CN101308566B - Digital image watermarking method against geometric attack based on contourlet transform - Google Patents

Abstract

The invention discloses a contourlet-transform-based anti-geometric attack digital image watermarking method, mainly solving the problem of poor robustness of the existing similar method. The invention is technically characterized in that the method completes watermark synchronization through a geometric moment method to realize watermark embedding and extraction in a contourlet transform domain,that is, to calculate a first-order origin moment containing a watermark image I when the watermark is embedded, and reserve the first-order origin moment for the estimation of the geometric attack parameters during extraction; during watermark extraction, to estimate the geometric transformation parameters through the attacked first-order origin moment and a second-order center moment containingwatermarks, and to recover the attacked watermarks through contourlet transform. The method described by the invention has the advantages of strong anti-geometric attack ability, good watermark extracting effect and self-positioning watermark embedded location, which can be used for safety protection of digital multimedia product copyright.

Description

Digital image watermarking method against geometric attack based on contourlet transform

technical field

The invention belongs to the field of image information processing, and relates to a digital image watermarking method, in particular to a geometric attack-resistant digital watermarking method based on contourlet transformation, which can be used to provide technical guarantee for the security of digital multimedia product copyright. the

Background technique

The digital watermarking technology based on image is one of the key research points of current watermarking technology. In recent years, the research of digital watermarking technology has made great progress, especially for image data, there are various watermarking methods, which have been widely studied, but the robustness of watermarking system has always been an obstacle restricting its application. Among them, the watermark synchronization under the geometric transformation attack becomes the decisive factor of whether the watermark technology can be commercialized, and thus gradually becomes the focus of current research in the field of digital watermarking. In order to solve this problem, many scholars have studied and proposed some methods, such as: (1) Embedding the watermark into the geometric invariant domain, and embedding the watermark in the invariant domain of the image. For example, the watermark is embedded in the magnitude space of the Fourier transform domain, which has rotation, scaling and translation invariance. However, this method has high complexity, and the ability of the watermark to resist lossy compression, filtering and other signal processing is also poor. (2) In addition to embedding the watermark, a template is embedded in the image to resist geometric attacks. A mask may consist of extreme points in the DFT magnitude spectrum. But the shortcoming of this kind of method is that in the case of a certain image embedding capacity, the embedding of the known template will reduce the robustness of the watermark information to a certain extent; and once the template is destroyed by the attacker, the watermark detection process cannot be carried out. (3) Embed the watermark into the carrier data with an identifiable structure. The main disadvantage of this class of methods is that they are susceptible to compression. the

To sum up, most of the existing digital watermarking methods that can resist geometric transformation operations are more complex in their implementation process and have poor robustness. Therefore, people have proposed a digital watermarking method based on contourlet transformation. the

Contourlet transform is a new extension of wavelet transform. It is a multi-scale, localized and multi-directional image representation method, which has developed into a "true" two-dimensional signal representation method that can capture geometric structures. Because the contourlet transformation has the advantages of good detail capture effect, anti-noise attack, fast transformation speed, etc., it has been applied to watermarking technology at home and abroad and some researches have been carried out. The currently published digital watermarking methods based on contourlet transformation mainly include:

(1). Li Haifeng [1] proposed a method of directly superimposing the watermark on the large coefficient in the contourlet domain and using the blind signal separation technology to realize the watermark extraction. The embedding effect of this method is good, but the robustness is not good;

(2).Zhao Xu[2] encrypted and embedded the watermark on this basis,

(3).Nadia Baaziz[3] introduced the wavelet domain visual model into the contourlet domain, and proposed a contourlet domain watermarking algorithm based on HVS characteristics after modification, so that the embedding of the contourlet domain watermark can be self-adaptive;

(4). Jayalakshmi M. [4] compared the wavelet domain watermark with the contourlet domain watermark, and proved that the contourlet domain does have a better effect in terms of masking and robustness. the

The above methods involving watermarking in the contourlet domain have a common defect in that they are less effective against geometric attacks, that is, they cannot resist attacks of large-angle rotation, large-scale scaling, and affine transformation. the

[1] Li Haifeng, Song Weiwei, Wang Shuxun, Robust Image Watermarking Algorithm Based on Contourlet Transform, Journal of Communications, Volume 4, 2007, 87-94

[2]Zhao Xu, Ke Wang and Xiao-hua Qiao, Novel Watermarking Scheme in ContourletDomain Based on Independent Component Analysis, computer society, 2006

[3] Nadia Baaziz, Adaptive Watermarking Schemes Based On A Redundant ContourletTransform, IEEE, 2005

[4]Jayalakshmi M., S.N.Merchant and U.B.Desai, Blind Watermarking in ContourletDomain with Improved Detection, computer society, 2006

content of the invention

The purpose of the present invention is to address the shortcomings of the above-mentioned prior art, and propose a digital image watermarking method based on contourlet transformation to resist geometric attacks, so as to realize reliable protection of digital product copyright. the

The technical key to realize the purpose of the present invention is to calculate the first-order origin moment of the watermarked image I when the watermark is embedded, and leave it as the estimation of the parameters of the geometric attack suffered during the extraction, so as to realize the ability of resisting the geometric attack. During the watermark extraction, the geometric transformation parameters are estimated by the first-order origin moment and second-order central moment of the watermarked image after the attack, and the watermark after the attack is restored by using the contourlet transformation. The specific method is as follows:

1. Watermark embedding process

(1) performing contourlet decomposition on the host image I to obtain sub-bands B(i) of low frequency, mid-low frequency, mid-high frequency and high-frequency levels and the transformation coefficient x _i of each sub-band;

