WATERMARKING BASED ON THE HYPERANALYTIC WAVELET TRANSFORM

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1 Voume 49, Number, 8 Eectronics and Teecommunications WATERMARKING BASED ON THE HYPERANALYTIC WAVELET TRANSFORM Corina NAFORNITA Aeandru ISAR Eectronics and Communications Facuty, Poitehnica University, Bd. V. Parvan, Timisoara, 45647, (corina.nafornita, aeandru.isar)@etc.upt.ro Abstract: In this paper we present a new technique for watermarking in the hyperanaytic waveet domain. This has the advantage of better directiona seectivity and shift invariance compared to the cassica waveet fiter. The watermark is inserted into a eves of decomposition using a perceptua mask conceived by the authors. We compare this technique with two systems in the DWT domain, and show its superiority. We study the resistance of the three watermarking systems against various signa processing attacks. A three methods successfuy detect the watermark. This eads to the concusion that the Hyperanaytic Waveet Transform (HWT) increases the payoad and robustness. Key words: HWT, watermarking, perceptua mask, capacity. I. INTRODUCTION The idea of digita watermarking has been proposed as a means to protect the copyright of digita images. The majority of watermarking agorithms operate based on the spread spectrum (SS) communication principe. A pseudorandom sequence is added to the host image in some criticay samped domain and the watermarked image is obtained by inverse transforming the modified image coefficients. Typica transform domains are the Discrete Waveet Transform (DWT), the Discrete Cosine Transform (DCT) and the Discrete Fourier Transform (DFT). Numerous Waveet Transforms (WTs) can be used to embed the watermark. The first one was the DWT, [1]. The DWT based agorithms usuay produce watermarked images with the best baance between visua quaity and robustness due to the absence of bocking artifacts. It has three main disadvantages, []: ack of shift invariance, ack of symmetry of the mother waveets and poor directiona seectivity. Due to the poor directiona seectivity for diagona features of the DWT the watermarking capacity is sma. The most important parameters of a watermarking system are robustness against attacks and capacity. These parameters must be maimized. The DWT s disadvantages can be diminished using a compe waveet transform [, ]. In this paper, we present a new technique based on a simpe impementation of the HWT, and compare its performance with two DWT based methods. We study the resistance of the three mutiresoution framework watermarking systems against various signa processing attacks. The paper is organized as foows. In the net two sections are described the HWT impementation and the proposed watermarking system. Net section is dedicated to simuation resuts and finay some remarks are drawn. II. THE HWT IMPLEMENTATION If the function ψ is a mother waveets then the functions jh{ψ} and ψ a ψ+ jh{ψ} are aso mother waveets, where H is the Hibert transform. This waveet pair (ψ, jh{ψ}) defines a compe discrete waveet transform (CDWT). A compe waveet coefficient is obtained by interpreting the waveet coefficient from one DWT tree as its rea part, whereas the corresponding coefficient from the other tree is imaginary part. The dua tree compe waveet transform (DTCWT) [] is a quadrature pair of DWT trees, simiar to the CDWT. The DTCWT coefficients may be interpreted as arising from the DWT associated with a quasi-anaytic waveet. Both DTCWT and CDWT are invertibe and quasi shift-invariant; however the design of these quadrature waveet pairs is quite compicated and it can ony be done through approimations. In the new impementation of the HWT we first appy a Hibert transform to the data. The rea waveet transform is then appied to the anaytica signa associated to the input data, obtaining compe coefficients. The two impementations of the CDWT are equivaent: d m, n t, ψ t + ih ψ t { } [ ] ( ) m, n ( ) m, n ( ) ( t), ψ m, n ( t) i ( t), H{ ψ m, n ( t) } ( t), ψ m, n ( t) + i H{ ( t) }, ψ m, n ( t) ( t) + ih{ ( t) }, ψ ( t) d [ m, n] DTCWT m, n HWT In the foowing we wi use the definition of the anaytic signa associated to a D rea signa named hyper-compe signa. So, the hyper-compe mother waveet associated to (1) Manuscript received September 1, 8; revised October 5, 8 1

