Keywords

1 Introduction

With the rapid development of the Internet and the media, data hiding [1] has drawn increasing attention. People have begun to study the various “non-visible reality” technology. In principle, although classical encryption method can be used for image information hiding, but not necessarily suitable, because of the large amount of image data, and has autocorrelation, and the classic encryption method will not be considered for this, but in fact, these characteristics of the image itself can not be ignored and we can use. Further, the image information as an intuitive expression, can be used also with confusing, i.e., if some important information (voice, image, text, etc.) can be hidden in the image, but the image loss is very small, then an attacker it is possible to image the surface were fooled, even though he knew the image in the possession of some important information extraction algorithms do not know nothing.

In recent years, the rise of digital watermarking technology is one such image information hiding technology, it is mainly used in the field of copyright protection of electronic publications [2, 11, 13]. If the copyright owner of the electronic publication of certain key image with its own identity implants, then, we can extract features of the image at the appropriate time to prove their ownership of the works. This feature is what we call digital image watermark, in general, the digital watermark is mainly required for the following two

  1. (1)

    hidden, which is a basic requirement for digital watermarking, digital watermark embedding that image, the human eye can not identify it out directly from the image.

  2. (2)

    Robustness, anti-attack capability that the watermark, the watermark can still be extracted require After general transformation, filtering, image processing operations.

It also requires the digital watermark embedding the original image after the impact is not too large.

Currently, the watermark embedding method is mainly from the position of the space of the original image, color space, frequency space starting watermark embedding position, and seek representation.

Digital watermarking technology is to add an imperceptible in digital works of identity information, the algorithm proposed identity information where it is needed to validate the technology. In watermark detection, the original digital domain watermark image through the process of printing and scanning which makes the digital image watermark not only lost some information, but also introduced a number of noise, the watermark information to a large extent by the damage, It is very difficult to design of a robust watermarking algorithm. Lin has used Fourier transform has characteristics such as rotation, shear, translation studies watermarking algorithm against printing and scanning [3, 4]. Li used photoshop itself brought function were studied aspects of printed information hiding [5]. Zhang Chongxiong presents a design of portable instrument for detecting trademark with digital watermark [6]. The digital watermarking technology for trademark counterfeiting. Robustness algorithms not only to the scanning process, but also to achieve the scanning process, the double printing of good robustness.

In this paper, we analyzed the commercial trademark common situation, we proposed a algorithm based on First degree orthogonal function of U system.

2 First Degree Orthogonal Function of U System

Around 1983, Professor Qi Dongxu and Professor Feng Yuyu establish a class of complete orthogonal function system [7, 9, 10, 12]. It consists of a series of segments composed of odd polynomial, named as K degree U system. K = 0, 1, 2, ….. It is composed of both smooth functions and all levels of discontinuous function, and Walsh function is a special case of k = 0 in U system.

Following is a brief definition of the First degree orthogonal function of U system are given:

$$ \begin{aligned} & U_{0} (x) = 1,\,0 \le x \le 1 \\ & U_{1} (x) = \sqrt 3 (1 - 2x),\,0 \le x \le 1 \\ & \mathop U\nolimits_{2}^{(1)} (x) = \left\{ {_{{\sqrt 3 (4x - 3),\,\frac{1}{2} < x \le 1}}^{{\sqrt 3 (1 - 4x),\,0 \le x < \frac{1}{2}}} } \right. \\ & \mathop U\nolimits_{2}^{(2)} (x) = \left\{ {_{{5 - 6x,\,\frac{1}{2} < x \le 1}}^{{1 - 6x,\,0 \le x < \frac{1}{2}}} } \right. \\ & \mathop U\nolimits_{3}^{(1)} (x) = \left\{ {_{{\mathop U\nolimits_{2}^{(1)} (2 - 2x)x,\,\frac{1}{2} < x \le 1}}^{{\mathop U\nolimits_{2}^{(1)} (2x),\,0 \le x < \frac{1}{2}}} } \right. \\ & \mathop U\nolimits_{3}^{(2)} (x) = \left\{ {_{{\mathop { - U}\nolimits_{2}^{(1)} (2 - 2x)x,\,\frac{1}{2} < x \le 1}}^{{\mathop U\nolimits_{2}^{(1)} (2x),\,0 \le x < \frac{1}{2}}} } \right. \\ & \mathop U\nolimits_{3}^{(3)} (x) = \left\{ {_{{\mathop U\nolimits_{2}^{(2)} (2 - 2x)x,\,\frac{1}{2} < x \le 1}}^{{\mathop U\nolimits_{2}^{(2)} (2x),\,0 \le x < \frac{1}{2}}} } \right. \\ & \mathop U\nolimits_{3}^{(3)} (x) = \left\{ {_{{ - \mathop U\nolimits_{2}^{(2)} (2 - 2x)x,\,\frac{1}{2} < x \le 1}}^{{\mathop U\nolimits_{2}^{(2)} (2x),\,0 \le x < \frac{1}{2}}} } \right. \\ & \vdots \\ & \mathop U\nolimits_{n + 1}^{(2k - 1)} (x) = \left\{ {_{{\mathop U\nolimits_{n}^{(k)} (2 - 2x)x,\,\frac{1}{2} < x \le 1}}^{{\mathop U\nolimits_{n}^{(k)} (2x),\,0 \le x < \frac{1}{2}}} } \right. \\ & \mathop U\nolimits_{n + 1}^{(2k - 1)} (x) = \left\{ {_{{\mathop { - U}\nolimits_{n}^{(k)} (2 - 2x)x,\,\frac{1}{2} < x \le 1}}^{{\mathop U\nolimits_{n}^{(k)} (2x),\,0 \le x < \frac{1}{2}}} } \right. \\ & k = 1,\,2,\,3, \ldots ,\quad \mathop 2\nolimits^{n - 1} ,\,n = 2,\,3, \ldots \\ \end{aligned} $$
(1)

