RDWT and Image Watermarking

Transcription

1 RDWT and Image Watermarkng L Hua and James E. Fowler Engneerng Research Center Msssspp State Unversty Techncal Report MSSU-COE-ERC-0-8 December 00. Introducton Image watermarkng s a technque for labelng dgtal mages by embeddng electronc stamps or so-called watermarks nto the mages for the purpose of copyrght protecton. Due to the exploson n use of the dgtal meda, watermarkng has been attractng sgnfcant nterest from both academc and ndustry recently. In order to be effectve, a watermark should exhbt a number of desrable characterstcs [,, 3, 4]. Unobtrusveness: The watermark should be perceptually nvsble and should not degrade the qualty of the mage content. Robustness: The watermark must be robust to transformatons ncludng common sgnal processngs, common geometrc dstortons and subterfuge attacks. Blndness: The watermark should not requre the orgnal nonwatermarked mage for watermark detecton. Unambguousness: Retreval of the embedded watermark should unambguously dentfy the ownershp and dstrbuton of data. number of technques have been developed for watermarkng. wdely used technque s spreadspectrum watermarkng whch embed whte Gaussan nose onto transform coeffcents []. The watermark s detected by computng a correlaton between the watermarked coeffcents and the watermark sequence whch s compared to a properly selected threshold. The DWT s an appealng transform for spread-spectrum watermarkng because ts space-frequency tlng exhbts a strong smlarty to the way the human vsual system (HVS processes natural mages [4]. Therefore, watermarkng technques n the wavelet doman can largely explot the HVS characterstcs and effectvely hde a robust watermark. Unlke the DWT, the redundant dscrete wavelet transform (RDWT gves an overcomplete representaton of the nput sequence and functons as a better approxmaton to the contnuous wavelet transform. The RDWT s shft nvarant and ts redundancy ntroduces an overcomplete frame expanson. It has been proved that frame expansons add numercal robustness n case of addng whte nose [5, 6] such n the case of quantzaton. Ths property makes RDWT-based sgnal processng tends to be more robust than DWT. It s well known that RDWT s very successful n nose reducton and feature detecton, and pror work has proposed usng the RDWT for mage watermarkng [7]. Intally, one mght thnk that, snce frame expansons such as the RDWT offer ncreased robustness to added nose, such overcomplete expansons outperform tradtonal orthonormal expanson n the watermarkng problem. In ths report, we offer analyss that contradcts ths ntuton. Specfcally, we present analyss that shows that, although watermarkng coeffcents n a tght-frame expanson does produce less mage dstorton for the same watermarkng energy, the correlaton-detector performance of tght-frame based watermarkng s dentcal to that obtaned by usng an orthonormal expanson. Techncal Report MSSU-COE-ERC-0-8, Engneerng Research Center, Msssspp State Unversty, December 00

2 . RDWT & Frame. RDWT The RDWT removes the decmaton operators from DWT flter banks. To retan the multresoluton characterstc, the wavelet flters must be adusted accordngly at each scale. Specfcally, h J [k] h[k] where J s the start scale, h J [k] s the RDWT scalng flter at scale J. h[k] s a normal DWT scalng flter. Flters at later scales are upsampled versons from the flter coeffcents at the upper stage, h [k] h + [k] and smlar defntons s appled to g [k], the wavelet flter of the orthonormal DWT. The RDWT multresoluton analyss can be mplemented va the flter bank equatons: nalyss: c [k] h + [ k] c + [k] ( d [k] g + [ k] d + [k] ( Synthess: c + [k] [ ] h + [k] c [k] + g + [k] d [k] (3 The lack of downsamplng n the RDWT analyss yelds a redundant representaton of the nput sequence; specfcally, two vald descrptons of the coeffcents exst after one stage of RDWT analyss.. Frames DEFINITION. [5] famly of functons (ψ J n a Hlbert space H s called a frame f there exst > 0, and B < so that, for all f n H, f J < ψ, f > B f (4 and B are called the frame bounds. The dual frame ( ψ of (ψ s an expanson set n Hlbert space H and for all f n H, B f < ψ, f > f (5 ny functon f H can be expanded as f α ψ β ψ < ψ, f > ψ (6 < ψ, f > ψ (7 If two frame bounds are equal, B, the frame s called a tght frame. In a tght frame, for all f H, < ψ, f > f (8 J ψ ψ (9 f ψ, f ψ (0 In ths case, >, and gves the redundancy rato, a measure of the degree of overcompleteness of the expanson. Techncal Report MSSU-COE-ERC-0-8, Engneerng Research Center, Msssspp State Unversty, December 00

