Robust digital image watermark scheme on wavelet domain using fuzzy rough sets

Abstract

Digital watermark technology has been proposed as an effective method to protect the copyright of digital image. By exploiting the characteristic of human visual system (HVS) and considering properties of discrete wavelet transform (DWT) coefficient, a novel image watermark scheme is designed in this paper. For embedding watermark, it selects wavelet detail coefficients to utilize HVS characteristic by fuzzy rough sets (FRS). The proposed scheme provides to select wavelet coefficients method using the fuzzy lower and upper approximations, which is partitioned by the FRS according to HVS. The experimental results confirm that the embedded watermarks by the proposed scheme are robust against common image processing attacks, such as JPEG compression, Gaussian noise, Salt & Pepper noise, cropping, mean filtering, wiener filtering, histogram equalization, and scaling etc. Proposed scheme can provide very good results in terms of image imperceptibility, too.

Keywords

Digital image watermark human visual system imperceptibility fuzzy rough sets robustness wavelets

1 Introduction

In recent years, the digital watermark technology has been widely used as one of the tools for copyright protection [1 –4]. According to how the watermark is inserted, a watermark technology can be classified into two categories, i.e., the space domain and transform domain techniques respectively. For example, discrete wavelet transform (DWT) is a commonly tool for inserting watermarks [5]. In general, transform domain technique is more robust than the space domain [5].

In practical applications, digital watermark technology is desirable to have some characteristics, including imperceptibility, robustness, and security [6]. We notice that the robustness of watermarks depends on the characteristic of the original image. When the watermark is embedded, an important problem must be considered, i.e., the choice of watermark embedding positions. For transform domain watermark, if selecting high frequency coefficients, it can guarantee good watermark imperceptibility, but the watermark data may be lost through some common image processing such as low pass filter. Therefore, when we choose some coefficients for embedding watermark, we should achieve a trade-off between imperceptibility and robustness.

Here, “imperceptibility”, “robustness”, “low”, and “high” are all fuzzy concepts. They have fuzzy boundary sets that cannot be precisely characterized using the available set of attributes. Therefore, to describe these concepts using fuzzy sets [7] are very suitable.

Besides, the theory of rough sets, proposed by [8 –11], is an extension of set theory for the study of intelligent systems characterized by insufficient and incomplete information. Using the concepts of lower and upper approximations in rough sets, knowledge hidden in information systems may be unraveled and expressed. Unlike other soft computing methods, rough set analysis requires no external parameters and uses only the information presented in the given data. It has been used successfully to deal with vagueness and uncertainty fields, for example knowledge discovery [12], clustering [7, 13] and pattern recognition [14] etc.

Based on above analyzes, we introduce the FRS, which combines the fuzzy sets and rough sets for designing a novel still image watermark technique. In fact, FRS has been successfully applied in many fields, such as attribute selection [15], data mining [16] and cancer detection [17] etc. In the proposed scheme, we embed the watermark in DWT domain, where the significant coefficients are selected adaptively using FRS.

The proposed scheme differs from previous works [18 –20] in two aspects. First, the image information is usually very complex, and it has characteristics of insufficient, fuzzy and incomplete. Through using the fuzzy lower and upper approximations, insufficient, fuzzy and incomplete information hidden in the image may be described, which makes that the proposed scheme is more suitable to image digital watermark. It will be considered more sufficiently than the previous works how to describe this insufficient, fuzzy and incomplete information hidden in the image.

Second, the previous works [18 –20] usually use the thresholds scheme to select the wavelet coefficients for embedding watermark, however there is stronger relativity between image pixels generally, and it is very difficult only to use the thresholds scheme for mining relativity information. The proposed scheme applied the equivalence classes to select the wavelet coefficients. How to mine this relativity information is researched more effectively than the previousworks.

The rest of this paper is organized as follows. Section 2 introduces some preliminary knowledge. In Section 3, we describe the select method of the wavelet coefficients according to the equivalence classes in FRS for inserting watermark. Section 4 describes the watermark embed and extraction method. Experimental results are presented in Section 5. Finally, conclusions are given in Section 6.

2 Preliminary

2.1 Rough sets

For representing incomplete knowledge, rough set is considered to be an extension of classical set theory. In this paper, we will use the definitions of the FRS that introduced by [11].

