Research Article Image Watermarking in the Linear Canonical Transform Domain

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1 Mathematical Problems in Engineering, Article ID 5059, 9 pages Research Article Image Watermarking in the Linear Canonical Transform Domain Bing-Zhao Li an Yu-Pu Shi School of Mathematics an Statistics of Beijing Institute of Technology, Beijing 0008, China Corresponence shoul be aresse to Bing-Zhao Li; li bingzhao@bit.eu.cn Receive December 0; Revise 9 February 0; Accepte 9 February 0; Publishe March 0 Acaemic Eitor: Juan J. Trujillo Copyright 0 B.-Z. Li an Y.-P. Shi. This is an open access article istribute uner the Creative Commons Attribution License, which permits unrestricte use, istribution, an reprouction in any meium, provie the original work is properly cite. The linear canonical transform, which can be looke at the generalization of the fractional Fourier transform an the Fourier transform, has receive much interest an prove to be one of the most powerful tools in fractional signal processing community. A novel watermarking metho associate with the linear canonical transform is propose in this paper. Firstly, the watermark embeing an etecting techniques are propose an iscusse base on the iscrete linear canonical transform. Then the Lena image has been use to test this watermarking technique. The simulation results emonstrate that the propose schemes are robust to several signal processing methos, incluing aition of Gaussian noise an resizing. Furthermore, the sensitivity of the single an ouble parameters of the linear canonical transform is also iscusse, an the results show that the watermark cannot be etecte when the parameters of the linear canonical transform use in the etection are not all the same as the parameters use in the embeing progress.. Introuction Over the past several ecaes, igital watermarking become more an more important in the application of copyright protection for igital meia as image, vieo, an auio [ ]. A igital watermark is a coe which embes copyright information incluing sequence number, a picture, an text into the multimeia for copyright protection. The watermark must be easily etecte by the copyright owner, the creator of the work, an the authorize consumer while is harly rea by the people who want to counterfeit the copyright of the ata without authorization. Digital watermarking is an emerging technology in signal processing an communications which is uner active evelopment. The methos use to embe the watermark influence both the robustness an the etection algorithm. One of the hottest irections of the watermarking metho is the watermarking in the transform omain, for example, in the iscrete Fourier transform (DFT) omain [ ] an in the iscrete cosine transform (DCT) omain [7, 8], an the watermark propose in [7] is two Gaussian sequences an it is embee in the magnitue of the DCT transformation coefficients. A wealth of information an references can be foun on the site of Watermarking Worl [9]. Recently, with the evelopment of the fractional signal an processing technologies, the research results of the fractional Fourier transform (FRFT) an fractional Fourier operators have shown that the fractional omain signal processingcanbelookeatasoneofthehottestresearch topics for nonstationary signals processing [0 5]. Several igital watermarking methos are propose in the FRFT Domain [ 9] base on these novel results of the FRFT. A nonsensical watermark embee in the FRFT omain was propose in [], an it has a more security because of the free parameter of the FRFT. Bultheel [8] escribes the implementation of a watermark embeing technique in the FRFT omain in etail an also iscusses the embeing several watermarks at the same time for images. The practical etecting threshol propose in [8] is one of the most important contributions of the paper. All of these results, which come from the igital watermarking technology in the FRFT omain, have shown that the watermarking metho in these transform omains can be more secure an har to be etecte compare to the traitional metho in the classical DFTanDCTomain. The linear canonical transform (LCT) [0], which can be looke at as the further generalization of the fractional

