Research ArticleA New Double Color Image Watermarking AlgorithmBased on the SVD and Arnold Scrambling
Ying Li1 Musheng Wei12 Fengxia Zhang1 and Jianli Zhao1
1College of Mathematical Sciences Liaocheng University Shandong 252000 China2College of Mathematics and Science Shanghai Normal University Shanghai 200234 China
Correspondence should be addressed to Ying Li liyingld163com
Received 12 July 2016 Revised 6 October 2016 Accepted 26 October 2016
Academic Editor M Montaz Ali
Copyright copy 2016 Ying Li et al This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
Wepropose a new imagewatermarking scheme based on the real SVDandArnold scrambling to embed a colorwatermarking imageinto a color host image Before embedding watermark the color watermark image119882 with size of119872times119872 is scrambled by Arnoldtransformation to obtain a meaningless image Then the color host image 119860 with size of119873 times119873 is divided into nonoverlapping119873119872 times 119873119872 pixel blocks In each (119894 119895) pixel block 119860 119894119895 we form a real matrix 119862119894119895 with the red green and blue components of119860 119894119895 and perform the SVD of 119862119894119895 We then replace the three smallest singular values of 119862119894119895 by the red green and blue values of 119894119895with scaling factor to form a new watermarked host image 119894119895 With the reserve procedure we can extract the watermark fromthe watermarked host image In the process of the algorithm we only need to perform real number algebra operations which havevery low computational complexity and are more effective than the one using the quaternion SVD of color image
1 Introduction
Security and copyright protection are important issues inmultimedia applications and services Among the protectiontechniques digital watermarking is considered a powerfulmethod which is a technique that hides the secret infor-mation in a host image without affecting its normal usageThe watermark can be extracted and used for authenticationand verification of ownership Digital watermarking has beenconcerned since it has been proposed in [1] in 1994 Overthe past 20 years many watermarking methods have beendeveloped [2ndash5] One can use a sequence of randomnumbersor a recognizable binary pattern or an image as a digitalwatermark Most of the existing color techniques use binaryor greyscale images [6ndash9] as watermarks Due to the rapidapplications of the color image technique on the Internetduring the recent years scientists and researchers have turnedtheir attention to color imagewatermarks [10ndash15] which havebecome one of the hot research topics
Compared with the binary and greyscale image the colorimage can greatly improve the capacity and fidelity of theinformation First the color image can hide a great amount
of data Second the color perception relies not only onthe luminance but also on the chrominance Color imagecan be broken into multiple prime color channels so thecolor image watermarking is more challenging compared tosingle channel greyscale images There are various breakuptechniques available in the literature for color image like YIQYCbCr RGB and HSI out of which RGB is apparently themost popular space because this channel format is a naturalscheme for representing real world color and each of thethree channels is highly correlated with the other two
The typical algorithms for color image watermarking canbe summarized as follows
(1) Single Channel Embedding In this approach the water-mark is embedded into the blue component part of thehost image [16] Such an approach might not be completelyadequate since it does not take the implication of the HumanVisual System into consideration and in particular it issensitive to color brightness and perception
(2) Multichannel Embedding In this approach the host andwatermark color images are first decomposed into single
Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2016 Article ID 2497379 9 pageshttpdxdoiorg10115520162497379
2 Journal of Applied Mathematics
color channels and then treated independently [13] Whethersingle channel processing or multichannel synthesis itsessence is to treat greyscale image Therefore this approachcannot reflect the connection between the channels very well
(3) Algorithms Based on Quaternion A color image canbe considered as a quaternion matrix of pure imaginarynumbers In this way the color image can be processed ina holistic manner without losing color information Noticethat these kinds of algorithms require more quaternionoperations
An effective watermarking algorithm shouldmeet certainrequirements including transparency robustness adequateinformation capacity and low computational complexity