Let U ≠ φ be a universe of discourse and X be a subset of U. An equivalence relation, R, classifies U into a set of subsets U/R = {X₁, X₂, …, X_n} in which the following conditions are satisfied: (1) X_i ⊆ U, X_i ≠ φ for any i; (2) X_i ∩ X_j = φ for any i ≠ j; (3) ∪_iX_i = U, i = 1, 2, …, n . Where, φ is a empty set. Any subset X_i, called a category, represents an equivalence relation of R. A category in R containing an object x ∈ U is denoted by [x] _R. An indiscernibility relation IND (R) is defined as follows $xIND (R) y = {(x, y) \in U^{2} | (x, y) \in P, P \in U / R}$ (1)

For a family of equivalence relation P ⊆ R , IND ( P ) is defined as follows $IND (P) \underset{R \in P}{\cap} IND (R)$ (2)

Suppose we have an equivalence relation R and a set of objects x ∈ U. The R-lower and R-upper approximation of X are defined as $\underline{R} X = {x \in U | [x]_{R} \subseteq X},$ (3) $\bar{R} X = {x \in U | [x]_{R} \cap X \neq φ} .$ (4)

The lower approximation $\underline{R} X$ contains sets that are certainly included in X, and the upper approximation $\bar{R} X$ contains sets that are possibly included in X.R-positive, R-negative and R-boundary regions of X are defined respectively as follows. ${POS}_{R} (X) = \underline{R} X,$ (5) ${NEG}_{R} (X) = U - \bar{RX},$ (6) ${BN}_{R} (X) = \bar{RX} - \underline{R} X .$ (7)

Figure 1 provides an illustrative example of the concept of approximation.

2.2 FRS model based triangle modular

We denote two sets Y = {x_m : m = 1, 2, …, M} and Z = {x_n : n = 1, 2, …, N}. F (Y) and F (Z) is a set generated by all fuzzy sets on Y and Z, respectively. R ∈ F (Y × Z) is a fuzzy relation on Y × Z, R : Y × Z → [0, 1] .

Lemma 2.1. Let I ∈ [0, 1] , the function T : I × I → I is called a triangle modular on I × I, if T satisfy following conditions:

T (a, 1) = a, a ∈ I .

T (a, b) = T (b, a) , a, b ∈ I .

T (T (a, b) , c) = T (a, T (b, c)) , a, b, c ∈ I .

a ≤ c, b ≤ d ⇒ T (a, b) ≤ T (c, d) , a, b, c, d ∈ I .

Definition 2.1. Let T be a triangle modular on I × I. The operator θ_T : I × I → I is called the residuation implication of T if it satisfies following conditions: $θ_{T} (a, b) = sup {c \in I | T (a, c) \leq b}, a, b \in I .$ (8)

Definition 2.2. Let $F_{Y}^{i} \in F (Y), F_{Z}^{j} \in F (Z), i = 1, 2, \dots, M, j = 1, 2, \dots, N .$ For i = 1, 2, …, M, j = 1, 2, …, N, y ∈ Y, z ∈ Z, we have $F_{Y}^{i} (y_{i}) = 1, F_{Z}^{j} (z_{j}) = 1 .$ $(Y \times Z, \underset{i, j}{\cup} F_{Y}^{i} \times F_{Z}^{j})$ is called a fuzzy T approximation space. For fuzzy relation R : Y × Z → [0, 1] , the $\underline{R}$ -lower and $\bar{R}$ -upper approximation with respect to fuzzy T approximation space $(Y \times Z, \underset{i, j}{\cup} F_{Y}^{i} \times F_{Z}^{j})$ are a pair fuzzy set defined on Y × Z, and their membership function are defined respectively as $\underline{R} (y, z) = \underset{i \in Y, j \in Z}{\land} θ_{T} (F_{Y}^{y} (i) \land F_{Z}^{z} (j), R (i, j)),$ (9) $\bar{R} (y, z) = \underset{i \in Y, j \in Z}{\lor} θ_{T} (F_{Y}^{y} (i) \land F_{Z}^{z} (j), R (i, j)),$ (10) $y = 1, 2, \dots, M, z = 1, 2, \dots, N$ $(\underline{R}, \bar{R})$ is called a fuzzy rough set, $\underline{R} (y, z)$ and $\bar{R} (y, z)$ are called the fuzzy T lower approximation and the fuzzy T upper approximation respectively. The rough set ${BN}_{R} (Y, Z) = \bar{R} - \underline{R}$ is called the boundary region of binary relation R.

