Joural of Network Itelligece c 16 ISSN 21-8105 (Olie) Taiwa Ubiquitous Iformatio Volume 1, Number 1, February 16 Ivariability of Remaider Based Reversible Watermarkig Shao-Wei Weg School of Iformatio Egieerig Guagdog Uiversity of Techology wswweiwei@126.com Jeg-Shyag Pa Iovative Iformatio Idustrial Research Ceter Shezhe Graduate School Harbi Istitute of Techology jegshyagpa@gmail.com Jie-Hag Deg School of Computers Guagdog Uiversity of Techology Received February 15; revised December 15 Abstract. A ovel ad simple iteger trasform is proposed i this letter. This trasform ca also be cosidered a predictio process, i which the mea value of a -sized block is used to predict each pixel i this block so that two bits are embedded ito each of ( 1) pixels. By employig the ivariability of the remider produced by the mea value divided by a give embeddig parameter, we ca achieve reversibility. The embeddig distortio ca be greatly cotrolled by embeddig 2( 1) bits ito blocks with strog itra-block correlatio while keepig the others ualtered. Experimetal results reveal the proposed algorithm is effective. Keywords: Ivariability of remaider, Reversible watermarkig. 1. Itroductio. Reversible watermarkig based o differece expasio was proposed by Tia [1]. The differeces betwee two pixels were expaded to carry watermark iformatio if either overflow or uderflow occurred. Alattar [2] geeralized the DE techique by takig a set cotaiig multiple pixels rather tha a pair. I Thodi s work [3], histogram shiftig was icorporated ito Tia s method to produce a ew algorithm called Alg. D2 with a overflow map. Weg et al. proposed a iteger trasform based o ivariability of the sum of pixel pairs []. I Wag et al. s method [5], a geeralized iteger trasform ad a payload-depedet locatio map were costructed to exted the DE techique to the pixel blocks of arbitrary legth. Whe the embeddig rule of Alattar s method is reformulated as a iteger trasform, this trasform ca be deemed to cotai a additioal term ad a predictio process which uses the mea value of a block to predict each pixel i this block. This additioal term is ecessarily a guaratee of ivariability of the mea value before ad after embeddig. However, its existece will result i great decrease of PSNR (peak sigal to oise ratio) value i Alattar s method. This is also the reaso that the performace of Alattar s method is ot capable of exceedig that of Wag et al. s. 16
Ivariability of Remaider Based Reversible Watermarkig 17 To solve this problem above, a ovel ad simple iteger trasform that ca remove the redudat term is proposed i this letter. Therefore, this iteger trasform ca be cosidered a pure predictio process, i which the mea value of a -sized block is used to predict each pixel i this block so that two bits are embedded ito each of ( 1) pixels. By employig the ivariability of the remider of the mea value divided by a give embeddig parameter, we ca achieve reversibility. Sice the proposed iteger trasform ca embed 2( 1) (rather tha (-1)) bits ito a -sized block which itroduces less distortio, we ca obtai a better performace tha Wag et al. at lager embeddig rates. 2. The related methods. The iteger trasform defied i Eq. (3) of the paper [5] (proposed by Wag et al.) is listed i Eq. (1). y 1 = 2x 1 a(x) y 2 = 2x 2 2f(a(x)) + w 1 = 2x 2 (a(x) + LSB(a(x))) + w 1 y = 2x 2f(a(x)) + w 1 = 2x (a(x) + LSB(a(x))) + w 1 where x = (x 1,, x ) Z ad y = (y 1,, y ) Z respectively represet a -sized pixel array ad its correspodig watermarked oe, x = 1 { x i, a(x) = x if x x < 0.5, f(x) = x otherwise x, ad w 2 i (i {0, 1,, 1}) deotes 1-bit watermark ad w i {0, 1}, ad LSB( ) represets the least sigificat bit (LSB). a(x) is actually the rouded value of x. Eq. (1) is rearraged as follows y 1 = a(x) + 2(x 1 a(x)) y 2 = a(x) + 2(x 2 a(x)) + w 1 LSB(a(x)) y = a(x) + 2(x a(x)) + w 1 LSB(a(x)) The iteger trasform defied i the paper [2] (proposed by Alattar) ca be summarized i Eq. (3). y 1 = x 2(x2 x 1 )+w 1 + +2(x x 1 )+w 1 = x 2 x i + 1 w i 2x 1 (3) y 2 = y 1 + 2 (x 2 x 1 ) + w 1 y = y 1 + 2 (x x 1 ) + w 1 Suppose k 2 = x i x, ad the k 2 {0,, 1}. Substitute x i ito Eq. (3), we have y 1 = x + 2(x 1 x ) 2k 2+ 1 y 2 = x + 2(x 2 x ) + w 1 2k 2+ 1 y = x + 2(x x ) + w 1 2k 2+ 1 Comparig Eq. (2) with Eq. (), we see that the iteger trasform proposed by Alattar has its ow disadvatage: it itroduces the higher distortio tha the trasform proposed (1) (2) ()