3 .3 RDWT and frame expanson Theorem. RDWT s a frame expanson wth frame bounds and B J, where J s the number of levels n the transform. Thus, for one level, the RDWT s a tght frame. Proof: One level RDWT nalyss c [0] c [0] c [] c [] c + [0] c + [] c + [] c + [3] d [0] d [0] d [] d [] Fgure : One scale of RDWT decomposton. s shown n Fg., for the lowpass coeffcents c, t s composed wth two parts, c and c, each of them s a vald DWT lowpass descrpton of c +. It s a smlar case for the hghpass coeffcents d. Thus, Parseval s theory holds for each of the descrptons: c + d c + c + d c + Thus, for the RDWT coeffcents all together, we have c + d c + ( Therefore, one-level RDWT decomposton s a tght frame wth. For decomposton level J >, suppose the decomposton starts at scale J. We have then cj ( c J + d J Whle, the energy for the RDWT coeffcents s: cj + dj + dj J J cj J + dj ( E c J J + J dj (3 3 Techncal Report MSSU-COE-ERC-0-8, Engneerng Research Center, Msssspp State Unversty, December 00

4 So that, On the other hand, J J c J E ( J d J (snce... J, we have J, so that J c J E 0 E J c J (4 E c J ( J c J J + J ( d J (snce J > and... J, we have, so that E c J 0 E c J (5 The bounds of and B J are the tghtest bounds snce we can fnd sequences that meet the bounds. Specfcally, for a constant sequence x [n], only the lowpass coeffcents are nonzero and all hghpass subbands are zero valued. That s, c J J 0 but d J 0, for... J. J J c J E ( J d J 0 E J c J (6 For an oscllatory sequence x [n] ( n, only the fnest detal coeffcents would be nonzero. That s, cj J 0 and d J 0, for... J. E c J ( J c J J + J ( d J 0 E c J (7 Therefore, for decomposton level J >, the energy of the decomposton coeffcents are well bounded. Thus, the RDWT s a frame expanson accordng to the defnton. 3. Robustness of ddng Whte Nose We are to compare the robustness of three dfferent transforms,.e. orthonormal bass, tght frame and frame bass, after addng whte Gaussan nose n the correspondng transform domans. Suppose the addtve nose s a zero mean, varance ɛ Gaussan nose, the robustness s measured as the mean square error (MSE of the reconstructed sgnal wth the orgnal sgnal. MSE E[ f ˆf ] E[< f ˆf, f ˆf >] E[< f, f >] E[< f, ˆf >] + E[< ˆf, ˆf >] In ths analyss, f s the orgnal sgnal, ˆf s the watermarked sgnal, and the MSE s the dstorton produced by watermarkng f wth watermark energy equal to ɛ. 4 Techncal Report MSSU-COE-ERC-0-8, Engneerng Research Center, Msssspp State Unversty, December 00

5 3. Orthonormal bass f ˆf where the Gaussan nose s n (0, ɛ. The error sgnal So that, 3. Tght frame α ψ ψ, f ψ (α + n ψ e ˆf f n ψ MSE E e E[ n ] E[n ] Nɛ (8 ( ( Snce f ˆf ψ, f ψ (α + n ψ α ψ MSE E[< f, f >] E[< f, ˆf >] + E[< ˆf, ˆf >] E[< f, f >] E E[< f, ˆf >] E α ψ, E[α α ] ψ, ψ α ψ, (α + n ψ α ψ E[α α ] ψ, ψ + E[α n ] ψ, ψ E[α α ] ψ, ψ 5 Techncal Report MSSU-COE-ERC-0-8, Engneerng Research Center, Msssspp State Unversty, December 00

6 (3 E[< ˆf, ˆf >] E (α + n ψ, (α + n ψ E[α α ] ψ, ψ E[n n ] ψ, ψ Combnng the three components: MSE E[ f ˆf ] E[< f, f >] E[< f, ˆf >] + E[< ˆf, ˆf >] E[n n ] ψ, ψ E[n ] ψ, ψ ɛ Nɛ (9 Note that, snce >, Nɛ does an orthonormal bass. < Nɛ, and watermarkng n the tght frame yelds less dstorton to f than 3.3 Frame expanson f ˆf α ψ < ψ, f > ψ (α + n ψ MSE E[< f, f >] E[< f, ˆf >] + E[< ˆf, ˆf >] ( E[< f, f >] E[< α ψ, α ψ >] E[α α ] < ψ, ψ > ( E[< f, ˆf >] E[< α ψ, (α + n ψ >] E[α α ] < ψ, ψ > + E[α n ] < ψ, ψ > E[α α ] < ψ, ψ > 6 Techncal Report MSSU-COE-ERC-0-8, Engneerng Research Center, Msssspp State Unversty, December 00