2 Mathematical Problems in Engineering Fourier transform, is introuce in the 970s with three free parameters an has been proven to be one of the most powerful tools for nonstationary signal processing. The wellknown signal processing operations, such as the Fourier transform (FT), the FRFT, the Fresnel transform, an the scaling operations, are all special cases of the LCT [0]. The igital computation methos of the LCT have been propose in [ ], an the sampling theories associate with the LCT have been stuie in [5 9], an the eigenfunction [0], the convolution an prouct function [, ], an the uncertainty principle [] havealsobeeninvestigatein etail. Therefore, unerstaning the LCT may help to gain more insight into its special cases an to carry the knowlege gaine from one subject to others [0]. However,forthebestofourknowlege,thereareno papers publishe about the watermarking in the LCT omain. So it is interesting an worthwhile to investigate the watermarkingmethoantechniqueassociatewiththelct. Focusing on this problem, a novel watermarking technique base on the iscrete LCT propose in [] isproposein this paper. The experiment results show that the embee watermarks are both perceptually invisible an robust to various image processing techniques. The remaining of this paper can be ivie into the following sections. The LCT is escribe in Section. Section evelops watermark embeing in LCT omain. Numerical examples an the iscussion of the simulation results are given in Section,an Section 5 is the conclusion.. The Linear Canonical Transform.. The Continuous LCT. The continuous LCT of a signal f(x) with parameter matrix A=( ab c ) can be efine as [0] + f A (y) = C A (f) (y) = f (x) C A (x, y) x, C A (x, y) = b e jπ/ exp {jπ [( a b )x ( b )xy+( b )y ]}, () where C A is the LCT operator an a, b, c, are real parameters. Furthermore the constraint a bc = must be satisfie to make the transform unitary. Actually the LCT has three free parameters; if we let a=γ/, b=/, c= +αγ/, =α/,thelctoff(x) canberewrittenas[] + f A (y) = C A (f) (y) = f (x) C A (x, y) x, () C A (x, y) = e jπ/ exp [jπ (γx xy+αy )], where parameter matrix γ a b A=( c )=( + αγ α ). () Two of interesting an important properties of LCT are reversibility an inex aitivity. Inex aitivity means that, if two LCTs with matrices A,A operateinasuccessive manner, then the equivalent transform is an LCT with the matrix A=A A. Because of the inex aitivity, the inverse of the LCT with matrix A is an LCT with the matrix A. With the evelopment of the fractional signal processing metho, the properties an applications of the LCT have been investigate in etail; for more information associate with the continuous LCT, one can refer to [, 5, 0]... The Discrete LCT. Besies the continuous LCT, we often encounter the computation of the iscrete LCT because we must process iscrete ata by computer. There are lots of iscrete an the fast LCT methos propose in the literature [,, ]. If we set δ x =δ y = (N ) /, x=nδ x, y=mδ y, an m,n = 0,,...,N,the N point iscrete LCT (DLCT) of f(n) canbeefineas[] where C A (m, n) = e (jπ/) N f A (m) = N n=0 f (n) C A (m, n), () exp [jπ N (αm mn+γn )]. (5) ThiskinofDLCTmethoisavailableforimageprocessing, because it is interval-inepenent an unitary. Moreover, it also has the property of inex aitivity. Following this metho, the two-imensional DLCT of a size H Nimage I(h, n) canberewrittenas I A (k, l) = N n=0 C A (l, n) H h=0 I (h, n) C A (k, m) () with k = 0,,...,H, l = 0,,...,N,anC A (k, m), C A (l, n) being the same as (). It is shown in [] thatthis kin of DLCT is analogous to the DFT an approximates the continuous LCT in the same sense that the DFT approximates the continuous Fourier transform. We will use this metho to compute the D LCT of an image in the following sections.. Watermark Embeing an Detecting It is well known that the watermarking process contains the watermark embeing an etecting steps; we propose a new kin of watermarking scheme following the iea of [8]inthis section... Watermark Embeing. The watermark itself is a sequence of M complex numbers [8], enote by s i =c i +j i, i =,,...,M, an the real an imaginary parts of s i are obtaine from a normal istribution with mean zero