2 Voume 49, Number, 8 Eectronics and Teecommunications the rea mother waveet (, ψ a (, ψ(, + ih (, + + jh (, + kh H { ψ(, } y ψ is defined as: { } y () where i j k and ij ji k, [7]. The HWT of the image f (, is: HWT{ f (, } f (,, ψ (,. () Taking into account reation () it can be written: a HWT + jdwt f a { f (, } DWT{ f (, } + idwt { H { f (, } { H y { f (, } + kdwt { H y { H { f (, } (,, ψ(, DWT{ f (, }. The D-HWT of the image ( a (4) f, can be computed with the aid of the D-DWT of its associated hyper-compe image. The new HWT impementation, [6, 8] (fig. 1), uses four trees, each one impementing a D-DWT. The first tree is appied to the input image. The second and the third trees are appied to 1D discrete Hibert transforms computed across the ines ( H ) or coumns ( H y ) of the input image. The fourth tree is appied to the resut obtained after the computation of the two 1D discrete Hibert transforms of the input image. The enhancement of the directiona seectivity of the D-HWT is reaized as in the case of the D- DTCWT, [, 4], by inear combinations of detai Figure 1. The new D-HWT impementation architecture. coefficients beonging to each subband of each of the four D-DWTs. The HWT detai images are oriented foowing the directions: ± atan(1/), ± π/4 and ± atan(). For eampe, this new impementation makes the difference between the two principa diagonas or between the directions ± atan(1/) whereas the DWT cannot make such differences. The coefficients z + r and z r are used in the foowing simuations for the watermark embedding. A possibe representation of the HWT s organization based on these two types of coefficients that highights the anges aready mentioned, for the first decomposition eve is given in Fig.a). This structure corresponds to the eampe in Fig.b), where the coefficients z + r and z r of the HWT Figure a. Representation of the rea coefficients in which the watermark is embedded (detai sub-images on. Figure b. An eampe for the representation in fig. a) in the case of test image Lena, for three eves of decomposition.

3 Voume 49, Number, 8 Eectronics and Teecommunications of the image Lena are represented. It must be mentioned that the sub-images corresponding to the category of coefficients z were horizontay mirrored. r III. WATERMARKING BASED ON THE NEW IMPLEMENTATION OF THE HWT The watermark capacity was studied in [9] where an information-theoretic mode for image watermarking and data hiding is presented. The ack of shift invariance of the DWT and its poor directiona seectivity are reasons to embed the watermark in the fied of another WT. To maimize the robustness and the capacity, the roe of the redundancy of the transform used must be highighted first. An eampe of redundant WT is represented by the tight frame decomposition. In [1] are anayzed the watermarking systems based on tight frame decompositions. The anaysis indicates that a tight frame offers no inherent performance advantage over an orthonorma transform (DWT) in the watermark detection process. This anaytica observation shoud be tempered with the fact that spread-spectrum watermarking is often depoyed in conjunction with an image-adaptive weighting mask to take into account the human visua system (HVS) and to improve perceptua performance. Another redundant WT, the DT-CWT, was aready used for watermarking, [11]. The authors of this paper prove that the capacity of a watermarking system based on a compe waveet transform is higher than the capacity of a simiar system that embeds the watermark in the DWT domain. Many authors (e.g. Daugman [1]) have suggested that the processing of visua data inside our visua corte resembes fitering by an array of Gabor fiters of different orientations and scaes. We have aready proved that the proposed impementation of HWT is efficient, has ony a modest amount of redundancy, it provides approimate shift invariance, it has better directiona seectivity than the D DWT and it can be observed that the corresponding basis functions cosey approimate the Gabor functions. So, the spread spectrum watermarking based on the use of an image adaptive weighting mask appied in the D-HWT domain is potentiay a robust soution that increases the capacity. We propose the embedding of the watermark in the images z +r and z -r, using a perceptua mask; whie the bind detection is made on the coupe (z +r, z -r ). This approach is compared with the cassica approach in the DWT domain, from Barni s method [1] and the method proposed in [14]. Perceptua Watermarks Embedded in the DWT Domain One of the quaities required to a watermark is its imperceptibiity. This can be achieved by embedding the watermark in coefficients of known robustness (usuay arge coefficients) or perceptuay significant regions, i.e. contours and tetures of an image. This can be done empiricay, seecting arger coefficients or using a