A New Statistical Detector for CT-Based Multiplicative Image Watermarking Using the t Location-Scale Distribution

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1 9th Internatonal Conference on Informaton and Knowledge Technology (IKT 07) October 8 & 9, 07, Amrkabr Unversty of Technology A ew Statstcal Detector for CT-Based Multplcatve Image Watermarkng Usng the t Locaton-Scale Dstrbuton Sadegh Etemad Dep. of Computer Engneerng and Info.Technology Amrkabr Unversty of Technology Tehran, Iran etemad.sadegh@aut.ac.r Maryam Amrmazlaghan Dep. of Computer Engneerng and Info.Technology Amrkabr Unversty of Technology Tehran, Iran mazlaghan@aut.ac.r Abstract In ths study, a new statstcal multplcatve watermark detector n contourlet doman s presented. The contourlet coeffcents of mages are hghly non-gaussan and a proper dstrbuton to model the statstcs of the contourlet coeffcents s a heavy-tal Probablty Dstrbuton Functon (PDF). In ths study, a multplcatve watermarkng scheme s proposed n the contourlet doman usng t locaton-scale dstrbuton (tls) Afterward, we used the lkelhood rato decson rule and tls dstrbuton to desgn an optmal multplcatve watermark detector. The detector showed hgher effcency than other watermarkng schemes n the lterature, based on the expermental results, and ts robustness aganst dfferent attacks was verfed. Keywords- t Locaton-Scale Dstrbuton; Contourlet transform; Multplcatve Image Watermarkng; Maxmum Lkelhood. I. ITRODUCTIO In recent years, due to ncreasng dgtal meda on the nternet, copyrght protecton turns nto an essental problem. One of the most popular approaches for the deal wth ths problem s dgtal watermarkng. Dgtal watermarkng has been proposed as a technology for copyrght protecton and content authentcaton. At the frst step of dgtal watermarkng, the secondary data (watermark) embedded nto dgtal meda such as mage, audo, vdeo, and text. Afterward, n the second step, the embedded watermark extract. The frst and second steps of dgtal watermarkng are called watermark embeddng and watermark extracton, respectvely. Some applcatons of dgtal watermarkng nclude broadcast montorng, tamper proofng, fngerprntng, content archvng, copyrght protecton and secret communcaton. In ths research, use of watermark detecton for copyrght protecton was examned. In the lterature, dgtal watermarkng methods categorzed n dfferent ways based on embeddng doman: spatal [] or frequency [], the embeddng methods: spread spectrum [3] or quantzaton based [4], and the extracton methods: detecton or decodng. Snce frequency doman technque more robust under dfferent types of attacks, t's favored to the spatal doman. Image watermarkng has been studed extensvely n the transform doman such as Dscrete Cosne Transform (DCT) [5,6], Dscrete Fourer Transform (DFT) [7], Dscrete Wavelet Transform (DWT) [8,9], Contourlet Transform (CT) [0,] and Rdgelet Transform []. At the early stage of the studes, some of transform doman watermarkng methods mplemented n the DCT, DFT. Later, the DWT-based watermarkng methods were present. ext, [3] demonstrates the outperformance of the contourlet-doman algorthms aganst the attacks compared wth other frequency-doman watermarkng schemes. There are some advances n contourlet doman over other drectonal representatons such as wavelet. The contourlet transform able to have several drectons whle obtanng nearly crtcal samplng. Also for computatonally effcent mplementaton, t uses the terated flter banks. Hence, n ths work, we focus on mage watermarkng n contourlet doman. Spread spectrum embeddng methods use two basc embeddng technques: addtve and multplcatve. In [0], we appled addtve embeddng rule for the watermarkng scheme. But snce the multplcatve embeddng method can model the Human Vsual System (HVS) preferred to addtve embeddng methods. Also, the multplcatve watermarks are mage content dependent [4]. So, n ths study, we mproved [0] by usng multplcatve embeddng n our watermarkng scheme. Accordng to [5], the contourlet coeffcents have large peaks and are hghly non-gaussan; they also have heaver tals compared to a Gaussan probablty densty functon, so, we can t use correlaton detector. Prevous works used varous PDFs as the pror dstrbuton for contourlet coeffcents. Some of the dstrbuton that used for modelng the contourlet coeffcents nclude generalzed Gaussan [6], ormal nverse Gaussan(IG) [7], Alpha-stable [8] and Bessel k Form (BKF) [] dstrbuton. In ths work, we propose a new statstcal multplcatve contourlet doman watermark detector usng t locaton-scale dstrbuton. The detector showed hgher effcency than other watermarkng schemes n the lterature, based on the expermental result, and ts robustness aganst dfferent attacks was verfed. We observed that the watermark detecton performance has enhanced as compared to other detectors such as addtve tls, multplcatve BKF, and GG detectors. The rest of the paper s as follows. Modelng of contourlet coeffcents s presented n secton. Secton 3 presents the multplcatve watermarkng method n the contourlet doman /7/$ IEEE 75