(2) Perform contourlet decomposition on the watermark image W to obtain sub-bands B _w (i) of the frequency levels of low frequency, intermediate frequency and high frequency and the transformation coefficient y _i of each sub-band;

(3) Calculate the energy of each subband of each layer of the host image I according to the transformation coefficient x _i of the host image: E _i =∑(xi ₎ ² , i=1, 2, 3...n;

(4) For the energy E _i of each layer of sub-bands, mark them in descending order, and select the sub-band with the highest energy as the watermark embedding target sub-band B _max ;

(5) Arrange the transformation coefficients _yi of each layer of watermark from large to small according to the absolute value, and record the sequence matrix of each layer as the first key R extracted by the watermark;

(6) Arrange the transformation coefficients in the maximum energy sub-band B _max of each layer from large to small in absolute value, and select the first S coefficients, and superimpose according to the relationship corresponding to the size of each transformation coefficient y _i of each watermark, That is to embed the watermark in the host image, the value of the S is the number of coefficients of the watermark image W corresponding to the sub-band of the layer;

(7) Perform contourlet inverse transformation on the transformation coefficient of the host image embedded with the watermark to reconstruct the watermarked image I′;

(8) Calculate the first-order origin moments of the watermarked image I′: m _1,0 , m _0,1 and m _0,0 , and store them in a 1*3 matrix, record it as the second key X, and leave it as Used for parameter estimation when extracting.

2. Watermark extraction process

1) Calculate the first-order origin moment m _1,0 ′, m _0,1 ′, m _0,0 ′ and the second-order central moment μ _1,1 ′, μ _2,0 ′ of the watermarked image I″ after the attack, μ _{0, 2} ′, and use the following formula to estimate the geometric parameters of the geometric attack on the image, namely

angle parameter ${\mathrm{\θ}}_c=\frac{1}{2}{\mathrm{the\; tan}}^{-1}\frac{2{\mathrm{\μ}}_{\mathrm{1,1}}^{\′}}{{\mathrm{\μ}}_{\mathrm{2,0}}^{\′}-{\mathrm{\μ}}_{\mathrm{0,2}}^{\′}},$

Horizontal scaling parameter $a=\frac{m_{\mathrm{0,0}}{m}_{\mathrm{1,0}}^{\′}}{m_{\mathrm{0,0}}^{\′}{m}_{\mathrm{1,0}}},$ Vertical scaling parameter $b=\frac{{m_{\mathrm{0,0}}^{\′}}^2{m}_{\mathrm{1,0}}}{m_{\mathrm{0,0}}^2{m}_{\mathrm{1,0}}^{\′}}$

horizontal translation parameter $p=\frac{m_{\mathrm{1,0}}^{\′}-m_{\mathrm{1,0}}}{m_{0,0}},$ Vertical translation parameter $q=\frac{m_{\mathrm{0,1}}^{\′}-m_{\mathrm{0,1}}}{m_{0,0}}$

2) Reversely operate the attacked image I″ according to the geometric attack parameters of the image to obtain the restored attack image I _r ;

3) Perform contourlet decomposition on the attack restoration image I _r to obtain the sub-bands B _r (i) of low frequency, mid-low frequency, mid-high frequency and high-frequency levels and the transformation coefficient x _i ' of each sub-band;

4) Calculate the energy E _i '=∑(xi _' ) ² of each sub-band in each layer of the attack restoration image I _r , and then arrange the sub-bands B _r (i) in descending order, i=1, 2, 3 ...n

5) Arrange the coefficients in the maximum energy sub-band B _rmax of each layer from large to small in absolute value, select the first S coefficients, that is, the number of coefficients of the watermark image W corresponding to the sub-band of the layer, the value of S and the embedding same time;

6) Subtract the selected coefficients of S from the coefficients in the maximum energy sub-band B _max of each layer in the original host image I after sorting, and obtain the transformed coefficients of each layer of the watermark after extraction.

y _i '=(I _r -I)/α

In the formula, α is the same as when embedding;

7) Restoring the position of the transformed coefficient y _i ' of each layer of the watermark after being attacked by the first key R;

8) Perform contourlet inverse transformation on the attacked watermark transformation coefficient y _i ' to obtain the extracted watermark.

Because the present invention adopts the digital watermark technology based on contourlet transformation, the watermark embedding has the advantages of fast calculation speed and good visual effect; at the same time, because the transformation coefficient suitable for embedding watermark is selected through the control of energy and coefficient size, the selection of watermark embedding position is solved problem, the appropriate coefficients can be independently selected to ensure the concealment of the watermark in security; in addition, by comparing and calculating the geometric moments between the original watermarked image and the attacked watermarked image, the attack on the image can be restored, and image synchronization can be achieved. Watermark extraction can effectively resist various geometric attacks such as rotation, scaling and translation. the

Description of drawings

Fig. 1 is a block diagram of the watermark embedding process of the present invention;

Fig. 2 is a block diagram of the watermark extraction process of the present invention;

Figure 3 is the watermark result map extracted from the unattacked image;

Figure 4 is the watermark result image extracted from the compressed and noise attack image;

Figure 5 is the watermark result map extracted from the rotation attack image;

Figure 6 is the watermark result map extracted from the 3 times enlarged attack image;

Figure 7 is the watermark result map extracted from the 0.8 times reduced attack image;

Figure 8 is the watermark result map extracted from the asymmetric amplification attack image;

Fig. 9 is a graph of watermarking results extracted from synthetic attack images. the

