Mathematical Problems in Engineering

Volume 2015 (2015), Article ID 931672, 14 pages

http://dx.doi.org/10.1155/2015/931672

## An Image Watermarking Scheme Using Arnold Transform and Fuzzy Smooth Support Vector Machine

^{1}College of Computer and Information Engineering, Henan Normal University, Henan 453007, China^{2}Engineering Technology Research Center for Computing Intelligence and Data Mining, Henan 453007, China^{3}Engineering Laboratory of Intellectual Business and Internet of Things Technologies, Henan 453007, China^{4}College of Information Science and Technology, Beijing Normal University, Beijing 100875, China

Received 10 October 2014; Revised 2 January 2015; Accepted 5 January 2015

Academic Editor: Daniel Zaldivar

Copyright © 2015 Lin Sun et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

With the development of information security, the traditional encryption algorithm for image has been far from ensuring the security of image in the transmission. This paper presents a new image watermarking scheme based on Arnold Transform (AT) and Fuzzy Smooth Support Vector Machine (FSSVM). First of all, improved AT (IAT) is obtained by adding variables and expanding transformation space, and FSSVM is proposed by introducing fuzzy membership degree. The embedding positions of watermark are obtained from IAT, and the pixel values are embedded in carrier image by quantization embedding rules. Then, the watermark can be embedded in carrier image. In order to realize blind extraction of watermark, FSSVM model is used to find the embedding positions of watermark, and the pixel values are extracted by using quantization extraction rules. Through using improved Arnold inverse transformation for embedding positions, the watermark coordinates can be calculated, and the extraction of watermark is carried out. Compared with other watermarking techniques, the presented scheme can promote the security by adding more secret keys, and the imperceptibility of watermark is improved by introducing quantization rules. The experimental results show that the proposed method outperforms many existing methods against various types of attacks.

#### 1. Introduction

Application and popularization of multimedia technologies and computer networks have made duplication and distribution much easier for multimedia contents [1, 2]. The digital media (such as video, image, audio, and text) can be modified easily by attackers who can claim their ownership [3]. Then copyright protection of intellectual properties has become an important and challenging task. One way for copyright protection is digital watermarking, which means embedding certain specific information about the copyright holder into the protected media [4]. In the last decade, digital watermarking methods for images from different purposes are usually categorized into two types: robust watermarking and fragile watermarking. And the watermarking techniques can also be classified into two groups based on the domain in which the watermark is inserted: spatial domain techniques and frequency domain techniques [2, 5–7]. The methods of spatial domain have advantages of easy implementation and low cost of operation, but they are generally not robust to geometrical attacks and image processing. Nowadays, there are many transform domain watermarking techniques [8, 9], such as discrete cosine transform, singular value decomposition, discrete Fourier transform, and discrete wavelet transform. In practice, the performances of watermarking methods are further improved by combining two or more transforms [5]. The image encryption using gyrator transform based on two-step phase-shifting interferometry and AT has been proposed in [10]. Liu et al. [11] proposed an image encryption algorithm by using AT and discrete cosine transform. Abuturab [12] proposed the color information encryption in the gyrator transform domain not only based on discrete cosine transform and radial Hilbert phase mask but also based on AT. Sui and Gao [10] presented a color image encryption scheme by using gyrator transform and AT. Chen et al. [13] designed a new image encryption algorithm based on singular value decomposition and AT. Chen et al. [14] offered a watermarking scheme based on Arnold cat map. AT has periodicity and the transform is simple, but the periodicity depends on image size. Moreover, it is the main concern of the watermarking schemes that the embedded watermark should not degrade the quality of carrier image and the inserted watermark must be as much invisible as possible. The distortion of the watermarked image should be negligible without degrading its robustness under attacking conditions. In order to generate the watermarked images of high quality and extract high fidelity watermarks in attacking environments, the overall watermarking procedure depends on a set of configuration parameters which need to be optimized. Most existing image watermarking algorithms take scaling factor as a single value which needs the proper fine tune. However, when each singular value has a different tolerance limit, a single scaling factor is not suitable in case of singular value decomposition. It is known that imperceptibility and robustness are two important issues of image watermarking schemes. Then the tradeoff between imperceptibility and robustness in watermarking problems is viewed as an optimization problem.