According to above definitions, we can immediately obtain following conclusions.

(1) For selecting the fuzzy set collection $F_{Y}^{i}, F_{Z}^{j},$ i = 1, 2, …, M, j = 1, 2, …, N, the following condition should be satisfied $F_{Y}^{y} (y) = 1, F_{Z}^{z} (z) = 1 .$ (11)

(2) If $F_{Y}^{i} (y) = 1, y \in {y_{1}, y_{2}, \dots, y_{M}}, i = 1, 2, \dots, M;$ $F_{Z}^{j} (z) = 1, z \in {z_{1}, z_{2}, \dots, z_{N}}, j = 1, 2, \dots, N,$ or $F_{Y}^{i} (y) = 0,$ y ∈ {y₁, y₂, …, y_i-1, $y_{i + 1}, \dots, y_{M}}, F_{Y}^{i} (y_{i}) = 1, i = 1, 2, \dots, M;$ $F_{Z}^{j} (z) = 0, z \in {z_{1}, z_{2}, \dots, z_{i - 1}, z_{i + 1}, \dots, z_{N}},$ $F_{Z}^{j} (z_{j}) = 1,$ j = 1, 2, …, N, then we have $\underline{R} (y, z) = R (y, z) = \bar{R} (y, z),$ (12) $y \in {y_{i} : i = 1, 2, \dots, M}, z \in {z_{j} : j = 1, 2, \dots, N} .$

In actual calculation process, for calculating fuzzy T upper approximation $\bar{R} (y, z)$ and fuzzy T lower approximation $\underline{R} (y, z)$ , it is not need that i and j come from all elements of the set Y and Z respectively, and only come from some neighborhoods of the element y and z respectively. In this paper, $F_{Y}^{i}, F_{Z}^{j},$ i = 1, 2, …, M, j = 1, 2, …, N, are fuzzy set, and their functional digraphs are in Fig. 2.

The functions of $F_{Y}^{i}$ and $F_{Z}^{j}$ can be represented as follows $f (x | μ, σ) = e^{- \frac{(x - μ)^{2}}{σ^{2}}}$ or $f (x | a, b) = {\begin{matrix} \frac{x - a + b}{b}, a - b \leq x \leq a \\ \frac{a + b - x}{b}, a < x \leq a + b \end{matrix} .$ (13)

3 Selection of DWT coefficients for inserting watermark

3.1 Watermark preprocessing

In recent years, chaotic maps have been used for increasing the security of the watermark [21, 22]. Chaotic phenomenon is a random procedure appeared in the non-linear dynamic system. This procedure has non-periodicity, non-convergence and extreme sensitivity to the initial conditions. We choose one of the simplest chaotic maps,described by $x_{k + 1} = μ \cdot x_{k} (1 - x_{k})$ (14)where 0 ≤ μ ≤ 4 . When 3.5699456< μ ≤ 4, the map is in the chaotic state. Using logistic map can generate a real value sequence {x_k : k = 1, 2, …}. In this paper, a logistic sequence is generated for watermark encryption. Assume μ = 4, the initial condition is 0.123.

A threshold ρ is predefined to get a binary logistic sequence u = {ρ (x_k) : k = 1, 2, …}. If x_k > ρ then x_k = 1, else x_k = 0. Assume w is a binary watermark image with size W_x × W_y, and we change w into a sequence {w (i) , 1≤ i ≤ W_x × W_y }. In this paper, let ρ = sum (x)/W_x × W_y. A new watermark image $\tilde{w}$ can be obtained after $\tilde{w} = w \oplus u$ arithmetic. To convenient, we change $\tilde{w}$ into another sequence w′ from the set {–1,1}, where if $\tilde{w} (i) = 1$ then w′ (i) =1, else w′ (i) = -1.

3.2 DWT coefficients

Due to the excellent time-frequency features and the well-matching to the HVS characteristic [23 –25], wavelet transform [5] has been widely used for digital watermark, especially after the wavelet transform becoming the basic technology in JPEG 2000 standards. Wavelet-based watermark schemes exploit the frequency information and spatial information of the transformed data in multiple resolutions to gain robustness. Digital watermark that use wavelet transforms has been experimented by some researchers [20, 26] recently. The advantages of wavelet transform compared to DCT and DFT were mentioned in [27]. Therefore, in this paper, the wavelet decomposition is used as the transformation tool.