Detailed ways

1. Introduction to basic theory

1. Contourlet transformation

Contourlet transform, a new extension of wavelet transform, is a multi-scale, localized, and multi-directional image representation method, which has been developed into a "true" two-dimensional signal representation capable of capturing geometric structures. The Contourlet transform employs a dual filter bank structure, which first uses a Laplacian decomposition to perform a multi-scale decomposition of the input image to capture point singularities. Each LP decomposition generates a low-frequency sub-band with half the resolution of the original image and a high-frequency sub-band with the same resolution as the original image. The high-frequency sub-band is the difference signal after sampling and filtering between the original image and the low-frequency sub-band. By continuing to use LP transform to iteratively decompose the low-frequency subbands, the original image can be decomposed into a series of low-frequency and high-frequency subbands on different scales. Subsequently, the directional filter bank DFB is used for directional analysis on the high-frequency subbands obtained by LP decomposition. The function of the DFB is to capture the directional high-frequency information of the image, and synthesize the singular points distributed in the same direction into a coefficient. The tree structure decomposition of layer 1 is used in the calculation. In each layer, the frequency segmentation in the fan direction is performed through the fan filter bank QFB, and then it is properly combined with the rotation resampling operation to realize the directional analysis of the high frequency information of the image. Capture line and surface singularities in images. The final result of DFB can be regarded as the high-frequency information of the image divides the frequency domain into 21 wedge-shaped regions. the

2. Geometric moments

Geometric moments include origin moments and central moments. For a two-dimensional function f(x, y)∈L(R ² ) defined on the o-xy plane, its (p+q) order mixed origin moments are defined as:

${\mathrm{<span\; class="notranslate">\; m</span>}}_{\mathrm{<span\; class="notranslate">\; pq</span>}}<span\; class="notranslate">\; =</span>\underset{<span\; class="notranslate">\; -</span><span\; class="notranslate">\; \&infin;</span>}{\overset{<span\; class="notranslate">\; +</span><span\; class="notranslate">\; \&infin;</span>}{<span\; class="notranslate">\; \&Integral;</span>}}\underset{<span\; class="notranslate">\; -</span><span\; class="notranslate">\; \&infin;</span>}{\overset{<span\; class="notranslate">\; +</span><span\; class="notranslate">\; \&infin;</span>}{<span\; class="notranslate">\; \&Integral;</span>}}{\mathrm{<span\; class="notranslate">\; x</span>}}^{\mathrm{<span\; class="notranslate">\; p</span>}}{\mathrm{<span\; class="notranslate">\; the\; y</span>}}^{\mathrm{<span\; class="notranslate">\; q</span>}}\mathrm{<span\; class="notranslate">\; f</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; the\; y</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; dxdy</span>}$

where p, q=0, 1, 2, . . . the

For a digital image f(x, y), its (p+q) order origin moment is defined as:

${\mathrm{<span\; class="notranslate">\; m</span>}}_{\mathrm{<span\; class="notranslate">\; pq</span>}}<span\; class="notranslate">\; =</span>\underset{\mathrm{<span\; class="notranslate">\; i</span>}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}\underset{\mathrm{<span\; class="notranslate">\; j</span>}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}{\mathrm{<span\; class="notranslate">\; i</span>}}^{\mathrm{<span\; class="notranslate">\; p</span>}}{\mathrm{<span\; class="notranslate">\; j</span>}}^{\mathrm{<span\; class="notranslate">\; q</span>}}\mathrm{<span\; class="notranslate">\; f</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; )</span>$

The central moment of the image is defined as:

${\mathrm{<span\; class="notranslate">\; \&mu;</span>}}_{\mathrm{<span\; class="notranslate">\; pq</span>}}<span\; class="notranslate">\; =</span>\underset{\mathrm{<span\; class="notranslate">\; x</span>}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}\underset{\mathrm{<span\; class="notranslate">\; the\; y</span>}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; -</span>\stackrel{<span\; class="notranslate">\; \&OverBar;</span>}{\mathrm{<span\; class="notranslate">\; x</span>}}<span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; p</span>}}{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; the\; y</span>}<span\; class="notranslate">\; -</span>\stackrel{<span\; class="notranslate">\; \&OverBar;</span>}{\mathrm{<span\; class="notranslate">\; the\; y</span>}}<span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; q</span>}}\mathrm{<span\; class="notranslate">\; I</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; the\; y</span>}<span\; class="notranslate">\; )</span>$

in $\stackrel{\&OverBar;}{x}=\frac{m_{\mathrm{1,0}}}{m_{\mathrm{0,0}}},$ $\stackrel{\&OverBar;}{\mathrm{the\; y}}=\frac{m_{\mathrm{0,1}}}{m_{\mathrm{0,0}}}$

Using the affine invariance of the geometric moments of the original image, one or more geometric moments of the original image are used to estimate the geometric transformation of the watermark image before watermark detection, and the watermark image is corrected and detected according to the estimated parameters. This method can be implemented in any domain, including the air domain and various frequency domains, and it is a relatively simple and effective method for resisting geometric attacks. the

2. Explanation of related symbols

I original host image

B(i) The i-th subband of each frequency level of the original host image

x _i original host image transformation coefficient

W watermark signal

B _w (i) the i-th subband of each frequency level of the watermark image

y _i watermark image transformation coefficient

E _i The energy of the i-th subband of each frequency level of the original host image

B _max The maximum energy subband in each frequency level of the original host image