7 (3 E[< ˆf, ˆf >] E[< (α + n ψ, (α + n ψ >] E[α α ] < ψ, ψ > + E[n n ] < ψ, ψ > E[α α ] < ψ, ψ > + E[n ] < ψ, ψ > Combnng all the components: MSE E[< f, f >] E[< f, ˆf >] + E[< ˆf, ˆf >] E[n ] < ψ, ψ > ɛ ψ (0 Theorem. s n the lterature of [6], but we derved n a dfferent way. For frame expanson, M B k ψ k M ( Proof: Substtute f ψ k nto nequaltes Eq. 5 and substtute f ψ k nto nequaltes Eq. 4, we have: B ψ k < ψ, ψ k > ψ k ψ k < ψ, ψ k > B ψ k Because for each k, we have ψ, ψ k and ψ, ψ k be postve, we have correspondngly: B ψ k < ψ, ψ k > k k k ψ k < ψ, ψ k > k k k ψ k B ψ k Snce the dmenson of the frame bass ψ s same wth the dmenson of the dual frame bass ψ, equals to M, t s obvous that: < ψ, ψ k > < ψ, ψ k > T k k Thus T must satsfy the two nequaltes smultaneously k k B ψ k T k ψ k ψ k T B ψ k 7 Techncal Report MSSU-COE-ERC-0-8, Engneerng Research Center, Msssspp State Unversty, December 00 k

8 Ths s only possble f k k ψ k ψ k ( k B ψ k B ψ k (3 k Snce frame ψ k are normalzed, ψ k, we have Eq. Smlarly, Eq. 3 k k ψ k ψ k k ψ k M k B ψ k B ψ k fracb k k ψ k M B k k Therefore, the bounds for the k ψ k are establshed. M B k ψ k M So that, MSE E[ f ˆf ] ɛ ψ Mɛ B Mɛ MSE Note that, snce the redundancy of r M N guarantee the lower bound of the MSE n the frame expanson case Mɛ B dstorton n the orthonormal bass case Nɛ. 4. Tght Frame and Watermarkng 4. Watermarkng s between the frame bounds and B(B >, we can rnɛ B be smaller than the In the spread-spectrum watermarkng problem, the sgnal s transformed usng an expanson bass, f α ψ 8 Techncal Report MSSU-COE-ERC-0-8, Engneerng Research Center, Msssspp State Unversty, December 00

9 and the watermark sequence, a whte Gaussan nose, n, s added to the coeffcents n the transform doman, to form the watermarked mage. f α ψ (α + ɛn ψ where n s a zero-mean, unt-varance whte Gaussan nose, and ɛ s a parameter that controls the watermark strength. The watermark can be detected assumng the watermark random sequence s known exactly. Ths s done by performng the forward transform on the watermarked sgnal, and then runnng the correlaton operaton on the coeffcents, ρ ˆα n where ˆα are the expanson coeffcents of the watermarked mage f. ˆα ψ, f For watermark detecton, the correlaton s compared to a threshold to decde the presence of the watermark. n optmal threshold can be set to mnmze the probablty of mssng the presence of the watermark error to a gven false-detecton error accordng to the Neyman-Pearson crteron [4]. Below, we compare the watermark performance of two procedures: usng an orthonormal bass and usng a tght-frame expanson. We ensure these two watermark procedures acheve the same MSE and the same false-alarm error P F, and the performance s measured by the mnmum mssed-detecton error, P M, as obtaned wth the Neyman-Pearson procedure. Three hypothess cases are possble for the watermark problem [4]. Case : mage s not watermarked. Case B: mage s watermarked wth a random sequence other than the detecton watermark, but s usng a same watermarkng approach. Case C: mage s watermarked exactly wth the detecton watermark. However, we wll not do a multple-hypothess testng. Instead we wll do two bnary hypothess testngs, on case v.s. case C and case B v.s. case C respectvely. 4. Neyman Pearson test[8] Neyman-Pearson test s also called the most powerful (MP test. The decson rule s obtaned by mnmzng the mssng error P M subect to the constrant on the false alarm error P F, P F α. The obectve functon s constructed based on the Lagrange multpler method. Suppose the decson rule s J P M + λ(p F α ρ H H 0 T ρ We derve the Neyman Pearson test on two bnary hypothess problems. 9 Techncal Report MSSU-COE-ERC-0-8, Engneerng Research Center, Msssspp State Unversty, December 00

