction Error Detection
The sender traverses all the vertices in faces data of 3D mesh in ascending order, and calculated “embedded” set C and “reference” set R according to topological information between vertices. Sender traverses the first vertex in the face data and add this vertex to C, find its adjacent vertices and add them to R. The “embedded” set C is used to embed additional data, and the “reference” set R is used to recover the mesh without modifying the vertices during the whole process.
As shown in Fig. 2, when traversing the Bunny’s faces sequence, sender first traverses to a vertex numbered 1, and add this vertex to C. Then, sender traverses the faces sequence to find the faces containing 1, and in this way sender find 2, 3, 4, 5, 6 as its adjacent vertices, and then all the adjacent vertices numbered 2, 3, 4, 5, 6 are added to R. For example, if the x coordinates of an “embedded” vertex numbered 1 has the MSB 0. Sender counts the number of 0 and 1 occurrences of the MSB of the “reference” vertex coordinates numbered 2, 3, 4, 5, 6. if the number of 0s is greater than or equal to the number of 1s, the MSB of the “embedded” vertex coordinates numbered 1 is predicted to be 0. We call vertex numbered 1 as the “embedded” vertex without prediction error in C. Otherwise, the vertex index information will be recorded as auxiliary information. After the mesh encryption and data embedding stage, the sender sends auxiliary information together with the mesh to recipient as auxiliary information on extract data and recover the mesh.
2.3 Encryption
The original 3D mesh is encrypted by an encryption key Ke. Firstly, after the vertex coordinates are preprocessed, the sender uses Eq. (4) to translate the integer coordinates to binary.
(4) 
where is a floor function and 1iN and jx,y,z, the bitlen of the coordinate can be obtained by Eq. (3).
Then, the sender uses a stream cipher function to generate pesudorandom bits , and encrypts the bitstream of the original 3D mesh to get the encrypted coordinate binary stream .
(5) 
where stands for exclusive OR.
Finally, we can get the encrypted integral mesh using Eq. (6)
(6) 
where are the integral value of coordinates.
2.4 Data Embedding
To prevent additional data from being detected, the data hiding key Kw is used to encrypt the tobeinserted data. Sender first calculates “embedded” set C and embed the data into the “embedded” vertex without prediction errors. Finally, the MSB of x, y, and z coordinate values of each available vertex is substituted by 1bit. That is to say, each vertex in “embedded” set C can be embedded with 3bits using Eq. (7). After data embedding stage, sender gets the encrypted mesh with additional data ,i.e., marked encrypted mesh E(M)w.
(7) 
Where w is additional data, vC is the vertex after preprocessing and encryption, v is the vertex of marked encrypted mesh.
2.5 Data Extraction and Mesh Recovery
After receiving the marked encrypted mesh E(M)w, since our method is separable, the recipient can use the data hiding key Kw to extract the additional data and use the encryption key Ke to recover the original mesh separately. There are three cases which are as follows:
2.5.1 Extraction with only Data Hiding Key
In this case, the MSB is extracted from the vertex coordinates of “embedded” set C without prediction errors, and then the corresponding plaintext additional data is obtained by using the data hiding key Kw.
(8) 
where v C is vertex of the marked encrypted mesh.
2.5.2 Mesh Recovery with only Encryption Key
With only Encryption Key Ke, the recipient can recover the marked encrypted mesh E(M)w to get the original mesh M. The original mesh M is recovered in two steps : mesh decryption and MSB prediction recovery.
The pseudorandom bits are generated by the encryption key Ke, and used to perform xor function with to decrypt the marked encrypted mesh E(M)w.
(9) 
Where is the binary stream of the marked encrypted mesh, is the binary stream of the decrypted mesh with additional data and u=0, 1…bitlen1.
After decrypting the mesh, the vertex coordinates of R have been recovered. However, in the data embedding stage, the MSB of the coordinates of the set of vertices in C is replaced by the additional data. After the mesh is decrypted, due to the spatial correlation of the original mesh, recipient can predict the MSB of the “embedded” vertex by the MSB of the adjacent vertices around the “embedded” vertex.
For example, the coordinate values of adjacent vertices 2, 3, 4, 5, and 6 have been correctly restored after decryption. Based on their MSB values, the coordinate values of vertex numbered 1 are predicted to be 0 or 1. When predicting the MSB of v, we count the MSB of x coordinates of vertex index numbers 2, 3, 4, 5, 6. If the highest bit 0 occurs more than or equals to the number of times 1 occurs, then the MSB of v is predicted to be 0.
2.5.3 Extraction and Mesh Recovery with Both Keys
If the recipient has both the data hiding key Kw and the encryption key Ke, the recipient can extract the additional data and recover the original 3D mesh perfectly. Note that data extraction step needs to be performed before mesh restoration.