2 Performance of tls detector s assessed n secton 4. Conclusons are fnally presented n secton 5. II. STATISTICAL MODELIG In the frst part of ths secton, we analyzed the locaton-scale famly dstrbutons and revewed the tls dstrbuton then n the second part, contourlet coeffcent modelng was examned usng tls dstrbuton. A. t locaton-scale Dstrbuton (tls) A x as a random varable (RV) follows a student's t dstrbuton [9] v + Γ( ) ( ; ) x f x v = ( + ) v v vπ Γ( ) v+ ( ) () Parameter v > 0 determnes the degree of freedom and Γ denotes the gamma functon, whch s defned as x a x e dx 0 Γ = () a ( ) Where a > 0. Student's t dstrbuton can be generalzed to a tls by applyng the lnear transformaton Y = μ + σ X. Through applcaton of a lnear transformaton, we can easly shft the center of the dstrbuton and affect the dvergence from ts mean. then Y follows the t locaton scale dstrbuton (or non-standard Student's t dstrbuton) wth the parameters μ, σ,v. The PDF of t locaton-scale dstrbuton s So, f X Student ' s ( v) and each resdual mage s fed nto the drectonal flter bank (DFB) to obtan drectonal nformaton. Fg. shows the results of applyng contourlet transform on the Barbara mage. At frst, the mage dsntegrated n two pyramdal levels. ext, the DFB appled on the mage to obtan four and eght drectonal subbands. Fgure. Barbara mage contourlet coeffcents for two fnest scale. Snce the contourlet coeffcents have heavy-taled, non- Gaussan dstrbutons, a proper dstrbuton s needed for modelng contourlet coeffcents, whch present large peaks and heaver tals compared to the Gaussan PDF. In ths research, tls dstrbuton s used to model the contourlet coeffcents. Fg. represents the log-scale hstogram of contourlet coeffcents and the log-scale pdf of the best ftted tls, student's t and generalzed Gaussan(GG) dstrbutons for Barbara mage (8th and 9th subband at fourth level of pyramdal decomposton). Ths fgure demonstrates that the tls dstrbuton s well ftted whle student's t dstrbuton fals n modelng contourlet coeffcents. Also, n comparson wth generalzed Gaussan dstrbuton, tls dstrbuton has better performance for modelng the contourlet coeffcents. v + Γ( ) v+ ( ) y μ _ ( ;,, ) = ( + ( ) ) f tls y μσ v σ vπγ v ( ) v σ (3) where < μ < +, σ > 0 and v > 0 are respectvely the parameters of locaton, scale and shape. The maxmum lkelhood estmator can be used to determne parameters of locaton, scale, and shape [9]. B. Contourlet Coeffcent Modelng wth tls In [3], Do and Vetterl ntroduced contourlet transform for obtanng sparse expansons. In ths transform, Laplacan pyramd (LP) s appled to the orgnal mage to reach multscale decomposton when the coarse mage s teratvely subsampled (a) 76

3 ( α ) y = x + w (5) Where W= [ w, w,..., w ] s the watermark sequence, α s (b) Fgure. The log-scale hstogram of contourlet coeffcents for Barbara mage at (a) 8th and (b) 9th subband n the fourth level of pyramdal decomposton and the best ftted t locaton scale, student's t and generalzed Gaussan dstrbutons. the embeddng power, Y = [ y, y,..., y ] refers to watermarked contourlet coeffcents. W represents the bpolar watermark wth smlar probablty (- and ). Accordng to watermark to document rato (WDR) value, the embeddng power (α ) value for each mage s calculated. ow, to nvestgate the mperceptblty of embeddng step, we have plotted Fg. 3. In the frst row of Fg.3 the orgnal mages (Barbara, Arplane, Boat and Lena) are presented. The correspondng watermarked mages and the dfference between them have been shown n mddle and bottom rows of Fg., respectvely. The peak sgnal-to-nose rato (PSR) and mean square error (MSE) are common ndces for analyzng the nvsblty of embeddng. The values of the MSE, PSR and α show n Table I. Smlar results were obtaned on dfferent subbands for other mages. III. WATERMARKIG SCHEME Watermark embeddng and detecton comprse a watermarkng scheme. So, n ths secton, we descrbed them. A. Watermark Embeddng In the watermark embeddng step, we use a contourlet doman multplcatve spread spectrum embeddng scheme. For ths purpose, frst, we appled the contourlet transform wth the fourth level of pyramdal decomposton and sxteen drectons of drectonal flter banks to the orgnal mages. So, for each mage, the contourlet coeffcents of the drectonal subband, X = [ x, x,..., x ], s calculated. Then, to embed the watermark bts nto the drectonal subband coeffcents, we select the subband that has maxmum energy. The energy of each drectonal subband calculated as follows: jk,, A B m = n = ( ) l j jk(, ) M E X m n X l j = (4) Where jk, refers to the subband mage at j-th decomposton level and k-th bandpass drectonal mage, decomposed by an j l -th level DFB. A B shows subband sze, and E jk, s the subband energyat j-th decomposton level and k-th bandpass. Later, we use multplcatve embeddng rule for embeddng watermark n the selected drectonal subband as followng Fgure 3. Orgnal, watermarked and dfference mages respectvely. the Barbara, Arplane, Boat and Lena wth sze 5 5 are test mages (WDR=-55). It can be seen from Fg.3 and Table I that the mages are ndstngushable wth hgh PSR values. Hence, we acheved the mperceptblty of the embedded watermark at the embeddng step. TABLE I. THE VALUE OF MSE, PSR AD α (WDR=-55) Image PSR MSE α Barbara Arplane Boat Lena