18 S. W. Weg, J. S. Pa, ad J. H. Deg Capacity vs. Distortio Compariso o Lea Prop. Alg. sigle embeddig Wag s Alg. sigle embeddig Alattar s Alg. double embeddig 55 Capacity vs. Distortio Compariso o Baboo Prop. Alg. sigle embeddig Wag s Alg., sigle embeddig Alattar s Alg., double embeddig Quality Measuremet PSNR (db) Quality Measuremet PSNR (db) 0 0.2 0. 0.6 0.8 1 1.2 1. 1.6 1.8 2 2.1 0 0.2 0. 0.6 0.8 1 (a) Lea (b) Baboo 60 55 Capacity vs. Distortio Compariso o Plae Prop. Alg. sigle embeddig Wag s Alg., sigle embeddig Alattar s Alg., double embeddig Quality Measuremet PSNR (db) 0 0.2 0. 0.6 0.8 1 1.2 1. 1.6 (c) Plae Figure 1. Performace compariso of our method with the other two methods. by Wag et al.. Alattar s method ca be deemed to cotai a additioal term ad a process which uses the mea value of a block to predict each pixel i the block. Specially, for each y i (i {0, 1,, }) i Eq. (), it has a term, i.e., 2k 2+ 1. Sice k 2 {0, 1,, 1} ad w i {0, 1}, 0 2k 2+ 1 3( 1) = 3 1 = 2, i.e., 2k 2+ 1 {0, 1, 2}. For istace, whe k 2 reaches its maximum value (i.e., 1), if all to-be-embedded bits are set to 1, the this term will reach its maximum value (amely 2). Similarly, Wag et al. s method cotais a term, i.e., LSB(a(x)) {0, 1} except the process which uses a(x) to predict each pixel i the block. Therefore, relative to this term i Wag et al. s method, the existece of the oe i Alattar s method will result i greater reductio of PSNR value. Abudat experimets also demostrate that Wag et al. s method has superior performace to Alattar s. For the purpose of icreasig algorithm performace, we propose a ovel ad simple iteger trasform that ca remove the redudat term. Therefore, this iteger trasform ca be cosidered a pure predictio process, i which the mea value of a -sized block is used to predict each pixel i this block so that two bits are embedded ito each of ( 1) pixels. By employig the ivariability of the remider of the mea value divided by a give embeddig parameter, we ca achieve reversibility. Sice the proposed iteger trasform ca embed 2( 1) (rather tha (-1)) bits ito a -sized block which itroduces less distortio, we ca achieve a better performace tha Wag et al. at lager embeddig rates. 3. The proposed method. 3.1. A iteger trasform. I the proposed method, a grayscale image is partitioed ito o-overlappig m m-sized sub-blocks, where = m m. A iteger trasform exploitig ivariability of the remaider produced by dividig the mea value by a give
embeddig parameter is proposed below Ivariability of Remaider Based Reversible Watermarkig 19 y 1 = x + (x 1 x ) + e 1 = x 1 3 x + e 1 y 1 = x + (x 1 x ) + e 1 = x 1 3 x + e 1 y = x + (x x ) = x 3 x where e i represets 2-bit watermark for each i {1,, 1}, amely e i {0, 1, 2, 3}, the umber idicates the give embeddig parameter. This iteger trasform ca also be cosidered as a predictio process, i which the mea value is used to predict each pixel i a -sized block so that two bits are embedded ito each of ( 1) pixels (see Eq. (5)). I this trasform, mod( x, ) remais ualtered before ad after embeddig, where the modulo fuctio mod(a, b) returs the remaider of a divided by b, which is expressed by the followig formula: mod(a, b) = a b a. This ivariability is the sigle b most importat elemet to retrieve the origial pixel values. Next, we will itroduce i detail how to correctly extract watermark bits. Firstly, subtractig y from y i for each i {1,, 1} is to obtai the differece values as follows y i y = (x i x ) + e i (6) Sice e i is a positive iteger smaller tha or equal to 3, i.e., e i {0, 1, 2, 3}, ad meawhile, (x i x ) is a multiple of, the remaider will remai ivariat whe e i ad (x i x ) + e i are divided by, respectively. That is to say, mod(y i y, ) = mod(e i, ) = e i. Cosequetly, watermark bits e i ca be correctly extracted. We subtract the correctly extracted bits e i from y i for each i {1,, 1} so as to be capable of correctly retrievig the origial block x. The differece value y i betwee y i ad e i is calculated via the followig equatio Eq.