The wavelet transform is a mathematical tool for decomposing. We briefly review the DWT model (Fig. 3), which shows a 1 level wavelet transform.

Wavelets are a class of functions used to localize a given function in both space and scaling [23]. A family of wavelets can be constructed from a basis function ψ (t), which is confined in a finite interval. A discrete wavelet basis function is defined by $ψ^{j, k} (t) = 2^{- \frac{j}{2}} ψ (2^{- j} t - k)$

Here, the scale factor j indicates the wavelet’s width (a power of 2) and the location index k its position (an integer). The ψ^j,k (t) are self-similar and are selected to be orthonormal, so $\int ψ^{j_{1}, k_{1}} ψ^{j_{2}, k_{2}} = 0, if j_{1} \neq j_{2} or k_{1} \neq k_{2}$ and it is thus possible to represent other functions as a linear combination of the ψ^j,k (t).

The basic idea of 2D DWT of image is described as follows. An image is firstly decomposed into four subimages denoting LL₁, LH₁, HL₁ and HH₁. LH₁, HL₁ and HH₁ contain the higher frequency detailed information. LL₁, the gross approximation, is the low frequency component containing most of the energy in the image. To obtain coarser-scaled wavelet coefficients, the sub-band LL₁ is further decomposed and critically sub-sampled. This process is repeated an arbitrary number of times, which is determined by the application at hand. If the process is repeated L times, we can obtain the subimages LL _L , LH _L , HL _L and HH _L through L levels wavelet transform.

Since the gross approximation of the wavelet-decomposed image includes most of the energy from the original image, it has a crucial effect on the image quality. As such, the proposed scheme does not utilize the gross approximation to retain the invisibility. Plus, the wavelet coefficients on the lowest level are also excluded in the proposed scheme because these coefficients can be easily eliminated and modified by lossy compression and common signal processing.

In this paper, f (m, n) is the original image with the size f_x × f_y. The f (m, n) is transformed into the wavelet domain. Specifically, the L-th level DWT of an original image produces a sequence of 3-L detail subimages corresponding to the horizontal, vertical and diagonal details at each of the L resolution levels, and a gross approximation at the coarsest resolution level. The value of L is user-specified and is set to a positive integer less than or equal to min (f_x, f_y) -1 . The different frequency orientations of the detail subimages are represented with the variable k for whichk = 1, 2, 3 corresponds to the horizontal, diagonal, and vertical details, respectively. The resolution level is denoted with the variable l, where a larger value of l corresponds to a coarser resolution level (i.e., larger scale). Thus, the detail coefficients for an l-th level DWT of an original image f are given by f_k,l (m, n), where k = 1,2,3, l = 1, … , L, and (m, n) corresponds to the spatial location at the l-th resolution. The gross approximation is represented with f_4,l (m, n) . Every subimage f_k,l is comprised of pixels representing the coefficient values for various (m, n) .

3.3 Selection of wavelet coefficients using FRS

In this section, the selection of the watermark embedding coefficients will be explained. We know that, in the HVS, the eye is not very sensitive to small changes in the edges or texture of the image, but is very sensitive to the smooth parts of the image. This is the texture sensitivity of the digital image. Edges and textures are well confined to the high frequency detailed information. According to FRS, we will make use of this property to select wavelet coefficients. For f_k,l (m, n), where k = 1,2,3, l = 2, … , L; m = 1, 2, … , s, n = 1, 2, … , t, the wavelet coefficients are determined according to following procedure.

Step 1. Fuzzify f_k,l (m, n) according to following function $R_{k, l} (m, n) = \frac{f_{k, l} (m, n) - α}{β - α} .$ (15)Where, $β = max_{1 \leq m \leq s, 1 \leq n \leq t} f_{k, l} (m, n)$ , $α = min_{1 \leq m \leq s, 1 \leq n \leq t} f_{k, l} (m, n)$ . So, we may transform f_k,l (m, n) into fuzzy relation R_k,l (m, n).