4 B. Watermark Detecton Snce, the applcaton of paper s copyrght protecton, we should verfy the exstence of a known watermark. In ths secton, to detect the watermark n the contourlet subband coeffcents, we develop an optmum detector based on the tls PDF. The watermark detecton can be represented as bnary hypothess test: H : y x =,,..., 0 = (6) H : y = x ( +αw ) =,,..., (7) H 0 and H represent respectvely the null (the watermark does not exst) and alternatve (the watermark exsts) hypotheses. We assume the orgnal mage coeffcents ( x ) to be ndependent and dentcally dstrbuted (..d) and follow a tls dstrbuton wth the parameters (,, ) v μ σ as defned n (3). So, due to (6) and (7), the dstrbuton of the contourlet coeffcents under null and alternatve hypothess can be computed as: H : P( y H ) = f _ tls( y, μσ,, v) 0 0 = = (8) H : P ( y H ) = f _ tls( y, μ ( + αw), σ ( + αw), v) (9) To mathematcal smplfcaton, the log-lkelhood rato test (LLRT) nstead of LRT was used. So; we have LLRT ( y) = () = ln ( ) v + y μ + α w + ln v+ σ v + y ( + αw ) μ ln v + ( + αw ) σ Equaton () s Maxmum Lkelhood(ML) detector that we proposed. IV. SIMULATIO RESULTS In ths secton, we study the performance of tls watermark detector on many mages [0]. Snce we have lmted space for reportng the results, we only publsh the results of four grayscale mages, namely, Barbara, Arplane, Boat, and Lena (5 5). In the followng, we frst examne the performance of the t locaton-scale detector wthout any type of attack and compare t wth other contourlet doman detectors such as Bessel K Form (BKF) [] and Generalzed Gaussan (GG) [6] by recever operatng characterstc (ROC) plot. Fg.4 represents the ROCs of the three detectors for mages (Barbara, Arplane, Boat and Lena) wthout any attacks. Ths fgure demonstrates that the hgher performance of our detector versus other detectors. f _ tls ndcates the tls dstrbuton as determned n (3). the lkelhood rato test (LRT) was appled, consderng ts effcacy based on the eyman-pearson crteron. So, we employ the LRT for a subband as: H P( y H ) > Λ ( y) = < η (0) P( y H ) H0 0 Where η denotes threshold that s computed usng eyman- Pearson crtera. By ntegratng (8) and (9) n (0) and usng (), we have Λ ( y) = = ( + α w ) y ( + αw ) μ v + ( αw ) + σ y μ v + σ () v+ (a) 78

5 detectors under JPEG Compresson, Gaussan Flter and Rotaton s, respectvely. In each row of tables, the bold value s the best result. These tables demonstrate the outperformance of the proposed detector. (b) TABLE II. type JPEG (QF=45) AUROC VALUES UDER JPEG COMPRESSIO ATTACK (WDR=-55) Images t-ls Addtve BKF GG t-ls [0] [] [6] Barbara Arplane Boat Lena TABLE III. type Medan Flterng (44) AUROC VALUES UDER MEDIA FILTER ATTACK (WDR=- 55) Images t-ls Addtve BKF GG t-ls [0] [] [6] Barbara Arplane Boat Lena (c) TABLE IV. type Rotaton (=0) AUROC VALUES UDER ROTATIO ATTACK (WDR=-55) Images t-ls Addtve BKF GG t-ls [0] [] [6] Barbara Arplane Boat Lena (d) Fgure 4. ROC curve usng tls, BKF and GG based detectors wthout any attacks on test mages (a)barbara, (b)arplane, (c)boat and (d)lena (WDR=-55 db). Then, we examne the performance of the BKF, GG, addtve tls [0] and Multplcatve tls detectors under dfferent types of attacks by usng the area under the ROC (AUROC). Table II, Table III and Table IV represents the AUROC results of the multplcatve tls, addtve tls, BKF and GG V. COCLUSIO A detector was proposed for multplcatve watermarkng n the contourlet doman based on tls dstrbuton. Accordngly eyman-pearson crteron, the LRT s optmal for desgnng the watermark detector. So n ths work, we used LRT test. Performance of our watermark detector was examned based on several experments and compared wth addtve tls, BKF, and GG detectors n the contourlet doman. The results confrm that our watermarkng detector was superor to other schemes. Besdes, t provded hgh robustness aganst dfferent attacks. REFERECES [] T. Zong, Y. Xang, I. atgunanathan, S. Guo, W. Zhou, and G. Belakov, Robust hstogram shape-based method for mage watermarkng, IEEE Transactons on Crcuts and Systems for Vdeo Technology, vol. 0, no. 3, pp ,

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