S selects the number of coefficients to embed the watermark

Z _i contains watermarked transform coefficients

u watermark every bit of information

α Embedding Strength

The position matrix of R watermark transformation coefficient sorting

m _{p, q} p+q order origin moment

μ _{p, q} p+q order central moment

X geometric moment parameter key

I'watermarked image

I″Watermarked image after attack

θ _c angle parameter

a Horizontal scaling parameter

b vertical scaling parameter

p horizontal translation parameter

q vertical translation parameter

i _r attack restore image

B _r (i) attack the ith subband of each frequency level of the restored image

B _rmax restores the maximum energy subband in each frequency level of the image

x _i 'Attack to restore image transformation coefficients

y _i 'Watermark image transformation coefficient after being attacked

3. Digital watermark embedding against geometric attacks based on contourlet transform

With reference to Fig. 1, the digital watermark embedding process of the present invention is as follows:

Step 1, perform contourlet decomposition on the host image I. the

Decompose the host image I into 4 layers of contourlet to obtain the sub-bands B(i) of low frequency, mid-low frequency, mid-high frequency and high-frequency levels and the transformation coefficient x _i of each sub-band; due to the poor anti-compression ability of the high-frequency part, and the human The eye is more sensitive to the low-frequency part, so we choose to decompose the image into 4 layers first, and then embed the watermark into the 2nd, 3rd, and 4th layers respectively. Such multi-layer embedding can improve the robustness of the watermark, and disperse the watermark energy so that even if a certain layer of the image is attacked, the watermark energy can still be preserved and restored.

Step 2, perform contourlet decomposition on the watermark image W. the

Decompose the watermark image W into a three-layer contourlet to obtain the subbands Bw(i) of the low frequency, intermediate frequency and high frequency levels and the transformation coefficient yi of each subband, and embed the coefficients of each layer of the watermark image W into the host image respectively Among the coefficients of layers 2, 3, and 4, the watermarked image has a better visual effect. the

Step 3, calculate the host image I subband energy. the

The energy Ei of each subband of each layer of the host image I can be obtained by summing the transform coefficients of the host image I, that is, Ei=Σ(xi) ² . Energy represents the complexity of the sub-band texture, which can be used to judge whether the sub-band is suitable for embedding watermarks.

Step 4, arrange the sub-bands according to the energy size. the

Since the maximum energy represents the richest texture details of the image in this direction, it also implies that the watermark embedded in it has the best concealment, so the maximum energy sub-band Bmax of each layer can be marked in descending order, and the maximum sub-band Target subband with Bmax embedded as watermark. the

Step 5, select the maximum energy sub-band coefficient. the

Arrange the coefficients in the maximum energy sub-band Bmax of each layer from large to small in absolute value, select the first S coefficients, the value of S is the number of coefficients corresponding to the sub-band of the watermark image W, and the important coefficient of contourlet transformation Another feature is that random noise will produce important wavelet coefficients similar to real edges, but will not produce contourlet important coefficients, so choosing to embed watermarks on these coefficients can effectively resist noise attacks. the

Step 6, process the watermark coefficients. the

The coefficients of the watermark are also arranged in descending order of absolute value, and the arrangement sequence matrix R is recorded as the first key. Watermark coefficient processing is to make large coefficients overlap with large coefficients, and small coefficients overlap with small coefficients in the process of watermark embedding and superposition, so as to effectively enhance the robustness of the watermark;

Step 7, watermark embedding. the

Use the superposition formula to superimpose each transformation coefficient y _i of each layer of watermark and the S transformation coefficients in the watermark embedding target subband B _max according to the size, that is, the watermark is embedded in the host image I, and the superposition formula is:

z _i = x _i +αu (u>0) (1)

z _i = x _i -αu (u<0) (2)

Among them, xi represents the transformation coefficient of the original host image, zi represents the transformation coefficient of the watermark, and u represents each bit of information of the watermark. The embedding strength α is selected through experiments. When the visual effect of the restored image after the watermark is superimposed has no obvious noise, the embedding strength can be accepted. In the experiment, α is selected as [0 0.1 0.2 0.3]. Since the low frequency belongs to the smooth area, the embedding strength needs to be selected to be low, and the high frequency belongs to the complex texture area, and the embedding strength can be increased appropriately. the

Step 8, reconstruct the watermarked image. the

By inverse contourlet transforming the coefficients embedded in the watermark, the image containing the watermark can be reconstructed, and the image I' containing the watermark can be output;

Step 9, record geometric moments. the

Calculate the first-order origin moments of the watermarked image I′: m _1,0 , m _0,1 , m _0,0 , store them in a 1*3 matrix as the second key X, and reserve them for parameter estimation during extraction . The specific calculation formula is:

${\mathrm{<span\; class="notranslate">\; m</span>}}_{\mathrm{<span\; class="notranslate">\; 1,0</span>}}<span\; class="notranslate">\; =</span>\underset{\mathrm{<span\; class="notranslate">\; i</span>}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}\underset{\mathrm{<span\; class="notranslate">\; j</span>}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}\mathrm{<span\; class="notranslate">\; i</span>}{<span\; class="notranslate">\; \&times;</span>\mathrm{<span\; class="notranslate">\; I</span>}}^{<span\; class="notranslate">\; \&prime;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 3</span>}<span\; class="notranslate">\; )</span>$

${\mathrm{<span\; class="notranslate">\; m</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; =</span>\underset{\mathrm{<span\; class="notranslate">\; i</span>}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}\underset{\mathrm{<span\; class="notranslate">\; j</span>}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}\mathrm{<span\; class="notranslate">\; j</span>}{<span\; class="notranslate">\; \&times;</span>\mathrm{<span\; class="notranslate">\; I</span>}}^{<span\; class="notranslate">\; \&prime;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 4</span>}<span\; class="notranslate">\; )</span>$