threshoding scheme in the transform domain [16, 17]. Another approach is to insert the watermark in a coefficients of a transform, using a variabe strength for each coefficient [1, 14]. Hybrid techniques, based on compression schemes, embed the watermark using a threshoding scheme and variabe strength [16]. Specificay we wi ony refer to the techniques presented in [1] and [14]. The watermark is masked according to the characteristics of the HVS, taking into account the teture and the uminance content of a the image subbands [1, 14]. For coefficients corresponding to contours of the image a higher strength is used, for tetures a medium strength is used and for regions with high reguarity a ower strength is used, in accordance with the anaogy water-fiing and watermarking proposed by Kundur in [15]. Barni s method [1] is quite robust against common signa processing attacks ike fitering, compression, cropping etc. However, because embedding is made ony in the highest resoution eve, the watermark information can be easiy erased by a potentia attacker. In [14], a new piewise mask was used in the DWT domain that modes the HVS behavior in a better way thus aowing embedding of the watermark in a resoution eves, ecept the ast one. At the detection, since the threshod is image dependent, the ratio between correation and threshod was used; hence the detection function becomes noninear with a fied detection threshod. Three types of detectors are being used, to take advantage of the waveet hierarchica decomposition. The watermark presence is detected 1) from a resoution eves, ) separatey, considering the maimum detector response from each eve and ) separatey, considering the maimum detector response from each subband. Evauating correations separatey per resoution eve or subband is sometimes advantageous. Layers or subbands with ower detector responses are discarded. This type of embedding combined with new detectors is more attack resiient to a possibe erasure of the three subbands. At the embedding process [1], the image I, of size M N, is decomposed into four eves using Daubechies-6 waveet mother, where I is the subband from eve {,1,,}, and orientation {,1,,} (corresponding to horizonta, diagona and vertica detai subbands, and approimation subband). A binary watermark j is embedded in a coefficients from the subbands from eve by addition: I I + αw, (5) where α is the embedding strength and w ( a weighting function. The mask (or equivaenty the weighting function) is buit pie by pie, and it gives the maimum amount of modifications that can be appied to the corresponding DWT coefficient in the detai band without compromising watermark invisibiity. This weighting function is a haf of the quantization step, which is computed in [1] as the product of three factors: q. Θ(, ) Λ(, Ξ(,, (6)

4 Voume 49, Number, 8 Eectronics and Teecommunications and the embedding takes pace ony in the first eve of decomposition, for. The first factor is the sensitivity to noise depending on the orientation and on the eve of detai: 1, 1. 1 Θ (, ) 1, otherwise.16.1 The second factor takes into account the oca brightness based on the gray eve vaues of the ow pass version of the image: where: and: L (, 1+ L' (,, (7) Λ (8) L(,, L(, (,, 1 <.5 L '(, (9) L otherwise 1 ( I ( 1+ i /,1 + j / ), (1) 56 The integer part operator from the right hand side is used to I modify the size of the image in accordance with the requirements of the previous decomposition eves. It produces an interpoation of the initia image by mutipe repetitions of each row and coumn. The third factor is computed as foows: Ξ (, Var I 1 1 k 16 I k y i 1+ y + k + j,1 + + i y + k, y,1 j, + k subimage from eve, denoted S. Normaization is aso required since the teture vaues obtained are too high. This is normaized to the oca mean, aso compressed in the waveet domain, U. Ξ S 1 k k k (, 16 [ I ( y + i /, + j / )] k, y / U k + (1) The second difference between [1] and [14] is that the uminance mask is computed on the approimation image from eve, where the watermark is embedded. The DWT of the origina image using decomposition eves was computed and the approimation sub-image corresponding at eve was separated, obtaining the image I. Reation (1) was repaced in [14] by: L (, I / 56. (1) Since both factors aready described are more dependent on the resoution eve in the method proposed in [14], the noise sensitivity function is aso repaced by: { }, 1 1,,1 Θ (, ) 1, otherwise.66, (14) In [1], detection is made using the correation between the marked DWT coefficients and the watermarking sequence to be tested