Step 2. For each spatial location (m, n) in R_k,l (m, n), we let Gausses functions y = exp {- (x - μ) ²/σ} be horizontal and vertical fuzzy set collections respectively, where μ and σ are mean and variance respectively.

We notice that, in every location (m, n), two Gausses functions calculating with R_k,l (m, n), if according to Equation (9), the lower approximation in this location can be obtained; if according to Equation (10), the upper approximation in this location can be obtained.

Step 3. Using inverse function of Equation (15), we may transform obtained the lower and upper approximations back to wavelet domain. So, we can get the fuzzy lower and upper approximations of f_k,l (m, n).

Step 4. The wavelet coefficients to correspond the difference between the fuzzy upper and lower approximation are selected for inserting watermark.

4 Proposed scheme

We need to generate a key K for inserting watermark, and this K is a matrix. For every coefficient in the wavelet domain, the key K has a corresponding value of one or zero to indicate if the coefficient is to be marked or not, respectively, i.e., for any f_k,l (m, n) is selected for inserting watermark, K (m, n) =1, else K (m, n) =0. The number of ones in the key K must be greater or equal to the size of the watermark. This request can be realized by adjusting the parameters μ and σ of Gausses functions.

Because the length of the watermark is less than the number of ones in the key K, the watermark values may be repeatedly embedded in different coefficients selected by the key K.

4.1 Watermark embedding

After the original image is decomposed by L level DWT, the watermark embedding procedure has the two steps as follows:

Step 1. For every coefficient f_k, l (m, n) in the wavelet domain, if K (m, n) =1 then proceed as follows, otherwise the watermark do not embed.

Sort the detail coefficients in ascending order so that f_1,l (m, n), f_2,l (m, n), and f_3,l (m, n) are coefficients such that f_1,l (m, n) ≤ f_2,l (m, n) ≤ f_3,l (m, n) . To embed the watermark, we quantize f_2,l (m, n). The range of values between f_1,l (m, n) and f_3,l (m, n) is divided into bins of width Δ = (f_3,l (m, n) - f_1,l (m, n))/(2η - 1) . Where η is a user defined variable, and it may provide an appropriate trade-off between the perceptibility and robustness of the watermark. To embed a 1, f_2,l (m, n) is quantized to the nearest value shown with bold vertical bars in Fig. 4. Alternatively, to embed a –1, f_2,l (m, n) is quantized to the nearest value shown by dashed vertical lines.

Step 2. Performing an L level inverse wavelet transform on the modified wavelet coefficients, we get a watermarked image f^w.

4.2 Watermark extraction

L level DWT is operated on the possible marked image f^w (m, n). Let $f_{k, l}^{w} (m, n)$ denote a wavelet coefficient in the k-th detail image component of the l-th resolution level.

We use the key K to find the locations in which the watermark was embedded for every resolution level l. Watermark extraction procedure has the two steps as follows:

Step 1. Order the detail coefficients in ascending order as before so that $f_{1, l}^{w} (m, n) \leq f_{2, l}^{w} (m, n) \leq f_{3, l}^{w} (m, n),$ and estimate the watermark bit value from the relative position of $f_{2, l}^{w} (m, n)$ . Using the same constant η with embedding, a particular watermark bit is determined by finding the closest quantized value to $f_{2, l}^{w} (m, n)$ and determining if this quantized value was used to embed a 1 or a –1.

Step 2. If the watermarks were embedded several times in different locations, for extracted watermark, then the most common bit value is assigned. If an equal number of 1 and –1 were extracted, then the randomly selection was made to its bit value. In the end, decrypt it by doing an inverse XOR arithmetic, and we get extracted watermark denoted by w^* .

The block diagram proposed semi-blind watermark scheme is shown in Fig. 5.

5 Experimental results

5.1 Experiments

5.1.1 Experiment images

To evaluate the performance of the proposed semi-blind image watermark scheme, a set of experiments is performed under the following conditions.

5 standard images for grayscale 8-bit are chosen as the tested images. The these images with size 512×512 pixels are Baboon, Lena, Peppers, Goldhill, and Sailboat. These images are shown as Fig. 6.

We use the watermark image with size 64×64, shown in Fig. 7.