${\mathrm{<span\; class="notranslate">\; m</span>}}_{\mathrm{<span\; class="notranslate">\; 0,0</span>}}<span\; class="notranslate">\; =</span>\underset{\mathrm{<span\; class="notranslate">\; i</span>}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}\underset{\mathrm{<span\; class="notranslate">\; j</span>}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}{\mathrm{<span\; class="notranslate">\; I</span>}}^{<span\; class="notranslate">\; \&prime;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 5</span>}<span\; class="notranslate">\; )</span>$

Among them, i and j represent the coordinates of the pixel of the watermarked image I', and I'(i, j) represents the pixel value of the image at this position. the

4. Digital watermark extraction based on contourlet transform anti-geometric attack

Referring to Fig. 2, the digital watermark extraction process of the present invention is as follows

Step 1, estimate the attack parameters. the

First, use the first-order origin moment formulas (3), (4), and (5) to calculate the first-order origin moment m _1,0 , m _0,1 , m _0,0 of the watermarked image I" after the attack;

Then use the following second-order central moment formula to calculate the second-order central moment μ _1,1 ′, μ _2,0 ′, μ _0,2 ′,

${\mathrm{<span\; class="notranslate">\; \&mu;</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; 1</span>}}^{<span\; class="notranslate">\; \&prime;</span>}<span\; class="notranslate">\; =</span>\underset{\mathrm{<span\; class="notranslate">\; i</span>}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}\underset{\mathrm{<span\; class="notranslate">\; j</span>}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; -</span>\stackrel{<span\; class="notranslate">\; \&OverBar;</span>}{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; \&times;</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; -</span>\stackrel{<span\; class="notranslate">\; \&OverBar;</span>}{\mathrm{<span\; class="notranslate">\; j</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; \&times;</span>{\mathrm{<span\; class="notranslate">\; I</span>}}^{<span\; class="notranslate">\; \&prime;</span><span\; class="notranslate">\; \&prime;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 6</span>}<span\; class="notranslate">\; )</span>$

${\mathrm{<span\; class="notranslate">\; \&mu;</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; 0</span>}}^{<span\; class="notranslate">\; \&prime;</span>}<span\; class="notranslate">\; =</span>\underset{\mathrm{<span\; class="notranslate">\; i</span>}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}\underset{\mathrm{<span\; class="notranslate">\; j</span>}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; -</span>\stackrel{<span\; class="notranslate">\; \&OverBar;</span>}{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; \&times;</span>{\mathrm{<span\; class="notranslate">\; I</span>}}^{<span\; class="notranslate">\; \&prime;</span><span\; class="notranslate">\; \&prime;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 7</span>}<span\; class="notranslate">\; )</span>$

${\mathrm{<span\; class="notranslate">\; \&mu;</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; 2</span>}}^{<span\; class="notranslate">\; \&prime;</span>}<span\; class="notranslate">\; =</span>\underset{\mathrm{<span\; class="notranslate">\; i</span>}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}\underset{\mathrm{<span\; class="notranslate">\; j</span>}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; -</span>\stackrel{<span\; class="notranslate">\; \&OverBar;</span>}{\mathrm{<span\; class="notranslate">\; j</span>}}<span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; \&times;</span>{\mathrm{<span\; class="notranslate">\; I</span>}}^{<span\; class="notranslate">\; \&prime;</span><span\; class="notranslate">\; \&prime;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 8</span>}<span\; class="notranslate">\; )</span>$

Wherein, i, j represent the coordinates of the attacked image I "pixel, and I" (i, j) represents the pixel value of the position image, $\stackrel{\&OverBar;}{i}=\frac{m_{\mathrm{1,0}}^{\&prime;}}{m_{\mathrm{0,0}}^{\&prime;}},$

$\stackrel{<span\; class="notranslate">\; \&OverBar;</span>}{\mathrm{<span\; class="notranslate">\; j</span>}}<span\; class="notranslate">\; =</span>\frac{{\mathrm{<span\; class="notranslate">\; m</span>}}_{\mathrm{<span\; class="notranslate">\; 0,1</span>}}^{<span\; class="notranslate">\; \&prime;</span>}}{{\mathrm{<span\; class="notranslate">\; m</span>}}_{\mathrm{<span\; class="notranslate">\; 0,0</span>}}^{<span\; class="notranslate">\; \&prime;</span>}}<span\; class="notranslate">\; ;</span>$

Finally, use the first-order origin moments in the second key X: m _1,0 , m _0,1 , m _0,0 and the parameter estimation formula to estimate the geometric attack parameters of the image, where:

The angle parameter is: ${\mathrm{\&theta;}}_c=\frac{1}{2}{\mathrm{the\; tan}}^{-1}\frac{2{\mathrm{\&mu;}}_{\mathrm{1,1}}^{\&prime;}}{{\mathrm{\&mu;}}_{\mathrm{2,0}}^{\&prime;}-{\mathrm{\&mu;}}_{\mathrm{0,2}}^{\&prime;}}---\left(9\right)$

The horizontal scaling parameters are: $a=\frac{m_{\mathrm{0,0}}{m}_{\mathrm{1,0}}^{\&prime;}}{m_{\mathrm{0,0}}^{\&prime;}{m}_{\mathrm{1,0}}},---\left(10\right)$

The vertical scaling parameters are: $b=\frac{{m_{\mathrm{0,0}}^{\&prime;}}^2{m}_{\mathrm{1,0}}}{m_{\mathrm{0,0}}^2{m}_{\mathrm{1,0}}^{\&prime;}}---\left(11\right)$