Recently, attacks against image watermarking systems have become more sophisticated [15, 16]. A simple solution consists in embedding the watermark in a geometrical invariant subspace, and another strategy for coping with geometric distortions is to insert an additional watermark or template into the carrier image [17, 18]. However, the implementation difficulties hinder the research of image watermarking schemes based on this principle above. Nasir et al. [19] proposed feature-based image watermarking schemes. Wang et al. [20] proposed a feature-based digital watermarking method for halftone images. However, some drawbacks indwelled in current feature-based schemes restrict the performance of watermarking systems. To address the issues, support vector machine (SVM) is introduced to the image watermarking domain. Fu et al. [21] embedded template and watermark into original images; then the output of SVM models was constructed and the watermark was extracted. Tsai and Sun [22] employed the classification technique based on SVMs to extract a watermark in spatial domain. Li et al. [23] used support vector regression for watermark embedding and extracting in spatial domain. Peng et al. [24] proposed an image watermarking method in multiwavelet domain based on SVMs. Tsai et al. [25] presented a robust lossless watermarking technique based on -trimmed mean algorithm and SVM. Li et al. [26] introduced a semifragile watermarking scheme based on SVM. Yang et al. [17] proposed a new geometrically invariant image watermarking algorithm based on fuzzy SVM correction. However, by virtue of the good learning ability of SVM and AT, most of the existing watermarking schemes mentioned above still have some shortcomings as follows. (1) The standard SVM is always used as a learning scheme, while the speed and precision of training samples are not very ideal. Then, the distortion of final extracted watermark is more serious. (2) As many AT schemes only have single secret key, this characteristic has the limitations of being susceptible to one or several joint attacks and breaks, and then the difficulty of watermark extraction will be increased. (3) Because the extracted image feature is not very stable, hyperplane and generalization capability in SVM of the learned models are affected. Moreover, all samples in training data set are treated uniformly in the same class during the learning process of SVM, but this is not always true. (4) Due to the poor feature vectors, these methods are not very robust against some attacks. In watermark detection procedure, the original carrier image is usually needed, so it is unfavorable to practical application. Furthermore, some of them lack blind detection features; then they cannot balance imperceptibility and robustness effectively.

In this paper, embedding and extraction models of watermark based on AT and FSSVM are proposed with corresponding algorithms. The watermark image is first processed by IAT, and the pixel values are embedded by using quantization embedding rules. The watermarked carrier image can be obtained. Then the embedding method is presented by increasing secret keys to enhance the security degree. The quantization embedding rules can improve imperceptibility of watermark well. FSSVM model is constructed by training some embedding positions to find out positions embedded with watermark in the extraction process. The pixel values of watermark are extracted by introducing quantization extraction rules. FSSVM model not only enhances the training speed and precision of image characteristic values, but also realizes the blind extraction of watermark with the help of quantization extraction rules. The watermark coordinates are calculated by improving Arnold inverse transformation for embedding positions. The original watermark image is extracted, and the processing does not need original carrier image. Moreover, the combination of spatial domain and SVM can balance imperceptibility and robustness of watermark well.

The rest of this paper is organized as follows. In Section 2, AT and SVM techniques are briefly described. In Section 3, the proposed techniques are illustrated. In Section 4, the experimental results are presented and the comparative analysis of our scheme with other methods is given. Finally, the conclusion is drawn in Section 5.

#### 2. Preliminaries

##### 2.1. Arnold Transform (AT)

In practical applications, AT not only scrambles the pixel position by encoding the iterative number of the process, but also reduces the key spaces of storage and transmission. Although there are many ways for scrambling, here we will discuss only the AT in [10, 14], which is an iterative process of moving the pixel position. Suppose that the original image is a array and the coordinate of the pixel is . The generalized two-dimension AT is denoted bywhere and are the transformed coordinates corresponding to and after iterations, respectively, and are positive integers, and is the height or width of the square image processed.