5.1.2 Performance measures

Our goal is to measure the imperceptibility of the watermarked images and the robustness of the extracted watermarks against various attacks. The Peak Signal to Noise Ratio (PSNR) and the Normal Correlation (NC) shown in Equation (16) and Equation (17) are used to measure the imperceptibility and the robustness, respectively. $PSNR = 10 {log}_{10} \frac{(f_{x} \times f_{y}) \times 255^{2}}{\sum_{m = 1}^{f_{x}} \sum_{n = 1}^{f_{y}} (f (m, n) - f^{w} (m, n))^{2}},$ (16) $NC = \frac{\sum_{i = 1}^{W_{x}} \sum_{j = 1}^{W_{y}} w^{*} (i, j) w (i, j)}{W_{x} \times W_{y}} .$ (17)

Where w and w^* are original watermark and extracted watermark, respectively.

5.1.3 Implementation details

In experiments, we let the larger level L = 3, wavelet decomposition uses Daubechies-1.

After DWT is used to the experiment image, we first calculate the fuzzy lower and upper approximations of wavelet coefficient according to steps of Section 3.3. Then, we embed watermark accordance with steps of Section 4.1. The Sailboat watermarked image is depicted in Fig. 8. For the extraction of the watermark, the implementation details can be seen in Section 4.2.

5.2 Results

5.2.1 Imperceptibility

The goal is to measure the imperceptibility of the watermarked images. PSNR is used to measure the imperceptibility of the watermarked image, where the higher PSNR, the more transparency of the watermark. The five images, Baboon, Lena, Goldhill, Sailboat, and Peppers, are chosen as the tested images.

By using the proposed scheme, we find that the watermark is almost imperceptibility to the human eyes, and the values of PSNR are very high. Table 1 shows PSNR of tested images after embedding watermark.

5.2.2 Robustness

NC is used to measure the robustness of the watermarked image, where the higher NC, the more robustness of the watermark.

Attacks were realized to the watermarked image, to observe the behavior of the watermark. The realized attacks are JPEG compression, adding noise, filtering, scaling, and geometrical distortions. Robustness of the proposed scheme is evaluated for various types of images distortions as discussed below.

A. JPEG compression

We demonstrate the robustness of the proposed scheme for some practical applications. Since most digital images that are exchanged on digital networks are compressed, resistance against lossy compression is essential to a watermark algorithm. Realize the JPEG compression with quality factors (QF) from 90% to 10% to the five watermarked images respectively. After applying this attack, images are degraded and a lot of data is lost but the extracted watermark images are still recognizable. In these experiments, NC and PSNR show average value of five watermarked images’. The experiment results sufficiently demonstrated that the proposed scheme has great robustness against JPEG compression.

B. Gaussian noise

It is known that the addition Gaussian noise is also a very common signal attack for digital watermark. We introduce Gaussian noise to watermarked image, Gaussian noise is zeros mean, variance vary from 0.001 to 0.004. Figure 9 shows four Sailboat watermarked images attacked by Gaussian noise variance (GNV) with varying from 0.001 to 0.004.

The experiment results are shown in Table 3. We notice that the watermarked image is seriously damaged when variance is 0.004 but can still obtain high NC and PSNR values. The experiment results confirmed that the proposed scheme has great robustness against Gaussian noise.

C. Salt & pepper noise

Introduce Salt & Pepper noise to the watermarked image, and the intensity values are 0.01, 0.03, 0.05, and 0.07 respectively. Figure 10 shows four Sailboat watermarked image attacked by Salt & Pepper noise intensities (SPNI) with 0.01, 0.03, 0.05 and 0.07.

The experiment results are shown in Table 4. The experiment results show that the proposed scheme can availably resist Salt & Pepper noise attack.

D. Filtering

(1) Wiener filtering

Wiener filtering is a kind of the common signal attack for digital watermark. To illustrate the effect of the wiener filtering to an individual tested image, Fig. 11(a-c) show Sailboat watermarked images under 3×3, 5×5, and 7×7 window respectively.

Table 5 collects the results in terms of the NC and PSNR. The experiment results demonstrated that the proposed scheme has great robustness against wiener filtering.

(2) Mean filtering

Mean filtering is another a kind of common watermark attack operates. To illustrate the effect of the mean filtering to an individual tested image, Fig. 12(a-c) show Sailboat watermarked images under 3×3, 5×5, and 7×7 window respectively.