The horizontal translation parameters are: $p=\frac{m_{\mathrm{1,0}}^{\&prime;}-m_{\mathrm{1,0}}}{m_{0,0}}---\left(12\right)$

The vertical translation parameters are: $q=\frac{m_{\mathrm{0,1}}^{\&prime;}-m_{\mathrm{0,1}}}{m_{0,0}}---\left(13\right)$

Step 2, restore the attacked image. the

Restoring the attacked image is the reverse operation of the following steps on the attacked image:

a) Estimate the rotation attack angle _θc suffered by the attacked image I" by the estimation formula (9), and reversely rotate the attacked image I" by the angle _θc to restore the rotation attack suffered by the image;

b) Estimate the horizontal scaling parameter a and vertical scaling parameter b suffered by the attacked image I by estimating formulas (10) and (11), and perform a horizontal scaling of the attacked image I with a ratio of 1/a, and the ratio is 1/ The vertical scaling of b can restore the scaling attack on the image;

c) Estimate the horizontal translation parameter p and vertical translation parameter q suffered by the attacked image I by estimating formulas (12) and (13), and perform a horizontal translation of the size of -p on the attacked image I, and a size of -q Translate vertically to undo the translation attack on the image. the

Step 3, decompose the restored image

Decompose the restored image I _r into four layers of contourlet, and obtain sub-bands B _r (i) of low frequency, mid-low frequency, mid-high frequency and high frequency levels and the transformation coefficient _xi ' of each sub-band.

Step 4, calculate subband energy

Summing the transformation coefficients xi _' of the attack restoration image I _r , the energy E _i ' of each sub-band of each layer of the attack restoration image I _r can be obtained, that is, E _i '=∑(xi _' ); Arrange sub-bands B _r (i) in small order, i=1, 2, 3...n;

Step 5, select the largest subband and coefficient

Arrange the coefficients in the maximum energy sub-band B _rmax of each layer after the attack restoration image I _r is transformed from large to small in absolute value, select the first S coefficients, and the value of S is the number of coefficients in the sub-band corresponding to the watermark image W ;

Step 6, calculating the watermark transformation coefficients. the

After sorting and subtracting the coefficients of the selected S from the coefficients in the maximum energy sub-band B _max of each layer in the original host image I, the conversion coefficients of each layer of the watermark extracted after being attacked are obtained as follows:

y _i '=(I _r -I)/α (14)

where α is the same as when embedding;

Step 7, restore the watermark coefficient position. the

Through the coefficient position information recorded in the first key R, the watermark coefficient is restored to the original position, and the transformation coefficient yi' of each layer of the watermark is reconstructed;

Step 8, restore the watermark

Perform contourlet inverse transformation on the attacked watermark transformation coefficient y _i ' to obtain the extracted watermark.

The effect of the present invention can be further illustrated by the following experimental simulation. the

1. Simulation conditions

Select the 256*256 lena.bmp image as the host image, as shown in Figure 3a, and select the 64*64 binary image for the test, as shown in Figure 3b. The experimental software environment is Matlab7.0. A series of attack tests are designed, including Gaussian low-pass filtering, Wiener filtering, median filtering, salt and pepper noise, Gaussian noise, JPEG attack, shearing, rotation, scaling, translation transformation of various scales, etc., in the maximum attack In the case of strength, the quality of the extracted watermark is evaluated by normalized correlation coefficient NC, peak signal-to-noise ratio PSNR and mean square error RSM. the

2. Simulation results

The experimental results are shown in Fig. 3c, Fig. 3d, Fig. 4, Fig. 5, Fig. 6, Fig. 7, Fig. 8 and Fig. 9 respectively. the

Figure 3c is the result of the composite image after embedding the watermark. Its mean square error is 16.1703 and PSNR can reach 34.9178, which has good visual effect and concealment. the

Figure 3d shows the extracted watermark result without attack, it can be seen that the watermark is intact. the

Figure 4, Figure 5, Figure 6, Figure 7, Figure 8 and Figure 9 all reflect the ability of digital watermarking to resist various attacks. the

Referring to Figure 4, Figure 4a is the result when the JPEG compression quality factor is 10, it can be seen that the watermark can still be clearly identified, and its NC value is 0.9372; Figure 4b is the attack when the salt and pepper noise is added, and the mean square value is 0.03 Watermark extraction results; Figure 4c is the result extracted when the mean square value of Gaussian plus noise is 0.1. It can be seen from Figure 4b and Figure 4c that after the noise attack, the NC value of the watermark remains around 0.9, which can be clearly identified. This is due to the outstanding performance of contourlet transform in suppressing noise, which also brings strong anti-noise attack ability to watermark. the

Referring to Figure 5, Figures 5a, 5b, and 5c are the original image, restored image, and watermark extraction results after a 60-degree right-rotation attack, and Figures 5d, 5e, and 5f are the original image, restored image, and watermark extraction after a 45-degree right-rotation attack As a result, Figures 5g, 5h, and 5i show the results of the original image, the restored image, and the extracted watermark after the 30-degree right-rotation attack. It can be seen from Figure 5 that due to the use of geometric moment parameter estimation technology, the image can be automatically restored after being subjected to rotation attacks, and the NC value of watermark extraction can be kept above 0.8700, showing a strong ability to resist rotation attacks. the

Referring to Fig. 6, Fig. 6a is the original watermarked image, Fig. 6b is the watermarked image after the magnification of 3 times the attack, and Fig. 6c is the watermark result extracted after geometric attack restoration. It can be seen from Fig. 6c that the restored image is synchronized. The NC value of watermark extraction can reach 0.8798. the