for presence, for eve : ρ M / 1 N / 1 4 MN ( ) I i j (15) (11) and it gives a measure of teture activity in the neighborhood of the pie. This factor is composed by the product of two contributions; the first is the oca mean square vaue of the DWT coefficients in a detai subbands, whie the second is the oca variance of the ow-pass subband. Both these contributions are computed in a sma neighborhood corresponding to the ocation ( of the pie. The first contribution is the distance from the edges, whereas the second one the teture. This oca variance estimation is not so precise, because it is computed with a ow resoution. In [14], the second factor from Eq. (11) is repaced by the oca standard deviation of the image, computed on a window of 7 7, which has a higher resoution. This is compressed in the waveet domain to have the same size as the subband where the watermark is to be inserted. The required mask wi be the approimation 4 The correation is compared to a threshod T ρ(), computed to grant a given probabiity of fase positive detection, using the Neyman-Pearson criterion. For eampe, if P f 1-8, the threshod is Tρ( ).97 σρ( ), with σ ρ ( ) the variance of the waveet coefficients, if the image was watermarked with a code Y other than X: σ ρ 4 MN M / 1 N / 1 ( ) ( ) I i j. (16) In [14] was considered the ratio between the correation ρ() in Eq. (16) and the image dependent threshod T ρ(), hence the detector was viewed as a noninear function with a fied threshod. Three detectors are being used. The first detector evauates the watermark s presence on a resoution eves: d ρ d T (17) 1 1 / d1

5 Voume 49, Number, 8 Eectronics and Teecommunications where the correation ρ d1 is given by: ρ M / 1 N / 1 1 d1 I,, i j MN The threshod for P f 1-8 is T σ 4 d1 ( i ρd1.97 σ, with: M / 1 N / 1 1 ρd 1,, j MN 4 ( I ( i ) (18) (19) The second detector considers the responses from different eves, as d()ρ()/t ρ(), with {,1,}, and discards the detector responses with ower vaues: { d( ) } d ma () The third detector considers the responses from different subbands and eves, as d(,) ρ(,)/t(,), with, {,1,}, and discards the detector responses with ower vaues: d ma d, (1) { ( )}, where ρ(,) and T(,) are the correations and threshods for each subband. For P f 1-8, the threshod is T (, ).97 σρ(, ) ρ(, ) coefficients from the subband (,): and:, with σ the variance of waveet M / 1 N / 1 ρ () (, ) 4 I /( MN ) i j ( ) ( MN ) M / 1 N / 1, / ( ) 16 σ ρ i j I () Perceptua Watermarks Embedded in the HWT Domain Adapting the strategy aready described in the previous paragraph to the case of HWT, we have decided to use the first three waveet decomposition eves. We have embedded the watermark into the coefficients z + r and z r. In this case the reations aready described in the previous paragraph were used independenty for each of these two images. The same message was embedded in both images, using the mask from [14]. The difference is that the orientations or preferentia directions are in this case: atan(1/), π/4, atan() (respectivey for,1,), for the image z + and -atan(1/), -π/4, -atan(), (,1,) for the r image z r. At the detection side, we consider the pair of images ( z + r, z r ), thus having twice as much coefficients than the standard approach, and takes a the possibe vaues, ± atan(1/), ± π/4, ± atan(). IV. SIMULATION RESULTS Some simuation resuts, obtained using the image Lena, of size 5151, are reported in the foowing. For α 1.5, and a watermark inserted in a eves:, 1 and, the image watermarked in the HWT domain has a peak signa-to-noise ratio (PSNR) of 5.6 db, and a weighted PSNR of 5 db (see tabe 1). The peak signa-to-noise ratio, PSNR is defined as: 55 ( ) ( ) PSNR I, I 1og1 [db], 1 i I i Ii N where I is the host signa, I % is the watermarked signa, i is the particuar pie position, and N is the number of pies in I or I %. Another measure is the weighted PSNR (WPSNR); where weighting is performed according to the contrast sensitivity function, CSF, of vision as defined in [18]. The origina image, the corresponding watermarked image and the difference image are presented in Figure. We have eposed our watermarked images at some common attacks: JPEG compression with different quaity factors (Q), shifting, median fitering with different window sizes WS, resizing with different scae factors, cropping with different areas remaining, gamma correction with different vaues of γ, burring with a specified point spread function (PSF) and perturbation with AWGN with different variances and we have studied the robustness of the new watermarking method. The