CN103247017A - Mesh segmentation-based non-blind watermark realization method for three-dimensional mesh model - Google Patents

Abstract

The invention provides a mesh segmentation-based non-blind watermark realization method for a three-dimensional mesh model, which comprises a watermark information embedment process and a watermark information detection process. The watermark information embedment process comprises the steps of significantly segmenting a three-dimensional mesh model, then calculating the robust gravity center of every segmentation block, taking every gravity center as a center to convert rectangle coordinates of the corresponding segmentation block into spherical coordinates, and finally, embedding the watermark sequence through modulating climax norm distribution of every segmentation block. The watermark information detection process comprises the steps of allowing a model to be detected and a master model to be subjected to aligning and re-sampling operation, and performing watermark extraction after the segmentation block information of the model to be detected is determined. The method provided by the invention can resist single convention attack, such as simplification, shearing and noise, and shows stronger robust in resisting combined attack.

Description

The non-blind watermark implementing method of cutting apart based on grid of three-dimensional grid model

Technical field

The present invention relates to the multi-media information security technical field, particularly a kind of non-blind watermark implementing method of cutting apart based on grid of three-dimensional grid model.

Background technology

Because the watermarking algorithm of the overall situation is not high to the robustness of shearing attack, therefore people have proposed the watermarking algorithm based on piecemeal, main thought is earlier three-dimensional model to be cut apart or extracted several meaningful or insignificant piecemeals, then watermark data is embedded into respectively in the different piecemeals, be that each piecemeal has one group or some groups of watermark datas, when watermark extracting, as long as there have some piecemeals to preserve to be complete, so just can from these piecemeals, extract complete watermark information.

Usually there is following problem in this type of 3D grid watermark based on piecemeal: 1) after model is under attack, the grid piecemeal that carries out when extracting watermark can not obtain a piecemeal result consistent with master pattern, especially when the grid shearing takes place; 2) the grid block division method generally can not guarantee the boundary problem of sectional pattern, because many points or few point be not to visually can impacting that grid is cut apart on the border, therefore can cause the watermark data that extracts inaccurate because boundary segmentation is inconsistent slightly for the watermarking algorithm that relies on the model partitioning boundary.

Summary of the invention

For solving deficiency of the prior art, the present invention is intended to provide a kind of non-blind watermark implementing method of cutting apart based on grid of three-dimensional grid model, this method can be resisted conventional single attacks such as simplification, shearing, noise, and also shows stronger robustness on opposing gangs up against.

For achieving the above object, the invention provides a kind of non-blind watermark implementing method of cutting apart based on grid of three-dimensional grid model, comprise watermark information telescopiny and watermark information testing process, wherein: the watermark information telescopiny is: three-dimensional grid model is carried out significant cutting apart, then each piecemeal is calculated the robust center of gravity, and centered by this center of gravity, convert piecemeal to spherical co-ordinate by rectangular coordinate, distribute the embed watermark sequence by the summit norm of modulating in each piecemeal again; The watermark information testing process is: treat that detection model heavily aligns with master pattern, re-sampling operations, and determine to carry out watermark extracting again behind the branch block message of model to be detected.

Further, in the described method, the implementation of watermark information telescopiny is as follows:

Step 1.1 is carried out meaningful cutting apart based on the partitioning algorithm of shape diameter function to model, removes the very few piecemeal of dough sheet, gets k piecemeal embed watermark, and then each piecemeal can be expressed as M _α=(V _α, F _α), α=1,2 ..., k, V _αAnd F _αBe expressed as α summit and the dough sheet information in the piecemeal;

Step 1.2 is calculated each piecemeal M _αRobust center of gravity m _α

Step 1.3 is calculated corresponding summit norm for each piecemeal.With robust center of gravity m _α(x _α, y _α, z _α) centered by with each vertex v _i(x _i, y _i, z _i) ∈ M _αBe transformed into spheric coordinate system v by lineal coordinate system _i(ρ _i, θ _i, φ _i), get norm: ρ to the limit after the unitization _i=(ρ _i-ρ _Min)/(ρ _Max-ρ _Min), wherein, ρ _MaxWith ρ _MinBe piecemeal M _ρIn maximal value and the minimum value of the ρ that has a few;

Step 1.4 is with piecemeal M _ρIn all summits according to the value of ρ after the unitization be divided into L interval, each interval table is shown