10 4.. No watermarkng vs. correct watermarkng ccordng to [4], modelng the correlaton ρ as a normally dstrbuted nose s realstc and ft the central lmt theorem. H 0 : Case P (ρ H 0 H : Case C P (ρ H πσρ e ρ σ ρ πσ e (ρ µ σ False larm Error: P F P (D H 0 T ρ Q( T ρ σ ρ T ρ πσρ e P (ρ H 0 dρ ρ σ ρ dρ erfc( T ρ σρ (4 Mssng Error: P M P (D 0 H 4.. Watermarkng Dscrmnaton Tρ Tρ P (ρ H dρ e (ρ µ σ dρ πσ G( T ρ µ σ H 0 : Case B P (ρ H 0 H : Case C P (ρ H Q( T ρ µ σ + erf( Tρ µ σ T ρ µ erfc( µ T ρ (5 σ T ρ < µ πσρb e ρ σ ρb πσ e (ρ µ σ False larm Error: Mssng Error: T ρ P F P (D H 0 erfc( (6 σρb P M P (D 0 H + erf( Tρ µ σ T ρ µ erfc( µ T ρ (7 σ T ρ < µ 0 Techncal Report MSSU-COE-ERC-0-8, Engneerng Research Center, Msssspp State Unversty, December 00

11 4.3 Correlatons In order to obtan the Neyman-Pearson test for each watermarkng procedure, the probablty dstrbutons of the watermark correlaton under each hypothess must be obtaned frst. The correlaton dstrbuton s modeled as a Gaussan nose Orthonormal bass Case no watermark ρ α n µ ρ E[ρ] 0 (8 σρ E[ρ µ ρ ] E[ρ ] E[ α n ] E[ α n ] + E[ α α n n ] α E[n ] + 0,, α f (9 Case B watermarked wth another random sequence ρ (α + ɛ m m n µ ρb E[ρ] E[α n ] + E[ɛ m m n ] 0 σρb E[ρ µ ρb ] E[ρ ] E[ (α + ɛ m m n ] E[ (α + ɛ m m n ] + E[ (α + ɛ m m n (α + ɛ m m n ],, E[α + ɛ mm + ɛ m m α ]E[n ] + 0 (30 α + ɛ m f + Nɛ m f + Nɛ (3 Techncal Report MSSU-COE-ERC-0-8, Engneerng Research Center, Msssspp State Unversty, December 00

12 Case C watermarked wth the correct watermark ρ (α + ɛn n µ E[ρ] ɛ N (3 σ E[ρ µ ] E[ρ ] µ E[ (α + ɛn n ] ɛ N E[ (α n + ɛn ] + E[ (α n + ɛn (α n + ɛn ] ɛ N,, E[α n + ɛ n 4 + ɛα n 3 ] + E[ (α n α n + ɛα n n + ɛα n n + ɛ n n ] ɛ N,, α E[n ] + ɛ E[n 4 ] + ɛα E[n 3 ] + ɛ E[n ]E[n ] ɛ N,, Snce n s zero mean, unt varance whte Gaussan nose, we have: E[n ] E[n 3 ] 0 E[n 4 ] 3 σ α + 3Nɛ + ɛ (N N ɛ N f + Nɛ (33 Techncal Report MSSU-COE-ERC-0-8, Engneerng Research Center, Msssspp State Unversty, December 00

13 4.3. Tght frame Case no watermark ρ α n µ ρ E[ρ] 0 (34 σρ E[ρ µ ρ ] E[ρ ] E[ α n ] α E[n ] α f (35 Case B watermarked wth another random sequence 3 Techncal Report MSSU-COE-ERC-0-8, Engneerng Research Center, Msssspp State Unversty, December 00