Table 6 collects the results in terms of the NC and PSNR. Experiment results show that the proposed scheme has very robustness against mean filtering.

E. Histogram equalization

Histogram equalization (HE) is a kind of the common signal processing operation. Figure 13(a) shows Sailboat watermarked images after attacked by HE and Fig. 13(b) shows the corresponding extracted watermark.

Above experiment results confirm that proposed scheme is robustness to HE attack, but also has more imperceptibility.

F. Cropping

We cropped a part of Sailboat watermarked image, and the extraction results are shown in Fig. 14.

Above experiments show that proposed scheme is robustness to cropping attack.

G. Scaling

Sailboat watermarked image is scaled 256×256, the experiment results is shown in Fig. 15. It can also extract clear watermark image.

Above experiment results confirm that proposed scheme is not only still robustness to scaling attack, but also has more imperceptibility.

H. Others

In the following, other four images Baboon, Lena, Peppers, and Goldhill are tested. Table 7 shows the experiment results of them after different attacks. It demonstrates that the proposed scheme can resist above attacks for different images.

5.3 Comparison with other method

We compare proposed scheme to [28, 29] methods using the Lena, Peppers, and Goldhill images. These compared methods are in wavelet domain. For [28], the watermark was embedded the local maximum coefficient. For [29], proposed watermarking scheme was based on wavelet tree quantization. The experiment results after various attacked are shown in Tables 8 and 9. In Tables 8 and 9, “Proposed method” is average NC value to three images Lena, Peppers, and Goldhill.

As shown in Table 8, for QF = 50% –90% and 10% –20% , the proposed scheme provides the highest average NC values. When QF = 30% and 40% , although the proposed scheme does not obtain highest average NC values, but the difference is not significant, namely difference value is 0.0010 and 0.0014 respectively.

As shown in Table 9, for mean filter (3×3, 5×5, and 7×7), cropping 1/4, and scaling 256×256 attacks, the proposed scheme provides the highest average NC values. For HE attack, although the proposed scheme done not obtain highest average NC values, but the difference is not significant, namely difference value is 0.0007.

In addition, we compare proposed scheme to [30, 31] methods using the Lena, Baboon, and Barbara images. The size of these images still is 512×512. Barbara image is shown as Fig. 16, and the watermark image with size 64×64 is shown in Fig. 17.

In [30], watermarking scheme was used in undecimated discrete wavelet transform domain. And in [31], the special frequency band and property of image in multiwavelet domain are employed for the watermarking algorithm. In this paper, BER is used to measure the reliability of watermark scheme. BER is defined by $BER = \frac{B}{W_{x} \times W_{y}}$

Where, B is the number of erroneously extracted bits.

The experiment results after various attacked are shown in Table 10. In Table 10, “Proposed” is average bit error rate (BER) value of the thirty times to each image Lena, Baboon, and Barbara respectively.

Table 10 shows the comparison results with [30, 31]. From Table 10, we notice that the proposed scheme can achieve better robustness than [30, 31] for most attacks.

6 Conclusions

In this paper, a novel watermark scheme of digital image is proposed. In proposed scheme, FRS is introduced into watermark scheme to improve the imperceptibility and robustness of watermark. By the experimental results, the proposed scheme has the following advantages: (1) We know that there is stronger relativity between image pixels. It is very difficult only to use a threshold scheme for indicating relativity information. The proposed scheme can indicate this relativity information very effectively, which makes that the proposed scheme has very good robustness. (2) Usually, the image information is very complex, and it has characteristics of insufficient, fuzzy and incomplete. The proposed scheme can indicate this insufficient, fuzzy and incomplete information hidden in the image, and which makes that the proposed scheme is very suitable to digital image watermark. (3) We believe that the proposed scheme can resist to common image processing attacks. Furthermore, watermark has good imperceptibility, and which makes that the proposed scheme solves the conflict between invisibility and robustness better.

Disadvantages of the proposed scheme are related to the computation time for the fuzzy lower and upper approximations, and performing FRS computation. Therefore, as long as no more restrictions on computation time, the proposed scheme is applicable. In addition, the proposed scheme is fragile to some geometrical distortions, such as rotation and translation, etc. Future work will focus on these disadvantages.

Footnotes

Acknowledgments

This work was supported by the science and technology research program of Wuhan of China (Grant No. 201210121023).

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