Among them, the discontinuity point at the 1/2. The average value from both sides of the function value limit. So the definition of first degree orthogonal function of U system the first eight images as shown in Fig. 1.

Fig. 1.
figure 1

The 8 function of U system

Full size image

The linear U system has the following properties:

Standard Orthogonality

$$ \begin{aligned} \left\langle {\mathop U\nolimits_{m}^{(i)} ,\mathop U\nolimits_{n}^{(j)} } \right\rangle &= \int_{0}^{1} {\mathop U\nolimits_{m}^{(i)} (x)\mathop U\nolimits_{n}^{(j)} (x)dx} \hfill \\ &= \mathop \delta \nolimits_{m,\,n} \mathop \delta \nolimits_{i,\,j} \hfill \\ \end{aligned} $$
(2)

Among them: m, n = 0, 1, 2,…, i = 1, 2, 3,…, 2 m−1,

$$ j = 1,\,2,\,3, \ldots ,\,2^{n - 1} ,\,\delta_{k,l} = \left\{ {\begin{array}{*{20}c} \begin{aligned} 1 \hfill \\ 0 \hfill \\ \end{aligned} & \begin{aligned} k = 1 \hfill \\ k \ne 1 \hfill \\ \end{aligned} \\ \end{array} } \right. $$

Convergence of Fourier-U

If a given function F Fourier-U series for F = \( \sum\limits_{i = 0}^{\infty } {\mathop a\nolimits_{i} \mathop U\nolimits_{i} } \), Ui is of the formula (1) in order to sort the functions, that \( \mathop U\nolimits_{i} = \mathop U\nolimits_{N}^{(K)} \), \( N = [\log_{2} i] + 1,\,k = i - \mathop 2\nolimits^{n - 1} + 1 \)

$$ a_{i} = \left\langle {F,U_{i} } \right\rangle = \int_{0}^{1} {F(x)U_{i} } (x)dx $$
(3)

And

$$ p_{n + 1} F = \sum\limits_{i = 0}^{n} {a_{i} U_{i} } $$
(4)

So

$$ \mathop {\lim }\limits_{n \to \infty } \left\| {\left. {F - P_{n} F} \right\|} \right._{2} = 0,F \in L_{2} \left[ {0,\,1} \right] $$
(5)
$$ \mathop {\lim }\limits_{n \to \infty } \left\| {\left. {F - P_{{2^{n} }} F} \right\|} \right._{\infty } = 0,F \in C\left[ {0,\,1} \right] $$
(6)

Equation (1) shows that Fourier-U Series L2 convergence and completeness. Equation (5) showed that Fourier-U series of “grouping” Uniform Convergence.

Fourier-U Series of Renewable

If the function F is a piecewise linear function, and discontinuities occur only in \( x = \frac{q}{{\mathop 2\nolimits^{r} }} \), where q and r is an integer, then F can use a limited term Fourier-U series to accurately express.

3 Commercial Trademark Counterfeiting Digital Watermarking Algorithm

Firstly, statistics orthogonal transformation matrix coefficients after the sign watermarking algorithm, which gives a U system based on digital watermarking algorithm of trademark counterfeiting.

Generate Orthogonal Matrix from First Degree Discrete U System

We generate a 32 × 32 orthogonal matrix in discrete first degree orthogonal function of U system, the interval [0, 1] is divided into 32 sub-section. Then the piecewise linear U system first 32 base discrete, unit orthogonal we can get the 32 × 32 transformation matrix.

Trademark Counterfeiting Digital Watermark Embedding Algorithm

Steps of the algorithm: first, through the scanner to scan the trademark to the computer, or select the digital trademark image already in computer, complete the import process of trademark image; U transform to digital trademark image, binding properties of the human visual masking; embedding a watermark bit to the U transformed matrix coefficients. The work is completed the watermark embedding.

Here we combine the process of embedding a specific experiment elaborated. Among them, the image size to be counterfeit trademark 256 × 512, binary watermark image size is 16 × 16 (Fig. 2).

Fig. 2.
figure 2

The original trademark image and the original watermark image.

Full size image
  1. (1)

    The trademark image into pieces of 32 × 32, 8 × 16–128 divided into blocks of each piece separately U transformation.