Referring to Figure 7, Figure 7a is the original watermarked image, Figure 7b is the watermarked image after being reduced by 0.8 times, and Figure 7c is the watermark result extracted after geometric attack restoration, the restored image is synchronized, and the watermark extracted NC The value can reach 0.9916. It can be seen from Figure 7c that this method has good resistance to scaling attacks, and the extracted watermarks can be clearly identified. the

Referring to Fig. 8, Fig. 8a, Fig. 8b, Fig. 8c are the attack results enlarged horizontally by 3 times, and Fig. 8c, Fig. 8d, Fig. 8e are the results of the vertically enlarged 2x attack. Above 0.88, showing the robustness against asymmetric amplification attacks. the

Referring to Figure 9, Figure 9a is the image after being enlarged twice and rotated by pi/4, Figure 9b is the restored image after geometric attack, and Figure 9c is the result of extracting the watermark. It can be seen from Figure 9c that the NC value of the watermark can reach 0.8815, which proves that for any geometric attack combination, this method can restore the image well, complete the watermark extraction, and realize the protection and identification of digital image copyright. the

The experimental data of the watermark extraction results after all the above attacks are shown in Table 1:

Table 1 Watermark extraction result data after various attacks

Attack Type NC NC PSNR RSM

Gaussian filter 0.9983 28.3421 0.0015

Wiener filter 0.9929 20.8088 0.0083

Median filter 0.9946 21.3524 0.0073

Cut Attack(150*150) 0.9180 11.2240 0.0754

Translation 1.0000 36.1236 0

JPEG Attack(10) 0.9372 10.6459 0.0862

Salt and pepper noise (0.03) 0.9065 10.0490 0.0989

Gaussian noise (0.1) 0.8913 9.8499 0.1035

Spin Attack (pi/3) 0.8997 9.2216 0.1196

Spin Attack (pi/4) 0.9153 9.2662 0.1184

Spin Attack (pi/5) 0.9912 17.9282 0.0161

Expand 3 times 0.8798 8.4224 0.1438

Reduced by 0.8 times 0.9916 19.9958 0.0100

Asymmetric magnification 0.8815 8.5118 0.1409

Magnify twice rotate pi/4 0.9392 10.4769 0.0896

It can be seen from Table 1 that the watermarking algorithm based on the contourlet domain has the anti-filtering characteristics of the frequency domain algorithm itself, and the NC value of the watermark extraction under the traditional filtering attack remains above 0.9900. Due to the transformation domain embedding method, the embedding position is uniform, which can effectively resist the shearing attack. After subtracting the 150*150 pixels of the whole image, the NC value of the extracted watermark can reach 0.9180. The translation attack does not cause any damage to the watermark extraction, and the NC value is 1.0000. In addition, due to the use of geometric moment reduction geometric attack technology, this method has strong resistance to various geometric attacks. The experimental results prove that this is a comprehensive and robust watermark embedding extraction method. the

Claims (5)

1. one kind based on contourlet conversion resist geometric attacks digital image watermark embedding method, comprises following process:

(1) host image I is carried out contourlet and decompose, obtain the subband B (i) of low frequency, medium and low frequency, medium-high frequency and high frequency level and the conversion coefficient x of each subband _i

(2) watermarking images W is carried out contourlet and decompose, obtain the subband B of the frequency level of low frequency, intermediate frequency and high frequency _w(i) and the conversion coefficient y of each subband _i

(3) according to the conversion coefficient x of host image _i, the energy of every layer of each subband of calculating host image I: E _i=∑ (x _i) ², i=1,2,3 ... n;

(4) to the ENERGY E of every straton band _i, according to sequence notation from big to small, the subband of selecting the energy maximum is that watermark embeds target subband B _Max

(5) with the maximum subband B of every layer of energy _MaxIn conversion coefficient arrange from big to small by absolute value, and S coefficient before selecting, the value of this S is watermarking images W to coefficient number that should the straton band;

(6) with the conversion coefficient y of every layer of watermark _iArrange from big to small according to absolute value, every layer the matrix that puts in order is designated as the first key R of watermark extracting;

(7) with each conversion coefficient y of every layer of watermark _iEmbed target subband B with watermark _MaxIn S conversion coefficient according to the corresponding stack of size, be about to watermark and be embedded in the host image;

(8) the host image conversion coefficient with embed watermark carries out the contourlet inverse transformation, reconstruct the image I that contains watermark ';

(9) calculate the first moment about the origin that contains watermarking images I ': m _1,0, m _0,1, m _0,0, depositing them in the 1*3 matrix and be designated as the second key X, parameter estimation is used when staying to extraction.

2. watermark embedding method according to claim 1, wherein step (7) is described is embedded in watermark in the host image, is to carry out according to following formula:

z _i＝x _i+αu(u＞0)

z _i＝x _i-αu(u＜0)

Wherein, x _iRepresent original host image conversion coefficient, z _iRepresentative contains the watermark conversion coefficient, and u represents each information of watermark, and embedment strength α is selected by test.

3. watermark embedding method according to claim 1, wherein step (9) is carried out according to the following procedure:

$m_{\mathrm{1,0}}=\underset{i}{\mathrm{\&Sigma;}}\underset{j}{\mathrm{\&Sigma;}}i\&times;I^{\&prime;}(i,j)$

$m_{\mathrm{0,1}}=\underset{i}{\mathrm{\&Sigma;}}\underset{j}{\mathrm{\&Sigma;}}j\&times;I^{\&prime;}(i,j)$

$m_{0,0}=\underset{i}{\mathrm{\&Sigma;}}\underset{j}{\mathrm{\&Sigma;}}{I}^{\&prime;}(i,j)$

Wherein, i, j representative contains the coordinate of watermarking images I ' pixel, and (i j) represents the pixel value of this location drawing picture to I '.