3 Experimental Results and Analysis
We perform extensive experiments in MATLAB R2018b under windows 10 to test the performance of the proposed method. The system configurations are Intel(R) Core(TM) i78700 CPU 3.20 GHz and RAM 8GB. We download 3D mesh models with .off format from The Princeton Shape Retrieval and Analysis Group ^{1}^{1}1http://shape.cs.princeton.edu/benchmark/index.cgi. and those in .ply format from The Stanford 3D Scanning Repository^{2}^{2}2http://graphics.stanford.edu/data/3Dscanrep/.. The additional data embedded in the mesh is a randomly generated 0/1 sequence.
3.1 Geometric and Visual Quality
The data embedding process causes distortion to the original mesh models, which cannot be observed by the naked eye. Therefore, Hausdorff distance and signaltonoise ratio (SNR) are used to measure the geometric distortion of the mesh model.
Hausdorff distance is a measure describing the similarity between two sets of points, which is a definition of the distance between two sets of points. Assuming there are two sets of A=(a,a…a) and B=(b,b…b), the Hausdorff distance between these two sets of points is defined as:
(10) 
Where is the distance between point a of set A and point b of set B (such as L2), p and q are the number of elements in the set. The geometrical distortion of a mesh after adding some noise to the mesh content can be measured by the the signaltonoise ratio (SNR), which is defined as follows:
(11) 
Where ,, are the average of the mesh coordinates, v ,v ,v are the original coordinates, g, g, g are the modified mesh coordinates value, N is the number of vertices.
In order to study the influence of m on the quality of the recovered mesh, we set m from 2 to 9. Fig. 3 shows that if m is 4, a balance can be established between the quality of the recovered mesh and the computational cost of the process. Fig. 4 shows the visual effects of encrypted, data embedded and mesh recovery based on m=4. The difference between the original and recovered meshes is not visible to the naked eye, which means we get a higher quality recovered mesh.
When m is 1, the decimal retention accuracy is too low, resulting in poor quality of the recovered mesh, which is not suitable for most application scenarios and has no practical significance. Therefore, we set m from 2 to 9, and calculated the Hausdorff distance and SNR between recovered mesh and original mesh. As shown in Fig. 5, the Hausdorff distance decreases linearly with the increase of m, while the SNR increases linearly, indicating that the quality of the recovery mesh becomes higher with the increase of m. In other words, at the same m, we get a higher quality recovery mesh than the method [37].
3.2 Embedding Capacity
The embedding rate is measured by the number of bits per vertex (bpv), which is the ratio of the number of embedded bits to the number of vertices in the mesh model. We performed experiments on the Princeton Shape Retrieval and Analysis Group and computed the average embedding rate, average SNR and average data extraction error based on m=4 for the proposed method and Jiang et al.’s method. Table 1 shows the average embedding capacity of the method [37] was 0.35 bpv and SNR was 31.9723. The average embedding capacity of our method is 1.02 bpv and SNR is 62.2365. In the data extraction stage, the Jiang et al.’s [37] first decrypts the marked encrypted mesh, and then uses the spatial correlation to predict the least significant bits (LSB) of embedded vertex coordinates to recovery the original mesh. In order to improve the shortcoming of the error rate in data extraction is large of [37], our method can directly extract additional data from the MSBs of coordinates of encrypted vertices without any error in “embedded” set . As shown in Table 1, the average data extraction error rate of [37] is 4.22, while the extraction error rate of the proposed method is 0. Fig. 6 illustrates the comparison of embedded capacity between the proposed method and Jiang et al.’s method [37] on the test meshes, including Beetle, Mushroom, Mannequin and Elephant meshes.
Method  Embedding rate  Average SNR  Average error 
[37]  0.35bpv  31.9723  4.22 
Proposed  1.02bpv  62.2365  0 
We randomly selected two dense meshes in .ply consisting of tens of millions or hundreds of millions of triangle from the The Stanford 3D Scanning Repository to test the performance of our method, which are shown in Fig. 7. Experimental results of dense meshes listed in Table 2 illustrate that the embedding rate is fairly high and the data extraction error rate is 0. The experiments show the applicability and effectiveness of the proposed method to dense meshes.
Meshes 




Error  
Dragon  871414  437645  1.0420  908013  0  
Thai Statue  7500000  4999996  1.0708  8031000  0 
4 Conclusion
In this paper, a separable RDHED based on MSB prediction for 3D meshes is proposed. Our method highlights not only feasible and efficient RDHED in 3D meshes also a balance between capacity and distortion. Firstly, the sender maps vertex coordinates to integer values, and uses bitstream encryption algorithms to encrypt the 3D mesh. Then, the MSB of “embedded” vertex coordinate is replaced by additional data. Finally, due to the spatial correlation of the original mesh, recipient can perfectly predict the MSB of the “embedded” vertex by the MSB of the adjacent vertices around the “embedded” vertex. Since our method is separable, the recipient can use the data hiding key Kw to extract the data and use the encryption key Ke to recover the original mesh separately. Experiments show that our method has higher embedding capacity and higher quality recovery mesh and errorfree extraction compared with the existing stateoftheart method. However, Considering the selection of the “embedded” set is limited by the connectivity of the mesh, so the embedding capacity is not very ideal. Our next work is to design a more effective method for selecting the “embedded” set while guaranteeing the higher embedding capacity and highquality recovered of the meshes.
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