J=0,1 ..., L-1, L is corresponding to watermark sequence length;

Step 1.5 couple interval I _jIn each vertex v _p(ρ _p, θ _p, φ _p) ∈ I _jThe summit norm carry out unitization: ${\mathrm{\ρ}}_p=({\mathrm{\ρ}}_p-\frac{j}{L})\×L;$

Step 1.6 is for embed watermark position w _j=1 situation, computation interval I _jIn the mean value of all summit norms

If Then think interval I _jEmbed watermark position w _j, enter step 1.8, otherwise enter step 1.7, wherein, Δ is expressed as watermark strength, 0＜Δ＜0.5;

Step 1.7 is for interval I _jIn all

The summit, revise ρ _pMake

Forward step 1.6 to;

Step 1.8 is for interval I _jAll summits:

${\mathrm{\ρ}}_p={\mathrm{\ρ}}_p\×\frac{1}{L}+j\×\frac{1}{L};$

Step 1.9 is for piecemeal M _αEach interval I _jAfter all finishing dealing with, obtain the new spherical co-ordinate value v in each summit _i' (ρ _i', θ _i, φ _i), reduction ρ _i'=ρ _i' (ρ _Max-ρ _Min)+ρ _Min, and convert spherical co-ordinate to rectangular coordinate v again _i' (x _i', y _i', z _i').

Wherein, for embed watermark position w _j=-1 situation need only be with the Rule of judgment of step 1.6

Change into

And with the Rule of judgment in the step 1.7

Change into

And with the ρ in the step 1.7 _pValue

Be revised as and change ρ into _p=ρ _p ²Get final product, wherein, Δ is expressed as watermark strength, 0＜Δ＜0.5.

Further, in the described method, the implementation of watermark information testing process is as follows:

Step 2.1 uses the partitioning algorithm based on shape diameter function to carry out significant division according to original mesh model and the partitioning parameters of storage, removes the very few piecemeal of dough sheet and obtains k piecemeal, and mark is carried out on the summit of each piecemeal;

Step 2.2 treats detection model and master pattern carries out reorientation, re-sampling operations, resampling to the summit carry out mark according to the summit of step 2.1 mark, according to energy function border vertices is carried out final mark again after the resampling, obtain k final on a model to be detected piecemeal, reach and treat the result that detection model is cut apart;

Step 2.3 is for each piecemeal M _α ^d, α=1 ..., k calculates its robust center of gravity m _α ^d(x _α ^d, y _α ^d, z _α ^d), centered by the robust center of gravity with the rectangular coordinate v on each summit _i(x _i, y _i, z _i) ∈ M _α ^dBe converted to spherical co-ordinate v _i(ρ _i, θ _i, φ _i), get norm: ρ to the limit after the unitization _i=(ρ _i-ρ _Min)/(ρ _Max-ρ _Min), wherein, ρ _MaxWith ρ _MinBe piecemeal M _αIn maximal value and the minimum value of the ρ that has a few, again the summit norm is divided into L interval, then each interval can be expressed as

J=0,1 ..., L-1, L are watermark sequence length;

Step 2.4 hypothesis piecemeal M _α ^dIn the watermark sequence that comprises be W _α ^d=(w _{α 0} ^d, w _{α 1} ^d..., w _{α (L-1)} ^d), calculate interval I _jThe mean value of middle summit norm is

Then detected watermark bit is ${w_{\mathrm{\αj}}}^d=\left\{\begin{array}{cc}+1,& \stackrel{\‾}{{\mathrm{\ρ}}_j}>(j+0.5)\×\frac{1}{L}\\ -1,& \stackrel{\‾}{{\mathrm{\ρ}}_j}\≤(j+0.5)\×\frac{1}{L}\end{array}\right.,0\≤j\≤L-1;$

Step 2.5 is carried out aforesaid operations to each piecemeal can draw k watermark sequence { W _α ^d| α=1 ..., k} carries out sum operation Σ W to the same watermark bit of each watermark sequence _α ^d=(Σ w _{α 0}, Σ w _{α 1}..., Σ w _{α (L-1)}), then each watermark bit is:

${w_j}^d=\left\{\begin{array}{cc}+1,& \mathrm{\Σ}{w}_{\mathrm{\αj}}\≥0\\ -1,& \mathrm{\Σ}{w}_{\mathrm{\αj}}>0\end{array}\right.,0\≤j\≤L-1,$

Obtain final detected watermark sequence W ^d=(w ₀ ^d, w ₁ ^d..., w _L-1 ^d).