Watermarking algorithms utilizing the Singular ValueDecomposition (SVD) have become popular during the lastten years [14 17ndash19] because of the stability of the SVD SVD-based algorithms embed a watermark by modifying eitherthe singular values or the orthogonal matrices The workin [17] proposed a SVD-based watermarking scheme wherethe singular values of the watermark image were used tomodify the singular values of the subband images which wereobtained by using DWT to the host image The work in [18]proposed an algorithm to employ SVD on the blue channel ofthe host image to retrieve the singular values and embed thewatermark in these singular valuesThework in [19] proposeda SVD-based color imagewatermarking scheme inwhich thesingular values of original watermarkwere required to extractthe embedded singular values and then 119880 and 119881 orthogonalmatrices of the original watermark were utilized to recoverthe watermark The work in [14] proposed a block-SVD-based watermarking method to embed color watermark tocolor host image All these algorithms can preserve the visualquality of the image even for large singular value changes
In this paper we propose a new image watermarkingscheme based on the real SVD and Arnold scrambling toembed a color watermarking image into a color host imageBefore embedding thewatermark the color watermark image119882with size of119872times119872 is scrambled by Arnold transformationto obtain a meaningless image Then the color host image119860 with size of119873 times119873 is divided into nonoverlapping119873119872 times119873119872 pixel blocks In each (119894 119895) pixel block 119860 119894119895 we form areal matrix 119862119894119895 with the red green and blue components of119860 119894119895 and perform the SVD of 119862119894119895 We then replace the threesmallest singular values of 119862119894119895 by the red green and bluevalues of 119894119895 with scaling factor to form a new watermarkedhost image 119894119895 With the reserve procedure we can extractthe watermark from the watermarked host image
The remainder of this paper is organised as follows InSection 2 we present a brief overview regarding the SVDand image scrambling based on Arnold transformation InSection 3 we compare the SVD algorithms for color imagesIn Section 4 we propose embedding and extracting proce-dures of the new algorithm The experimental results andanalysis are presented in Section 5 Finally our conclusionsare stated in Section 6
2 Preliminaries
21The SVD The SVD [20] is one of themost powerful toolsand is widely applied to digital image processing [21]
The SVD of an119898 times 119899matrix 119860 is expressed as
119860 = 119880Σ119881119867 (1)
where both 119880119898times119898 and 119881119899times119899 are unitary matrices 119881119867 isconjugate transpose of 119881 and
Σ = [Σ1 00 0] (2)
where Σ1 = diag(1205901 1205902 120590119903) 1205901 ge 1205902 ge sdot sdot sdot ge 120590119903 gt 0 arepositive singular values of119860 and 119903 is the rank ofmatrix119860 If119860is a real matrix then both119880119898times119898 and119881119899times119899 are real orthogonalmatrices
119860 = 119880Σ119881119879 (3)
where 119881119879 is the transpose of 11988122 Arnold Scrambling Scrambling is a pretreatment stageof watermarking which transforms the meaningful imageinto another meaningless one and provides security in water-marking scheme Even if the aggressors obtain the embeddedwatermarking image they cannot extract the watermarkwithout the knowledge of the scrambling algorithm
In the proposed watermarking scheme the 2D Arnoldscrambling transformation [22] is used which shuffles thepixels positions of watermark image as follows
[] = 119860[119909119910] mod 119873[119909119910] = 119860minus1 [] mod 119873
119860 = [1 11 2]
119860minus1 = [ 2 minus1minus1 1 ]
(4)
where (119909 119910) is the old pixel coordinates of the originalimage and ( ) is the new pixel coordinates after iterativecomputation scrambling and 119873 is the size of the imageThe transformation is periodic and shows periodicity aftercompleting the period which is shown in Figure 1
3 Comparison of the SVD Algorithms forColor Images
31 Quaternion Representation of a Color Image A quater-nion 119902 isin 119876 is expressed as
119902 = 119886 + 119887119894 + 119888119895 + 119889119896 (5)where 119886 119887 119888 119889 isin 119877 and three imaginary units 119894 119895 and 119896satisfy
1198942 = 1198952 = 1198962 = 119894119895119896 = minus1 (6)When 119886 = 0 119902 is called pure quaternion
Journal of Applied Mathematics 3
(4) Scrambling times = 48(3) Scrambling times = 20
(2) Scrambling times = 1(1) Scrambling times = 0
Figure 1 Arnold scrambling
In [23] Sangwine proposed to encode the three channelcomponents of a RGB image on the three imaginary parts ofa pure quaternion that is
where 119903(119909 119910) 119892(119909 119910) and 119887(119909 119910) are the red green and bluevalues of the pixel (119909 119910) respectively Thus a color imagewith 119898 rows and 119899 columns can be represented by a pureimaginary quaternion matrix