Since the transformation is an iterative process, if the location () is transformed several times, it returns to its original position after iterations. is called the period of the transformation and depends on parameters , , and . These parameters (, , and ) can be used as secret keys in this paper. The pixels will continue to move until they return back to their original positions. Here, the moving time is , and the size of pixel space is . Pixels move with periodicity. , , , and (the size of original watermark) are correlated. Whenever the values change, it generates a completely different Arnold cat map. After being multiplied a few times, the correlation among the pixels will be completely chaotic. To get back to the original image, periodicity is required. Suppose that the scrambling has performed iterations; then one can get back to the original image by performing () iterations.

##### 2.2. Support Vector Machine (SVM)

SVM has been successfully applied to classification and function estimation problems introduced by Vapnik within the context of statistical learning theory and structural risk minimization [21, 22], and it can be used as regression prediction. Suppose that are training samples, and samples have nonlinear relationship in many cases. The regression function can be denoted by , where is an input vector, is a weight vector, is a nonlinearity mapping function, and is an offset.

In order to obtain the last two parameters and , the structural risk minimization rule is used and the original problem is transformed as follows:where describes the complexity of the function , is a constant to determine the complex rate of model and moderation of experimental risk, and are slack variables, is a sample, is a weight vector, is an offset, and is an insensitive loss function.

To solve the above convex optimization problem, the core idea is to transform the optimization question into dual form by using Lagrange multiplier method as follows:where , , , and are the weight coefficients and denotes the inner product operation. In the above formulas, kernel function is introduced to complete the inner product computation for inputting data of high-dimensional feature space while the is not known. The kernel function must satisfy Mercer theorem.

#### 3. Proposed Techniques

##### 3.1. Fuzzy Smooth Support Vector Machine (FSSVM)

In order to improve the efficiency and precision of prediction, SVM is introduced, and through combining with fuzzy mathematics and transforming the problem into unconstrained optimization problem, one can optimize the object function and transform the risk function into fuzzy dual extreme problems. Then, it can effectively reduce the errors between the predicted pixel values and the actual pixel values in the carrier image. The FSSVM model is used to train the specific pixels in some positions and find the embedding positions of watermark in carrier image, which is constructed in detail as follows.

*Step 1. *Select coordinates from embedding position coordinates at random, where . The pixel values corresponding to the original carrier image are denoted by , where .

*Step 2. *Take each position coordinate as the center of the original carrier image for each selected watermark position , select an image block with the size , and then receive image blocks.

*Step 3. *Calculate the corresponding eigenvalues of the image block in the carrier image for each position coordinate . Namely, the mean value of pixels except the central point is calculated as follows:And the pixel variance except the central point is calculated as follows:Thus, one can totally get groups of eigenvalues .

*Step 4. *Let each feature vector be a training data set and the corresponding pixel value of the original carrier image the target value of the training. It constitutes couples of training sample sets, shown as , to present a FSSVM training process for the above sample sets as follows.*Step 4.1*. Introduce fuzzy membership degree to each training sample, and blur the input sample set . Fuzzy membership degree is determined by the relationship between sample input sets and optimization values . *Step 4.2*. Divide the sets into two categories through fuzzy -means clustering to all eigenvalue sets , find out the center of the two classes, and then calculate the distance from the feature vector to the center of the corresponding class. Then the membership degree can be expressed aswhere represents the maximum distance from feature point to class center for the class of feature vector .*Step 4.3*. Introduce the kernel function to map the sample points into high-dimensional feature space. *Step 4.4*. Carry out linear regression in high-dimensional feature space, and obtain the nonlinear regression effect in the original space. Then its regressive function can be expressed asHere, the kernel function mainly uses Gaussian radial basis function as follows:where represent the input vector , and are the weight coefficients after training, is the deviation, is a nonlinearity function, and denotes the inner product operation.

*Step 5. *Use the principle of structural risk minimization to determine the parameters (, and ). Namely, transform the original regression equation into solving unconstrained programming problem. Then, the objective function is denoted bywhere , simply, is an adjustable parameter, and is a constant which is used to determine the complexity of the model and the folded moderation of empirical risk.

##### 3.2. Watermark Embedding Algorithm

The watermark embedding procedure, participating in the optimization flow chart of Figure 1, consists of the processing modules depicted as follows.