3 Mathematical Problems in Engineering an variance σ /. In orer to embe this watermark into an image I of size H N, we first compute the DLCT of this image I to erive the transform coefficients {S i :i= N}an then reorere the transform coefficients in nonincreasing sequence as follows: S i =C i +jd i : S i S i+, i=,...h N. (7) Similar with the metho in [], we chose the mile reorere transform coefficients to embe the watermarks; in other wors, we embe the watermark into the coefficients S i, i=l+,l+,...,l+m. This is because if we embee the watermarks in the lowest coefficients, they woul be sensitive to noise removing or compressing operations, while if we embee the watermarks in the highest coefficients, they woul significantly affect the imperceptibility of the watermarks. So, the watermarks were embee as follows: S w i =S i +c i C i +j i D i, i=l+,...,l+m, (8) where S w i isthewatermarkeimageofi an (c i, i ) is the watermarks sequence... Watermark Detecting. When the watermark is embee intheimage,thentheimageistransferretothewatermark etection process to see whether it contains watermark. The etection of the watermark can be escribe like this: given the watermarke image I a, maybe uner some attacks such as low pass an meian filtering, aition of Gaussian noise, an resizing, we compute the DLCT of I a an obtain the transform coefficients S a an then compute the etection value []: = L+M i=l+ (c i j i )S i. (9) The threshol can be achieve accoring to the statistical performance of the propose algorithm. The expecte value of is E [] = σ L+M i=l+ ( C i + D i ). (0) In [], Djurovic et al. propose a useful an simple threshol as E[]/; when the value of is larger than the threshol, it is ecie that a watermark has been etecte. Otherwise, there is no watermark. However, it is shown in [8] thatthiskin of threshol suffers from the false conclusion; therefore we useanaaptivethresholproposein[8], because it is more practical when we eal with the image after some attacks. Therefore, the threshol can be compute by the following steps. (i) First, we compute the value of of all the ranom watermarks (maybe 000 watermarks). (ii) Then, we compute the average (say μ) an the stanar eviation (say σ)ofthese. (iii) At last, we can achieve the threshol τ=μ+pσwhere p is a suitable number.. Simulation Examples.. Watermark Embeing an Detecting. The Lena (5 5) was chosen as the test image in the simulations. Accoring to some experiments, the value of p in threshol τ = μ+pσwas chosen to be 5. The D DLCT parameters are α = α = 0., = =0., γ =γ =0.an can be escribe as (α,,γ,α,,γ ) = (0., 0., 0., 0., 0., 0.). Therefore, the D DLCT parameter matrixes can be rewritten as A =A =( γ + αγ α )=( ), () an the D DLCT is performe base on (). The simulations performe using Matlab version in Winows 8 system an the processer of the system is Intel(R) Core(TM) i5-7u;thecpuantheramofthesystemare.80ghzan.00 GB, respectively. We chose L = 9000, M = 000, σ = 0 in the simulation. In orer to test the performance of the propose metho, we use the PSNR an the elapse time of the process to measure the performance of the watermarking technology [8]. The original an watermarke images are shown in Figures an,respectively.it is shown that the watermarke picture Figure is almost the same as the original Figure. The etection of the correct watermark from the watermarke image over the other 000 ifferent watermarks, which are also Gaussian white noise with variance σ G = σ / = 0. The etection result is plotte in Figure.Inthis case, the PSNR an the elapse time are 9.7 B an.7 secons, respectively. In Figure, we can easily fin that the etection value of the correct watermark is significantly larger than the threshol an other false watermarks. So, the watermark can be etecte by the comparison... The Robustness. In this subsection, we investigate the robustness of the algorithm after the following attacks: aing noise, upper cropping, central cropping, an central cropping after aing noise. These experiments have been performe as the following. Firstly, Figures an plot the robustness of the watermarking uner the Gaussian noise. Figure is the noisy image of the watermarke image in Figure by aing mean zero an variance 00 Gaussian noise, while thevarianceoffigure is 00. Figures an are etection results of these two situations, the PSNR are 9.7 B an 5.08 B, the elapse times are 9.75 an.8 secons, respectively. This result shows that the metho is robust against noise, because the watermark can be still etecte. Seconly, wecroppethewatermarkeimagefigure from the size 5 5 to an,anobtain Figures 5 an, respectively. The etection results are shown in Figures 5 an, respectively.itisshownin Figures 5 an that the watermark can also be etecte. In this situation, the PSNR are.5 B an 0.8 B, the elapse time are 8.70 an 9.5 secons, respectively.

4 Mathematical Problems in Engineering Figure : The original image of Lena, the watermarke image of Lena Figure : The etection result from the watermarke Figure Figure : The noisy Lena, var = 00. The etection of the noisy Lena.