By above technical scheme of the present invention as can be known, beneficial effect of the present invention is to select the partitioning algorithm based on shape diameter function that three-dimensional grid model is carried out meaningful division, the segmentation result that obtains is stable, can not cause the inconsistent of grid segmentation result because of problem such as unique point selection.And the distribution of choosing the summit norm of piecemeal embeds primitive as watermark, uses the method for statistics, makes algorithm can avoid effectively based on the watermarking algorithm of the piecemeal dependence to partitioning boundary.At last, use the method for non-blind Detecting to detect watermark, make algorithm more have with reference to property in the watermark detection stage, and in the resampling stage, by master pattern piecemeal mark is cut apart model to be detected, make the model of watermark embedding and detection-phase cut apart and keep consistency, on the whole, technical scheme of the present invention can be resisted conventional single attacks such as simplification, shearing, noise, and also show stronger robustness on opposing gangs up against on robustness.

Description of drawings

Fig. 1 is watermark information telescopiny synoptic diagram in the non-blind watermark implementing method of preferred embodiment of the present invention.

Fig. 2 is watermark information testing process synoptic diagram in the non-blind watermark implementing method of preferred embodiment of the present invention.

Embodiment

In order more to understand technology contents of the present invention, especially exemplified by specific embodiment and cooperate appended graphic being described as follows.

With reference to illustrated in figures 1 and 2, according to preferred embodiment of the present invention, the non-blind watermark implementing method of cutting apart based on grid of three-dimensional grid model comprises watermark information telescopiny and watermark information testing process, wherein: the watermark information telescopiny is: three-dimensional grid model is carried out significant cutting apart, then each piecemeal is calculated the robust center of gravity, and centered by this center of gravity, convert piecemeal to spherical co-ordinate by rectangular coordinate, distribute the embed watermark sequence by the summit norm of modulating in each piecemeal again; The watermark information testing process is: treat that detection model heavily aligns with master pattern, re-sampling operations, and determine to carry out watermark extracting again behind the branch block message of model to be detected.

The watermark information telescopiny specifically may further comprise the steps:

Step 1.1 is carried out meaningful cutting apart based on the partitioning algorithm of shape diameter function to the three-dimensional network model, the shape diameter value of elder generation's computing grid model, carry out GMM-EM (Gaussian mixture model exception-maximization) cluster again, remove the very few piecemeal of dough sheet, at last the cutting border is become more meticulous, obtain a final k piecemeal and carry out the watermark embedding, then each piecemeal can be expressed as M _α=(V _α, F _α), α=1,2 ..., k, V _αAnd F _αBe expressed as α summit and the dough sheet information in the piecemeal;

Step 1.2 is calculated the robust center of gravity of each piecemeal: for each piecemeal M _αCalculate corresponding robust center of gravity

Area (f _i) (A _i+ B _i+ C _i)/3, wherein, m _αRepresentation model piecemeal M _αCenter of gravity, A _i, B _i, C _iTri patch f is formed in expression _iThree apex coordinate values, the area of area () expression current region;

Step 1.3 is calculated corresponding summit norm: with robust center of gravity m for each piecemeal _α(x _α, y _α, z _α) centered by with each vertex v _i(x _i, y _i, z _i) ∈ M _αBe transformed into spheric coordinate system v by lineal coordinate system _i(ρ _i, θ _i, φ _i),

Wherein,

${\mathrm{\ρ}}_i=\sqrt{{(x_i-x_{\mathrm{\α}})}^2+{(y_i-y_{\mathrm{\α}})}^2+{(z_i-z_{\mathrm{\α}})}^2},$

${\mathrm{\θ}}_i={\tan }^{-1}\frac{(y_i-y_{\mathrm{\α}})}{(x_i-x_{\mathrm{\α}})},$

${\mathrm{\φ}}_i={\cos }^{-1}\frac{(z_i-z_{\mathrm{\α}})}{\sqrt{{(x_i-x_{\mathrm{\α}})}^2+{(y_i-y_{\mathrm{\α}})}^2+{(z_i-z_{\mathrm{\α}})}^2}},$

Unitization gets norm to the limit: ρ _i=(ρ _i-ρ _Min)/(ρ _Max-ρ _Min), wherein, ρ _MaxWith ρ _MinBe piecemeal M _αIn maximal value and the minimum value of the ρ that has a few;