4. one kind based on contourlet conversion resist geometric attacks digital image watermark extraction method, comprises following process:

(1) calculate after attacking and contain watermarking images I " first moment about the origin m _1,0', m _0,1', m _0,0' and second-order moment around mean μ _1,1', μ _2,0', μ _0,2', utilize the first moment about the origin among the second key X: m _1,0, m _0,1, m _0,0With the parameter estimation formula, estimate the suffered geometric attack parameter of image, wherein:

Angle parameter theta _c ${\mathrm{\&theta;}}_c=\frac{1}{2}{\tan }^{-1}\frac{{2\mathrm{\&mu;}}_{\mathrm{1,1}}^{\&prime;}}{{\mathrm{\&mu;}}_{2,0}^{\&prime;}{-\mathrm{\&mu;}}_{\mathrm{0,2}}^{\&prime;}}$

The horizontal scaling parameter $a=\frac{m_{\mathrm{0,0}}{m}_{\mathrm{1,0}}^{\&prime;}}{m_{\mathrm{0,0}}^{\&prime;}{m}_{\mathrm{1,0}}},$ The vertically scale parameter $b=\frac{{m_{\mathrm{0,0}}^{\&prime;}}^2{m}_{\mathrm{1,0}}}{m_{\mathrm{0,0}}^2{m}_{\mathrm{1,0}}^{\&prime;}}$

The horizontal translation parameter $p=\frac{m_{\mathrm{1,0}}^{\&prime;}-m_{\mathrm{1,0}}}{m_{\mathrm{0,0}}},$ The vertical translation parameter $q=\frac{m_{0,1}^{\&prime;}-m_{0,1}}{m_{\mathrm{0,0}}};$

(2) at these geometric attack parameters image I under fire " is carried out reverse operating, obtained attacking the reduction image I _r

(3) will attack the reduction image I _rCarry out contourlet and decompose, obtain the subband B of low frequency, medium and low frequency, medium-high frequency and high frequency level _r(i) and the conversion coefficient x of each subband _i';

(4) calculate attack reduction image I _rThe ENERGY E of every layer of each subband _i'=∑ (x _i') ², then by descending series arrangement subband B _r(i), i=1,2,3 ... n;

(5) with the maximum subband B of every layer of energy _RmaxIn coefficient arrange from big to small by absolute value, S coefficient before selecting is watermarking images W to coefficient number that should the straton band, the value of S is identical during with embedding;

(6) with every layer of maximum subband B of energy among each coefficient of selected S and the former host image I _MaxSubtract each other after the middle coefficient ordering, obtain extracting every layer of conversion coefficient of back watermark under fire

y _i’＝(I _r-I)/α

It is identical when wherein α is with embedding;

(7) by every layer of conversion coefficient y of watermark after the first key R reduction under fire _i' position;

(8) incite somebody to action back watermark conversion coefficient y under fire _i' carry out contourlet inverse transformation, the watermark that obtains extracting.

5. watermark extracting method according to claim 4, wherein the described calculating of step (1) contains watermarking images I after attacking " first moment about the origin m _1,0', m _0,1', m _0,0' and second-order moment around mean μ _1,1', μ _2,0', μ _0,2', be calculated as follows:

$m_{\mathrm{1,0}}^{\&prime;}=\underset{i}{\mathrm{\&Sigma;}}\underset{j}{\mathrm{\&Sigma;}}i\&times;I^{\&prime;\&prime;}(i,j)$

$m_{\mathrm{0,1}}^{\&prime;}=\underset{i}{\mathrm{\&Sigma;}}\underset{j}{\mathrm{\&Sigma;}}j\&times;I^{\&prime;\&prime;}(i,j)$

$m_{\mathrm{0,0}}^{\&prime;}=\underset{i}{\mathrm{\&Sigma;}}\underset{j}{\mathrm{\&Sigma;}}{I}^{\&prime;\&prime;}(i,j)$

${\mathrm{\&mu;}}_{\mathrm{1,1}}^{\&prime;}=\underset{i}{\mathrm{\&Sigma;}}\underset{j}{\mathrm{\&Sigma;}}(i-\stackrel{\&OverBar;}{i})\&times;(j-\stackrel{\&OverBar;}{j})\&times;I^{\&prime;\&prime;}(i,j)$

${\mathrm{\&mu;}}_{\mathrm{2,0}}^{\&prime;}=\underset{i}{\mathrm{\&Sigma;}}\underset{j}{\mathrm{\&Sigma;}}{(i-\stackrel{\&OverBar;}{i})}^2\&times;I^{\&prime;\&prime;}(i,j)$

${\mathrm{\&mu;}}_{0,2}^{\&prime;}=\underset{i}{\mathrm{\&Sigma;}}\underset{j}{\mathrm{\&Sigma;}}{(j-\stackrel{\&OverBar;}{j})}^2\&times;I^{\&prime;\&prime;}(i,j)$

Wherein, i, j represent image I under fire " coordinate of pixel, I " (i j) represents the pixel value of this location drawing picture, and:

$\stackrel{\&OverBar;}{i}=\frac{m_{\mathrm{1,0}}^{\&prime;}}{m_{\mathrm{0,0}}^{\&prime;}}$ $\stackrel{\&OverBar;}{j}=\frac{m_{\mathrm{0,1}}^{\&prime;}}{m_{\mathrm{0,0}}^{\&prime;}}.$

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