14 ρ ˆα n ψ, f n ψ, (α + ɛ m m ψ n [ (α + ɛ m m ψ, ψ ]n (α + ɛ m m n ψ, ψ µ ρb E[ρ] 0 (36 σρb E[ρ µ ρb ] E[ρ ] E[ (α + ɛ m m n ψ, ψ ] [ E[ α n ψ, ψ ] + E[ ɛ m m n ψ, ψ ] + E[ α n ψ, ψ ] ɛ m m l n k ψ k, ψ l ] k l E[ α n ψ, ψ ] + E[ ɛ m m n ψ, ψ ] E[n ][ α ψ, ψ ] + ɛ me[m n ] ψ, ψ α ψ, ψ ] + ɛ m ψ, ψ [ Snce for the tght frame, we have: f ψ, f ψ α ψ So that, [ α ψ, ψ ] nd also because tght frame has the property of: ψ, f f 4 ψ, f ψ, f α (37 Techncal Report MSSU-COE-ERC-0-8, Engneerng Research Center, Msssspp State Unversty, December 00

15 Then we have the second tem of the expresson: ψ, ψ σ ρb α + ɛ mn ψ N α + ɛ m N f + ɛ mn f + ɛ N (38 Case C watermarked wth the correct watermark ρ α n ψ, ψ + ɛn n ψ, ψ µ E[ρ] ɛ N (39 σ E[ρ µ ] E[ρ ] µ [ E[ α n ψ, ψ ] + E[ ɛn n ψ, ψ ] + E[ α n ψ, ψ ] ɛn l n k ψ k, ψ l ] ɛ N k l [ E[n ][ α ψ, ψ ] + E[ ɛn n ψ, ψ ɛn l n k ψ k, ψ l ] k l + E[ α n ψ, ψ ] ɛn l n k ψ k, ψ l ] ɛ N For each of the tem n the expresson, k l ( From the Eq. 37, we have E[n ][ α ψ, ψ ] [ α ψ, ψ ] α 5 Techncal Report MSSU-COE-ERC-0-8, Engneerng Research Center, Msssspp State Unversty, December 00

16 ( The second tem n the expresson E[ ɛn n ψ, ψ ɛn l n k ψ k, ψ l ] E[ + E[ + E[ ɛ n 4 ψ k, ψ l ] ɛ n ψ, ψ n k ψ k, ψ k ] k ɛ n n ψ, ψ ] + 0 else ɛ E[n 4 ] + ɛ,k, k k l k l E[n ]E[n k ] + ɛ 3Nɛ + ɛ ( N N + ɛ [ N, k l, k k, l, or l, k,,, 3Nɛ + ɛ ( N N + ɛ ( N N ɛ N + ɛ N ɛ (N + N E[n ]E[n ] ψ, ψ ψ, ψ (3 The thrd tem n the expresson E[ α n ψ, ψ ɛn l n k ψ k, ψ l ] k l E[ ɛα n 3 ψ, ψ ψ, ψ ] k l + 0 else 0 ψ, ψ ] σ [ ] α + ɛ (N + N ɛ N α + ɛ (N + N ɛ N f + ɛ N ( Performance comparson We know the MSE ncurred by the watermarkng process: Orthonormal Bass: Tght Frame: Frame Bass: MSE E[ f f ] Nɛ D MSE E[ f f ] Nɛ Mɛ B MSE E[ f f ] Mɛ 6 Techncal Report MSSU-COE-ERC-0-8, Engneerng Research Center, Msssspp State Unversty, December 00

17 To acheve same MSE s for the three expansons, the watermarkng strength ɛ must be adusted as: D Orthonormal Bass: ɛ (4 N D Tght Frame: ɛ (4 N D D Frame Bass: M ɛ M B (43 The statstcal parameters of the Gaussan modeled correlatons are lsted n Table 4.4. Table : Mean and standard devaton of the watermark correlaton. Orthonormal Bass Tght Frame Case µ ρ 0 µ ρ 0 σρ f σ ρ f Case B µ ρb 0 µ ρb 0 σρb f + Nɛ f + D σ ρb f + Nɛ f + D Case C µ ɛ N ND µ ɛ N ND σ f + Nɛ f + D σ f + Nɛ f + D 4.4. No watermarkng vs. correct watermarkng H 0 : Case H : Case C For Orthonormal Bass: For Tght Frame: T ρ P F erfc( σρ erfc( T ρ f T ρ P F erfc( σ ρ erfc( T ρ f In order for the false alarm error be the same, P F P F, we need T ρ f T ρ f So that T ρ T ρ Snce the mssng error s: P M + erf( Tρ µ σ T ρ µ erfc( µ T ρ σ T ρ < µ 7 Techncal Report MSSU-COE-ERC-0-8, Engneerng Research Center, Msssspp State Unversty, December 00