Step 1.4 is with piecemeal M _αIn all summits according to the value of ρ after the unitization be divided into L interval, each interval table is shown

J=0,1 ..., L-1, L is corresponding to watermark sequence length;

Step 1.5 couple interval I _jIn each vertex v _p(ρ _p, θ _p, φ _p) ∈ I _jThe summit norm carry out unitization: ${\mathrm{\ρ}}_p=({\mathrm{\ρ}}_p-\frac{j}{L})\×L;$

Step 1.6 is for embed watermark position w _j=1 situation, computation interval I _jIn the mean value of all summit norms

If

Then think interval I _jEmbed watermark position w _j, enter step 1.8, otherwise enter step 1.7;

Step 1.7 is for interval I _jIn all

The summit, revise ρ _pMake Forward step 1.6 to;

Step 1.8 is for interval I _jAll summits:

For embed watermark position w _j=-1 situation need only be with the Rule of judgment of step 1.6

Change into

And with the Rule of judgment in the step 1.7 $\stackrel{\&OverBar;}{{\mathrm{\&rho;}}_j}<=0.5+\mathrm{\&Delta;}$ Change into $\stackrel{\&OverBar;}{{\mathrm{\&rho;}}_j}>=0.5+\mathrm{\&Delta;}$ With ${\mathrm{\&rho;}}_p=\sqrt{{\mathrm{\&rho;}}_p}$ Change ρ into _p=ρ _p ²Get final product.Wherein, Δ is expressed as watermark strength, 0＜Δ＜0.5.

Step 1.9 is for piecemeal M _αEach interval I _jAfter all finishing dealing with, obtain the new spherical co-ordinate value v in each summit _i' (ρ _i', θ _i, φ _i), reduction ρ _i'=ρ _i' (ρ _Max-ρ _Min)+ρ _Min, and convert spherical co-ordinate to rectangular coordinate v again _i' (x _i', y _i', z _i').

Handle all model piecemeals, can obtain behind the embed watermark model M ', the three-dimensional model of issue be behind the embed watermark model M ', then master pattern and partitioning parameters are preserved together, the foundation during as watermark extracting is used in order to extracting.

The watermark information testing process is: use the method for non-blind extraction to detect watermark, treat that detection model heavily aligns with master pattern, re-sampling operations, and carry out watermark extracting again behind the branch block message of definite model to be detected, testing process is similar to the inverse process of watermark embed process.Concrete steps are as follows:

Step 2.1 is carried out as step 1.1 identical operations master pattern according to original mesh model and the partitioning parameters of storage, obtains k piecemeal, and mark is carried out on the summit in each piecemeal;

After step 2.2 is used S-ICP (scaled iterative closet point) algorithm to treat detection model and master pattern to carry out reorientation and operate, carry out re-sampling operations again, resampling to the summit carry out mark according to the summit of step 2.1 mark, the resampling step is carried out final mark according to energy function to border vertices after finishing again, obtain k final on a model to be detected piecemeal, reach and treat the result that detection model is cut apart;

Step 2.3 is for each piecemeal M _α ^d, α=1 ..., k carries out step 1.2,1.3 operation equally, calculates robust center of gravity m _α ^d(x _α ^d, y _α ^d, z _α ^d), with the rectangular coordinate v on each summit _i(x _i, y _i, z _i) ∈ M _α ^dBe converted to spherical co-ordinate v _i(ρ _i, θ _i, φ _i), and with the summit norm after the unitization be divided into L interval, then each interval can be expressed as

J=0,1 ..., L-1, L are watermark sequence length;

Step 2.4 hypothesis piecemeal M _α ^dIn watermark sequence to be detected be W _α ^d=(w _{α 0} ^d, w _{α 1} ^d..., w _{α (L-1)} ^d), calculate interval I _jThe mean value of middle summit norm is

Then detected watermark bit is:

${w_{\mathrm{\&alpha;j}}}^d=\left\{\begin{array}{cc}+1,& \stackrel{\&OverBar;}{{\mathrm{\&rho;}}_j}>(j+0.5)\&times;\frac{1}{L}\\ -1,& \stackrel{\&OverBar;}{{\mathrm{\&rho;}}_j}\&le;(j+0.5)\&times;\frac{1}{L}\end{array}\right.,0\&le;j\&le;L-1;$