  2. (2)

    We get the small piece’s IF {u + v = 27, 28, 29, 30, 31, 32, 33, 34, 35} (u, v is the image pixel position). Each intermediate frequency coefficient can in turn corresponds to 26, 27, 28, 29, 30, 29, 28, 27, 26 pixel position, total of 250 position, then select 6 position from 36 intermediate frequency coefficient {(16, 20) (17, 19) (18, 18) (19,17) (20, 16) (21, 15)}. A total of 256 positions were selected (for the embedded watermark 16 × 16 = 256 bits of information selected).

  3. (3)

    We use pixel position (1, 26) for example. Statistical sign U coefficient 128 pieces in that position when embedding. Order pos (u, v) is positive number, nega (u, v) is the number of negative. Let d for embedding strength, experiments take 126.

When embedding watermark bits to 0, if

$$ posi\,\left( {u,v} \right) - nega\,(u,v) > d $$
(7)

There is no need to change the coefficient U of the position of the 128 pieces, otherwise the adjustment coefficient U. The absolute value of the smallest positive number sign negated, until compliance with the formula (7); When the embedded watermark bit is 1, if

$$ nega\,\left( {u,v} \right) - posi\,(u,v) > d $$
(8)

There is no need to change the coefficient U of the position of the 128 pieces, otherwise the adjustment coefficient U. The absolute value of the smallest positive number sign negated, until compliance with the formula (8). When the 256 turn after the watermark information processing, embedding the work completed.

Watermark Extraction Algorithm

Watermarking algorithm is the inverse process of the embedding algorithm. Trademark image has been printed is scanned to obtain a digital trademark image. and to make 32 × 32 block, U transformation, get the matrix coefficients after U transformation, embedding position for each extraction, the extraction coefficient matrix when the number is greater than the number of positive negative watermark bit is 0; otherwise extract the watermark bit is 1. Complete extraction of all embedded position to obtain a complete watermark information.

4 Algorithm Analysis and Experimental Trademark Legend

Algorithm Analysis

(1) U transform the selected IF coefficients.

We have come through the coefficient matrix after U transformed by experiment. Most of the energy in the low frequency coefficients are concentrated reflection. Watermark embedding quality images in low frequency would be a greater impact, there will be a significant degradation. Two high-frequency coefficients are reflected image detail, many compression algorithms in today’s high-frequency coefficients are discarded because the embedded watermark in the high frequencies will not be able to resist compression attack. Taking these two cases, our embedding algorithm selected frequency coefficients.

(2) D embedding strength selection.

D embedding strength ranges from 0 to n, N is the small pieces of trademark image. Select the embedding strength of the appropriate D, can also according to different frequency coefficients adaptively selects the D, ensure the algorithm’s advantage (Fig. 3).

Fig. 3.
figure 3

The figure is a trademark of the embedded algorithm to generate an image with the watermark, the next figure is extracted from the graph algorithm extracted watermark with the watermark.

Full size image

Trademarks Algorithm Simulation and Attack Renderings

Figure 4(A1)–(H1) were broken watermarked image trademark, trademark of watermarked image smear, irregular cutting, cutting quarter, JEPG .80 % and 50 % compression, 0.02 and 0.1 plus noise images.

Fig. 4.
figure 4figure 4

Figure 4 pairs trademarks watermarked image into line effect of the image after attacks.

Full size image
Fig. 5.
figure 5

Figure 5(A2)–(H2), respectively Fig. 4(A1)–(H1) from the extracted watermark image in Fig. 4.

Full size image

We use the Normailized Correlation [8] as the similarity measure reference the original watermark W and extracted watermark between EW. Which is defined as

$$ {\text{NC}} = \frac{{\sum_{i} \sum_{j} w\left( {i,j} \right)EW\left( {i,j} \right)}}{{\sum_{i} \sum_{j} \left[ {w\left( {i,j} \right)} \right]^{2} }} $$
(9)

NC values closer to 1, indicating that the closer to the extracted watermark the original watermark.

We also once the watermarking algorithm based on U systems were compared with DCT-based watermarking algorithm, due to limited space, this does not give the image and image watermark extraction various attacks DCT-based watermark image. Table 1, respectively, using two algorithms to extract the watermark after N values from the test results, the use of a U watermarking algorithm based system can be achieved with the DCT watermarking algorithm is basically the same effect on, which basically can be completely extracted watermark information is not If the visual impact of the watermark, which shows that the algorithm can resist a certain degree of attack, the robustness of the algorithm is better

Table 1. Watermark NC value U algorithm and DCT algorithm comparison
Full size table

5 Conclusion

In this paper, digital watermarking technology, through the commercial trademark in the course of the analysis of a variety of circumstances, we propose a new class-based orthogonal functions of commercial trademark counterfeiting digital watermarking algorithm. The algorithm for trademark counterfeiting has the following characteristics: (1) algorithm is simple and easy to implement, low cost and high yield. (2) execution speed, quick watermark embedding and detection. (3) robust, resistant to breakage trademark, cropping, JPEG compression, noise and other attacks to some extent. (4) wide application areas, suitable for all types of trademarks.