5 Mathematical Problems in Engineering Figure : The noisy Lena, var = 00. The etection of the noisy Lena Figure 5: The upper croppe image of Figure. The etection of upcroppe image. Thirly, we perform the upper cropping of the noisy imageinfigures an in the same way as in Figure 5 an obtain Figures 7 an 8. The etection results are plotte in Figures 7 an 8, respectively.itisshown in Figure 7 that the watermark can also be etecte for the upper croppe noisy watermarke image of variance 00. We can still etect the watermark for the upper croppe noisy imageofvariance00asshowninfigure 8.Inthissituation, the PSNR are.50 B an.8 B, an the elapse times are 8.70 an 9.88 secons, respectively. Lastly,wecentralcropthenoisyimageinFigures an in the same way as in Figure an obtain Figures 9 an 0. The etection results are plotte in Figure 9 an Figure 0,respectively.ItisshowninFigure 9 that the watermark can also be etecte for the central croppe noisy watermarke image of variance 00. We can still etect the watermark for the central croppe noisy image of variance 00 as shown in Figure 0. Inthissituation,thePSNRare 0.8 B an 0.78 B, an the elapse times are 9.5 an 9.0 secons, respectively. From these simulations, it can be conclue that the propose metho is robust uner the common image attacks, such as the noise, crops, an the crops of the noisy image. It shoul be also notice from Figures 8 an 0 that the propose metho still works uner the attack of cropping if the variance of the aing noise is about 00.

6 Mathematical Problems in Engineering Figure : The central croppe image of Figure. The etection of central croppe image Figure 7: The upper croppe noisy Lena of Figure. The etection of the upcroppe noisy Lena... The Parameters Sensitivity. As compare to the traitional watermarking metho, for example, the DFT an DCT omain metho [5 8], the avantage of the propose metho is that it has three more free parameters, an this can enhance the security an robustness of the watermarking images. It is well known that the parameters of the LCT are two more than the parameters of the FRFT, an for the D-LCT there are six parameters. So, when we nee to etect the watermarks, we not only nee the watermarke keys but also nee the six parameters which is three times the number of the FRFT s parameter. Therefore, it is more ifficult for the unauthorize person to etect the watermark an estroy it. In orer to show the avantage of the LCT base watermarking metho propose in this paper, the sensitivity of the parameter (α,,γ,α,,γ ) is iscusse in this subsection.weusethewatermarkeimageinfigure as teste image, we set (α,,γ ) = (0., 0., 0.), an o not know the value of α,,anγ in simulations; the value of is sensitive with the α,,anγ asplotteinfigure. It is shown in Figure that the value of is significantly larger when the value of α,,anγ are more correct than the false values of the parameters. For example, when the unauthorize people know (,γ,α,,γ ) = (0., 0., 0., 0., 0.), the correct place of the watermark, an the correct watermark but not sure about the value of α,the watermark still cannot be etecte because only the value of correct α can reach the peak accoring to Figure.We can also see that the sensitivity of α an is goo, while

7 Mathematical Problems in Engineering Figure 8: The upcroppe noisy Lena of Figure. The etection of the upcroppe noisy Lena Figure 9: The central croppe noisy Lena of Figure. The etection of the central croppe noisy Lena. the sensitivity of γ is not so gratifying especially when γ is between an in Figure (c). 5. Conclusion A novel watermarking technique base on the iscrete LCTisproposeinthispaper.Inthiskinofmetho, the watermarks are embee in the mile coefficients in the transform omain, an the etecting threshol is etermine aaptively. The simulations for the robustness of the propose metho uner the common image processing are performe, an the simulation results fit the theories well.theproposewatermarkingismoresecurethanthe watermarking base on FRFT or DCT omain because it has more free parameters. We also iscusse the parameter s sensitivity of the propose metho in the paper an showe that this kin of watermarking metho is sensitive to the parameters of the LCT. Conflict of Interests The authors eclare that there is no conflict of interests regaring the publication of this paper. Acknowlegments This work was supporte by the National Natural Science Founation of China (no an no. 795) an

8 8 Mathematical Problems in Engineering Figure 0: The central croppe noisy Lena of Figure. The etection of the central croppe noisy Lena (c) Figure : The sensitivity of parameters. The sensitivity of α,thesensitivityof, an (c) the sensitivity of γ.

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