Step 2.5 is carried out aforesaid operations to each piecemeal can draw k watermark sequence { W _α ^d| α=1 ..., k} carries out sum operation Σ W to the same watermark bit of each watermark sequence _α ^d=(Σ w _{α 0}, Σ w _{α 1}..., Σ w _{α (L-1)}), then each watermark bit is:

${w_j}^d=\left\{\begin{array}{cc}+1,& \mathrm{\&Sigma;}{w}_{\mathrm{\&alpha;j}}\&GreaterEqual;0\\ -1,& \mathrm{\&Sigma;}{w}_{\mathrm{\&alpha;j}}>0\end{array}\right.,0\&le;j\&le;L-1,$

Obtain final detected watermark sequence W ^d=(w ₀ ^d, w ₁ ^d..., w _L-1 ^d).

In sum, the present invention is based on the partitioning algorithm that the non-blind watermark implementing method of three-dimensional grid model that grid cuts apart chooses based on shape diameter function three-dimensional grid model is carried out meaningful division, the segmentation result that obtains is stable, can not cause the inconsistent of grid segmentation result because of problem such as unique point selection.And the distribution of choosing the summit norm of piecemeal embeds primitive as watermark, uses the method for statistics, makes algorithm can avoid effectively based on the watermarking algorithm of the piecemeal dependence to partitioning boundary.At last, use the method for non-blind Detecting to detect watermark, make algorithm more have with reference to property in the watermark detection stage, and in the resampling stage, by master pattern piecemeal mark is cut apart model to be detected, make the model of watermark embedding and detection-phase cut apart and keep consistency, on the whole, technical scheme of the present invention can be resisted conventional single attacks such as simplification, shearing, noise, and also show stronger robustness on opposing gangs up against on robustness.

Though the present invention discloses as above with preferred embodiment, so it is not in order to limit the present invention.The persond having ordinary knowledge in the technical field of the present invention, without departing from the spirit and scope of the present invention, when being used for a variety of modifications and variations.Therefore, protection scope of the present invention is as the criterion when looking claims person of defining.

Claims (3)

1. non-blind watermark implementing method of cutting apart based on grid of three-dimensional grid model, it is characterized in that, comprise watermark information telescopiny and watermark information testing process, wherein: the watermark information telescopiny is: three-dimensional grid model is carried out significant cutting apart, then each piecemeal is calculated the robust center of gravity, and centered by this center of gravity, convert piecemeal to spherical co-ordinate by rectangular coordinate, distribute the embed watermark sequence by the summit norm of modulating in each piecemeal again; The watermark information testing process is: treat that detection model heavily aligns with master pattern, re-sampling operations, and determine to carry out watermark extracting again behind the branch block message of model to be detected.

2. the non-blind watermark implementing method of cutting apart based on grid of three-dimensional grid model according to claim 1 is characterized in that the implementation of described watermark information telescopiny is as follows:

Step 1.1 is carried out meaningful cutting apart based on the partitioning algorithm of shape diameter function to model, removes the very few piecemeal of dough sheet, gets k piecemeal embed watermark, and then each piecemeal can be expressed as M _α=(V _α, F _α), α=1,2 ..., k, V _αAnd F _αBe expressed as α summit and the dough sheet information in the piecemeal;

Step 1.2 is calculated each piecemeal M _αRobust center of gravity m _α

Step 1.3 is calculated corresponding summit norm for each piecemeal.With robust center of gravity m _α(x _α, y _α, z _α) centered by with each vertex v _i(x _i, y _i, z _i) ∈ M _αBe transformed into spheric coordinate system v by rectangular coordinate system _i(ρ _i, θ _i, φ _i), get norm: ρ to the limit after the unitization _i=(ρ _i-ρ _Min)/(ρ _Max-ρ _Min), wherein, ρ _MaxWith ρ _MinBe piecemeal M _αIn maximal value and the minimum value of the ρ that has a few;

Step 1.4 is with piecemeal M _αIn all summits according to the value of ρ after the unitization be divided into L interval, each interval table is shown

J=0,1 ..., L-1, L is corresponding to watermark sequence length;

Step 1.5 couple interval I _jIn each vertex v _p(ρ _p, θ _p, φ _p) ∈ I _jThe summit norm carry out unitization: ${\mathrm{\&rho;}}_p=({\mathrm{\&rho;}}_p-\frac{j}{L})\&times;L;$