(7) (5) y i = x i 3 x (7) For the coveiece of descriptio, we use y = (y 1,, y 1, y ) represets the pixelvalue-array via Eq. (7). Suppose x = k 1 + k 2, where k 1 R ad k 2 {0, 1, 2, 3}. Notice that, mod( x, ) = k 2. Substitute x ito Eq. (7) ad y, we have y i = (x i 3k 1 ) 3k 2 = x i 3k 2 y = (x 3k 1 ) 3k 2 = x 3k 2 (8) We use x i to replace (x i 3k 1 ) so as to further simplify Eq. (8). We will prove that mod(x i 3k 2, ) is idetical to k 2 whe x i is set as some iteger usig the followig equatio mod(x i 3k 2, ) = mod( 3k 2, ) = mod(k 2 3k 2, ) = mod(k 2, ) = k 2 (9) I a word, mod(x i 3k 2 ) = mod( x, ). O the decodig side, after we ca correctly get the remaider of x divided by, i.e., k 2, each pixel value of y is subtracted by k 2 to get the differece value i accordig to the followig equatio i = y i k 2 = x i k 2 = x i 3 x k 2 (10)
S. W. Weg, J. S. Pa, ad J. H. Deg After both sides of Eq.(10) is divided by, y i = x i 3 x +k 2 for each i {1,, }. We have It yields that 1 i = 1 i = x i 3 x + k 2 x i 3 x +k 2 = x k 2 x = 1 = k 1. So (11) i + k 2 (12) Substitute x via Eq. (12) ito Eq. (5), we ca correctly retrieve x. Ad the, x is substituted ito Eq. (7) so as to retrieve x i for each i {0, 1,, 1}. 3.2. Data Embeddig. To prevet the overflow/uderflow, each watermarked pixel value should be cotaied i [0, 5]. We defie { } x A : 0 xi 3 x 2(1 i 1), D = 0 x 3 x 5 where A = {x = (x 1,, x ) Z : 0 x i 5}. For a pixel-value array x A, it is classified ito oe of two sets: E t = {x D : v(x) T h } ad O t = {x A E t : v(x) > T h }, where T h is a give threshold, v(x) represets the variatio of a block, i.e., v(x) = 1 (x i x) 2, E t ad O t are used to deote the sets of pixels which are altered so as to carry 2( 1) bits or kept ualtered, respectively. A locatio map is geerated i which the locatios of the pixel arrays belogig to E t are marked by 1 while the others are marked by 0. The locatio map is compressed losslessly by a arithmetic ecoder ad the resultig bitstream is deoted by L. L S is the bit legth of L. For each x E t, it ca carry 2( 1) watermark bits, so the maximum hidig capacity is C ap = 2( 1) E t L S bits, where represets the cardiality of a set. Namely, the size of the payload equals C ap. For each x O t, the it is kept ualtered, i.e., y = x. For each x, if it belogs to E t, the 2( 1) bits are embedded ito it accordig to Eq. (5). After the first L S pixel arrays have bee processed (meaig either altered to carry 2( 1) bits or kept ualtered), the LSBs of the first L S pixels of their correspodig watermarked pixel arrays y are firstly appeded to the payload P, ad the replaced by the compressed locatio map L. After all the sub-blocks are processed, a ew marked image I w is obtaied. 3.3. Data Extractio ad Image Restoratio. The LSBs of the pixels i I w are collected ito a bitstream B accordig to the same order as i embeddig. B is decompressed by a arithmetic decoder to retrieve the locatio map. For each watermarked pixel array y = (y 1,, y ), if its locatio is associated with 0 i the locatio map, the it is igored, i.e., x = y. Otherwise, the watermark ca be extracted usig the followig formula: e i = mod(y i y, ) for each i {0,, 1}. Ad meawhile, the origial mea value is retrieved by Eq. (12). Fially, the origial image is recovered.. Experimetal results. The capacity vs. distortio comparisos amog the proposed method, Wag et al. s, Alattar s are show i Figs. 1 ad 2. Three stadard 512 512- sized grayscale images are used i our experimets: Lea, Baboo ad Plae. Suppose the thresholds correspodig to sigle embeddig ad double embeddig are T 1 ad T 2,