PSF is specified with the aid of two parameters: ength L and ange β. Resistance to unintentiona attacks, for watermarked image Lena, for our new method, can be compared to the resuts obtained using the watermarking methods reported in [1] and [14] anayzing tabe. In the simuation corresponding to the watermarking method proposed in [14], we use the same watermark strength, 1.5 and we embed the watermark in a three waveet decomposition eves, resuting in a PSNR of 6.86 db and a weighted PSNR of 5. db. For the simuation corresponding to the watermarking method proposed in [1], we use the watermark strength. and the embedding is made ony in the first resoution eve, resuting in a simiar quaity of the images in terms of cassica PSNR vaue (PSNR6.9 db, weighted PSNR. db). Specia attention was paid to the shifting attack. First the watermarked image was circuary shifted with i ines and co coumns, obtaining the attacked image ( I t ). Supposing that the numbers i and co are known, the messages at eve are circuary shifted with i/ ines and co/ coumns obtaining the new messages ( ) t. Net the 5

6 Voume 49, Number, 8 Eectronics and Teecommunications watermark was detected using the image ( I ) t and the messages ( ) t. The vaues obtained for i18 and co18 are presented in tabe. The robustness of the watermarking method against the shifting attack is given by two properties: the shift invariance degree of the WT used and the masking abiity. Anayzing tabe, the abiity of masking seems to be more important than the shift invariance degree of the WT used for the conception of counter-measures against the shifting attack, when the numbers of ines and coumns used for the attack are aready known. Finay, we have submitted the watermarked images to the recenty proposed oca desynchronization attack (DA), namey the oca permutation with canceation and dupication (LPCD) DAs [5]. The parameters used in the attack were the ones that visuay damage the image ess indicated by the authors, N5 and L6. The watermark is successfuy detected each time, for each method. Two eampes for the attack corresponding at the ast ine in tabe are presented in figures 4, 5 and 6. From the resuts, it is cear that embedding in the rea parts of the HWT transform yieds in a higher capacity at the same visua impact and robustness. In fact the resuts obtained for the new method are sighty better than the resuts obtained with the methods described in [1] and [14] for JPEG compression, median fitering with WS, resizing and gamma correction. For the other attacks the resuts, obtained using the new method, are simiar with the resuts of the watermarking methods based on DWT. Tabe 1. Comparison of invisibiity for the proposed method and the methods [1], [14]. Measure of Method Barni s Proposed invisibiity vs. methods proposed in [14] method [1] HWT method PSNR 6.86 db 6.9 db 5.6 db Weighted PSNR 5. db. db 5. db Figure. Origina (eft) and watermarked (midde) images with the new method, for α 1.5, PSNR5.6 db; (right) Difference image, ampified 8 times. Tabe. Comparison of robustness for different attacks. The HWT method proposed here, as we as the traditiona approaches from [1] and [14] are compared. Attacks vs. detector response A eves Method proposed in [14] ma eve Ma subband Barni s method [1] a eves Proposed HWT method ma eve ma subband Before attack JPEG, Q JPEG, Q JPEG, Q Shift, i18, co Median fitering, WS Median fitering, WS Resizing, Resizing, Cropping, Cropping, Cropping, Gamma correct., γ Gamma correct., γ Bur, L1, β LPCD attack, N5,L

7 Voume 49, Number, 8 Eectronics and Teecommunications Figure 4: (eft): Watermarked image for the method in [14], having a PSNR6.86 db and a weighted PSNR5. db compared to the origina Lena, (midde): Distorted watermarked with the LPCD attack N5 and L6 with PSNR9. db, weighted PSNR8.41 db. (right): Difference between the two images, magnified 1 times. Figure 5: (eft): Watermarked image for Barni s method [1], having a PSNR6.9 db and a weighted PSNR. db compared to the origina Lena, (midde): Distorted watermarked with the LPCD attack N5 and L6 with PSNR.68 db, weighted PSNR.8 db. (right): Difference between the two images, magnified 1 times. Figure 6: (eft): Watermarked image for the HWT based method, having a PSNR5.6 db and a weighted PSNR5 db compared to the origina Lena, (midde): Distorted watermarked with the LPCD attack N5 and L6 with PSNR8.16 db, weighted PSNR7.87 db, (right): Difference between the two images, magnified 1 times. V. CONCLUSION The HWT is a very modern WT as it has been formaized ony two years ago. In this paper we have