Step 1.6 is for embed watermark position w _j=1 situation, computation interval I _jIn the mean value of all summit norms

If

Then think interval I _jEmbed watermark position w _j, enter step 1.8, otherwise enter step 1.7, wherein, Δ is expressed as watermark strength, 0＜Δ＜0.5;

Step 1.7 is for interval I _jIn all The summit, revise ρ _pMake

Forward step 1.6 to;

Step 1.8 is for interval I _jAll summits:

${\mathrm{\&rho;}}_p={\mathrm{\&rho;}}_p\&times;\frac{1}{L}+j\&times;\frac{1}{L};$

Step 1.9 is for piecemeal M _αEach interval I _jAfter all finishing dealing with, obtain the new spherical co-ordinate value v in each summit _i' (ρ _i', θ _i, φ _i), reduction ρ _i'=ρ _i' (ρ _Max-ρ _Min)+ρ _Min, and convert spherical co-ordinate to rectangular coordinate v again _i' (x _i', y _i', z _i').

Wherein, for embed watermark position w _j=-1 situation need only be with the Rule of judgment of step 1.6

Change into

And with the Rule of judgment in the step 1.7

Change into

And with the ρ in the step 1.7 _pValue

Be revised as and change ρ into _p=ρ _p ²Get final product.

3. the non-blind watermark implementing method of cutting apart based on grid of three-dimensional grid model according to claim 2 is characterized in that the implementation of described watermark information testing process is as follows:

Step 2.1 uses the partitioning algorithm based on shape diameter function to carry out significant division according to original mesh model and the partitioning parameters of storage, removes the very few piecemeal of dough sheet and obtains k piecemeal, and mark is carried out on the summit of each piecemeal;

Step 2.2 treats detection model and master pattern carries out reorientation, re-sampling operations, resampling to the summit carry out mark according to the summit of step 2.1 mark, according to energy function border vertices is carried out final mark again after the resampling, obtain k final on a model to be detected piecemeal, reach and treat the result that detection model is cut apart;

Step 2.3 is for each piecemeal M _α ^d, α=1 ..., k calculates its robust center of gravity m _α ^d(x _α ^d, y _α ^d, z _α ^d), with the rectangular coordinate v on each summit _i(x _i, y _i, z _i) ∈ M _α ^dBe converted to spherical co-ordinate v _i(ρ _i, θ _i, φ _i), get norm: ρ to the limit after the unitization _i=(ρ _i-ρ _Min)/(ρ _Max-ρ _Min), wherein, ρ _MaxWith ρ _MinBe piecemeal M _αIn maximal value and the minimum value of the ρ that has a few, again the summit norm is divided into L interval, then each interval can be expressed as

J=0,1 ..., L-1, L are watermark sequence length;

Step 2.4 hypothesis piecemeal M _α ^dIn the watermark sequence that comprises be W _α ^d=(w _{α 0} ^d, w _{α 1} ^d..., w _{α (L-1)} ^d), calculate interval I _jThe mean value of middle summit norm is Then detected watermark bit is ${w_{\mathrm{\&alpha;j}}}^d=\left\{\begin{array}{cc}+1,& \stackrel{\&OverBar;}{{\mathrm{\&rho;}}_j}>(j+0.5)\&times;\frac{1}{L}\\ -1,& \stackrel{\&OverBar;}{{\mathrm{\&rho;}}_j}\&le;(j+0.5)\&times;\frac{1}{L}\end{array}\right.,$ 0≤j≤L-1；

Step 2.5 is carried out aforesaid operations to each piecemeal can draw k watermark sequence { W _α ^d| α=1 ..., k} carries out sum operation Σ W to the same watermark bit of each watermark sequence _α ^d=(Σ w _{α 0}, Σ w _{α 1}..., Σ w _{α (L-1)}), then each watermark bit is:

${w_j}^d=\left\{\begin{array}{cc}+1,& \mathrm{\&Sigma;}{w}_{\mathrm{\&alpha;j}}\&GreaterEqual;0\\ -1,& \mathrm{\&Sigma;}{w}_{\mathrm{\&alpha;j}}>0\end{array}\right.,0\&le;j\&le;L-1,$

Obtain final detected watermark sequence W ^d=(w ₀ ^d, w ₁ ^d..., w _L-1 ^d).

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