Ivariability of Remaider Based Reversible Watermarkig 21 Capacity vs. Distortio Compariso o Lea Prop. Alg. sigle embeddig Prop. Alg. double embeddig Wag s Alg. sigle embeddig Wag s Alg. double embeddig Capacity vs. Distortio Compariso o Baboo Prop. Alg. sigle embeddig Prop. Alg. double embeddig Wag s Alg. sigle embeddig Wag s Alg. double embeddig Quality Measuremet PSNR (db) Quality Measuremet PSNR (db) 0 0.2 0. 0.6 0.8 1 1.2 1. 1.6 1.8 2 2.1 (a) Lea 15 0 0.2 0. 0.6 0.8 1 1.2 1. 1.6 (b) Baboo Quality Measuremet PSNR (db) 60 55 Capacity vs. Distortio Compariso o Plae Prop. Alg. sigle embeddig Prop. Alg. double embeddig Wag s Alg. sigle embeddig Wag s Alg. double embeddig 0 0.2 0. 0.6 0.8 1 1.2 1. 1.6 1.8 2 (c) Plae Figure 2. Performace compariso of our method with Wag et al.s method, for sigle ad double embeddig. respectively. Suppose also that T 1 has a iitial value which is set to a value capable of esurig the embeddig rate larger tha 0. The, T 1 is gradually icreased from this iitial value util the give embeddig rate is achieved. Oce T 1 is determied, its correspodig PSNR value uder the give embeddig rate is obtaied experimetally. These obtaied umerical results o three test images are plotted i Fig. 1, respectively. For double embeddig, T 2 is set to half of T 1. Whe T 1 is gradually icreased from the iitial value, T 2 is also accordigly raised. Similarly, oce T 1 ad T 2 is determied, each correspodig data pair cotaiig a give embeddig rate ad its PSNR value is plotted i Fig. 2 o three test images, respectively. From Fig. 1, it ca be see that the two curves correspodig to the proposed method ad Wag et al. s are very close whe the embeddig rate is low. Whe the embeddig rate is icreased, the proposed method outperforms Wag et al. s for Lea ad Plae. Moreover, the embeddig capacity i Wag et al. s method is upper bouded by (1 1 ) bpp for a sigle embeddig process. However, we ca get a bpp close to 2(1 1 ) without multiple embeddig. As illustrated i Fig. 1, we get a sigificat performace icrease relative to Alattar s. Sice the correlatios betwee the eighborig pixels i Baboo ot as high as i the others so that little improvemet was made by the proposed method. So, the proposed method has almost the same performace as Wag et al. s, while provide a better performace tha Alatter s (see Figs 1 ad 2). Double embeddig is the process of embeddig data ito a always embedded image. We also perform double embeddig for three test images. As show i 2, we ca achieve the same performace as Wag et al. s whe the embeddig rate is smaller tha 1.6 bpp for Lea or 1. bpp for Plae. Our superiority becomes more obvious whe the embeddig rates gradually approach to 2.0 bpp, especially for Lea ad Plae. Whe the threshold
22 S. W. Weg, J. S. Pa, ad J. H. Deg value is set to a large iteger, ad correspodigly, the umber of pixel arrays used for embeddig is greatly icreases. Therefore, we ca obtai higher embeddig rates tha Wag et al. s. Sice double embeddig that embeds more data ito blocks with high correlatio will lead to lower distortio, double embeddig may be better tha sigle embeddig. 5. Coclusios. A ovel iteger trasform is proposed i this letter. This trasform ca also be cosidered a predictio process, i which the mea value of a -sized block is used to predict each pixel i this block so that two bits are embedded ito each of the ( 1) pixels. Experimetal results reveal the proposed algorithm is effective. Ackowledgmet. This work was supported i part by Natioal NSF of Chia (No. 611393, No. 612267), New Star of Pearl River o Sciece ad Techology of Guagzhou (No. 1J20085). REFERENCES [1] J. Tia, Reversible data embeddig usig a differece expasio, IEEE Tras. Circuits Syst. Video Techol., vol. 13, o. 8, pp. 890 896, 03. [2] A. M. Alattar, Reversible watermark usig the differece expasio of a geeralized iteger trasform, IEEE Tras. Image Process., vol. 13, o. 8, pp. 117 1156, 0. [3] D. M. Thodi ad J. J. Rodrguez, Expasio embeddig techiques for reversible watermarkig, IEEE Tras. Image Process., vol. 16, o. 3, pp. 721 7, 07. [] S. W. Weg, Y. Zhao, J. S. Pa, ad R. R. Ni, Reversible watermarkig based o ivariability ad adjustmet o pixel pairs, IEEE Sigal Process. Lett., vol., o., pp. 1022 1023, 08. [5] X. Wag, X. L. Li, B. Yag, ad Z. M. Guo, Efficiet geeralized iteger trasform for reversible watermarkig, IEEE Sigal Process. Lett., vol. 17, o. 6, pp. 567 570, 10.