18 ( f T ρ µ, then T ρ T ρ µ µ We have P M + erf(t ρ µ σ P M + erf(t ρ µ σ Snce T ρ µ σ Tρ µ f + Nσ Tρ µ σ T ρ µ σ Therefore, we have P M P M. ( Smlarly, f T ρ < µ, we have T ρ < µ, and P M erfc(µ T ρ σ P M erfc(µ T ρ σ Snce, µ T ρ σ µ T ρ σ we have P M P M also. That s, the probablty of mssed detecton error s the same regardless of whether an orthonormal or tght frame s used Watermarkng Dscrmnaton H 0 : Case B H : Case C For Orthonormal Bass: T ρ P F erfc( σρb erfc( T ρ f + D 8 Techncal Report MSSU-COE-ERC-0-8, Engneerng Research Center, Msssspp State Unversty, December 00

19 For Tght Frame: T ρ P F erfc( σ ρb erfc( T ρ f + D In order for P F P F, we need erfc( T ρ f + D erfc( f + D T ρ So that T ρ T ρ ( f T ρ µ, then T ρ T ρ µ µ we have P M + erf(t ρ µ σ P M + erf(t ρ µ σ T ρ µ Tρ µ σ σ T ρ µ σ Therefore, we have P M P M. ( f T ρ < µ, we have T ρ < µ, and P M erfc(µ T ρ σ P M erfc(µ T ρ σ µ T ρ µ T ρ σ σ µ T ρ σ So that we wll have P M P M also. 9 Techncal Report MSSU-COE-ERC-0-8, Engneerng Research Center, Msssspp State Unversty, December 00

20 gan, the performance s the same. 5. Conclusons fter the above theoretcal analyss, we can draw the followng conclusons:. RDWT expanson s a frame expanson, and one scale RDWT s a tght frame wth the redundancy.. Frame expanson s more robust than the orthonormal bass expanson when addng whte nose onto the transform doman;.e., less dstorton s obtaned when water-markng wth a fxed watermark energy. 3. However, tght frame doesn t show obvous performance advantages over the orthonormal bass when consderng the spread-spectrum watermarkng problems, as the addtonal robustness of the overcomplete expanson does not ad watermark detecton by correlaton operators. 0 Techncal Report MSSU-COE-ERC-0-8, Engneerng Research Center, Msssspp State Unversty, December 00

21 References [] I. J. Cox, J. Kllan, F. T. Leghton, and T. Sharmoon, Secure Spread Spectrum Watermarkng for Multmeda, IEEE Transactons on Image Processng, vol. 6, no., pp , December 997. [] I. J. Cox and M. L. Mller, Revew of Watermarkng and The Importance of Perceptual Modelng, n Human Vson and Electronc Imagng II, B. E. Rogowtz and T. N. Pappas, Eds., February 997, pp. 9 99, Proc. SPIE 306. [3] M. D. Swanson, M. Kobayash, and. H. Tewfk, Multmeda Data-Embeddng and Watermarkng Technologes, Proceedngs of the IEEE, vol. 86, no. 6, pp , June 998. [4] M. Barn, F. Bartoln, and. Pva, Improved Wavelet-Based Watermarkng Through Pxel-Wse Maskng, IEEE Transactons on Image Processng, vol. 0, no. 5, pp , May 00. [5] I. Daubeches, Ten Lectures on Wavelets, Socety for Industral and ppled Mathematcs, Phladelpha, P, 99. [6] V. K. Goyal, M. Vetterl, and N. T. Thao, Quantzed Overcomplete Expansons n IR N : nalyss, Synthess, and lgorthms, IEEE Transactons on Informaton Theory, vol. 44, no., pp. 6 3, January 998. [7] J.-G. Cao, J. E. Fowler, and N. H. Younan, n Image-daptve Watermark Based on a Redundant Wavelet Transform, n Proceedngs of the Internatonal Conference on Image Processng, Thessalonk, Greece, October 00, pp [8] M. D. Srnath, P. K. Raasekaran, and R. Vswanathan, Introducton to Statstcal Sgnal Processng wth pplcatons, Prentce-Hall, Upper Saddle Rver, NJ, 996. Techncal Report MSSU-COE-ERC-0-8, Engneerng Research Center, Msssspp State Unversty, December 00