used a very simpe impementation of this transform, which permits the epoitation of the mathematica resuts and of the agorithms previousy obtained in the evoution of waveets theory. It does not require the construction of any specia waveet fiter. It has a very feibe structure, as we can use any orthogona or bi-orthogona rea mother waveets for the computation of the HWT. The proposed impementation eads to both a high degree of shift-invariance and to an enhanced directiona seectivity in the D case. We have proposed a new type of pie-wise masking for robust image watermarking in the HWT domain. Modifications were made to two eisting watermarking techniques proposed in [1] and in [14], based on DWT. These techniques were seected for their good robustness 7 against the usua attacks. In fact, to be efficient, an attack must aso be transparent. The majority of the attacks presented in tabe are more severe, their parameters seection was made to highight the great robustness of the anayzed watermarking systems. The method proposed in [14] was inspired by the method proposed in [1], but it contains some modifications. The first modification is in computing the estimate of the variance, which gives a better measure of the teture activity. An improvement is aso owed to the use of a better uminance mask. The third improvement proposed in [14] is to embed the watermark in the detai coefficients at a resoutions, ecept the coarsest eve; this can be particuary usefu against erasure of high frequency subbands containing the mark in the watermarking system proposed in [1]. A noninear detector with fied threshod as ratio between correation and the image dependent threshod-has been conceived. Using it,

8 Voume 49, Number, 8 Eectronics and Teecommunications three watermark detectors were derived in [14]: 1) from a resoution eves, ) separatey, considering the maimum detector response for each eve and ) separatey, considering the maimum detector response for each subband. We have transformed here, a these usefu modifications proposed in [14], for the case of the HWT. A new embedding mechanism specific for this transform is proposed, epoiting the coefficients z + r and z r. The simuation resuts presented in this paper iustrate the effectiveness of the proposed agorithm. We tested the robustness of our method against a great diversity of attacks (with different parameters), and found out that it is very good being simiar or better than the robustness of the methods described in [1] and [14] in the case of the image Lena. Our watermarking method has superior capacity than the method proposed in [1] and even [14]. Of course, we wi repeat in the future these eperiments on an entire image database (ike for eampe that used in [14]), to statisticay vaidate the resuts obtained using the image Lena. There are good reasons to beieve that these resuts wi be good enough taking into account the high variabiity into different regions of this prototype image. Another future research direction is the anaysis of the new watermark detectors proposed in [14] and appied here in the fied of HWT. The anaysis of tabe shows, from the detectors point of view, an important difference between the DWT and the HWT. In the DWT domain the best is the ma subband detector whie in the HWT domain the best is the ma eve detector. The theoretica anaysis of those detectors seems to be very chaenging. The goa is to find the most robust detector that coud be based on a combination of a those detectors. Other embedding mechanisms can aso be conceived. Anayzing figure it can be observed that the coefficients z + i and z i are not epoited yet. So, the redundancy of the embedding can be increased epoiting the quaternions ( z+ r, z r, z+ i, z i ). Another embedding mechanism can use compe images of the form z + z+ r + jz+ i or z z r + jz i. The watermark can be embedded in the absoute vaue or in the phase of those images. We have aready tried the embedding in the absoute vaues, [19], obtaining simiar resuts with those presented in this paper. The observation that most of the information contained into a compe image is carried by its phase component can be taken into account in the future. REFERENCES [1] D. L. Donoho, I. M. Johnstone, Idea spatia adaptation by waveet shrinkage, Biometrika, 81() : , [] N. Kingsbury, Compe Waveets for Shift Invariant Anaysis and Fitering of Signas, Appied and Comp. Harm. Ana. 1, 1, 4-5. [] N. G. Kingsbury, A Dua-Tree Compe Waveet Transform with improved orthogonaity and symmetry properties, Proc. 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