Since then quaternion representation of a color imagehas attracted great attention Many researchers applied thequaternion matrix to study the problems of color imageprocessing due to the ability of quaternion matrices to treatthe three color channels holistically without losing colorinformation
32 Comparison of Real SVD and Quaternion SVD For aquaternion matrix 119860 isin Q119898times119899 of a color image we can per-form the quaternion SVD with different kinds of algorithmsFor example in [24] authors provided the function ldquosvdrdquoin Matlab Toolbox using quaternion arithmetics In [25 26]authors proposed a real structure-preserving algorithmbasedon the following results
For any quaternion matrix 119860 = 1198601 + 1198602119894 + 1198603119895 + 1198604119896where 1198601 1198602 1198603 1198604 isin R119898times119899 its real representation can bedefined as follows [25 26]
The properties of 119860119877 are as followsTheorem1 (see [27]) Let119860 119861 isin Q119898times119899 119862 isin Q119899times119904 and 119886 isin RThen one has the following
(1) (119860 + 119861)119877 = 119860119877 + 119861119877 (119886119860)119877 = 119886119860119877 (119860119862)119877 = 119860119877119862119877(2) (119860119867)119877 = (119860119877)119879(3) 119860 isin Q119898times119898 is a unitary matrix if and only if 119860119877 is an
orthogonal matrix
From (9) andTheorem 1 for the matrix119860119877 we only needto store the first column block of 119860119877 denoted as
119860119877119888 =[[[[[
1198601119860211986031198604
]]]]] (10)
From this notation andTheorem 1 we have the followingresult
Theorem2 Let119860 119861 isin Q119898times119899 119862 isin Q119899times119904 119902 isin Q119898 and 119886 isin RThen one has the following
(1) (119860 + 119861)119877119888 = 119860119877119888 + 119861119877119888 (119886119860)119877119888 = 119886119860119877119888 (119860119862)119877119888 = 119860119877119862119877119888 (2) (119860119867)119877119888 = [(119860119877)119879]119888(3) 119860119865 = 119860119877119888119865 1199022 = 119902119877119888 2In addition we have the following results about the SVD
of a quaternion matrix and its real representation
Theorem 3 Let 119860 isin Q119898times119899 Then the singular values of 119860119877appear in fours
Therefore if we want to compute the SVD of an 119898 times 119899quaternion matrix 119860 then we can deal with the SVD ofthe 4119898 times 4119899 real representation matrix 119860119877 In fact underthe orthogonal transformations the real representation of aquaternion matrix has the standard form stated in the nexttheorem
Theorem 4 (see [25]) Suppose that 119860 isin Q119898times119899 and 119860119877 is thereal representation of 119860 Then there exist orthogonal matrices119880 isin R4119898times4119898 and 119881 isin R4119899times4119899 such that
119880119879119860119877119881 = [[[[[
119863 0 0 00 119863 0 00 0 119863 00 0 0 119863
]]]]] (11)
where119863 isin R119898times119899 is a bidiagonal matrix
4 Journal of Applied Mathematics
Table 1 Computation amounts and assignment numbers for realSVD and quaternion SVD
FromTheorems 3 and 4 the SVD of 119860119877 can be obtainedby computing the SVD of bidiagonal matrix 119863 So weshould first get bidiagonalmatrix119863 usingHouseholder basedtransformation In [26] we listed three forms of Householderbased transformations that appeared in the literatures andproposed a new formof quaternionHouseholder based trans-formation And then we gave the real structure-preservingalgorithms for these quaternion Householder based trans-formations By comparison on computation amounts andassignment numbers we obtained the most flexible andefficient one which is described as follows
Theorem 5 Suppose that 0 = 119910 isin Q119899 is not a multiple of 1198901denote 119906 = (119910 minus 1205721198901)119910 minus 1205721198901 where
120572 = minus 1199101100381610038161003816100381611991011003816100381610038161003816
120572119872 = diag(120572|120572| 119868119899minus1) and 119867 = 120572119872(119868 minus 2119906119906119867) and then 119867maps 119910 to |120572|1198901
This kind of Householder based transformation is 1198674introduced in [26] For its specific real structure-preservingalgorithm refer to Algorithms 41 42 and 48 in [26]
After the bidiagonal matrix 119863 is calculated we canperform a sequence of iterations using Givens rotations onit and then the SVD of quaternion matrix is obtained
On the other hand we can extract the three imaginaryparts R G and B of matrix 119860 to rearrange as a new realrectangle matrix
119862 = [[[
RGB
]]] (13)
and then directly use the function ldquosvdrdquo on the real matrix119862Remark 6 Not only is the computation time related tothe number of floating points arithmetics but also it hasa lot to do with the assignment number In Table 1 welist the numbers of real flops and assignment numbers forcomputing real SVD for 119862 isin R3119898times119899 and quaternion SVDfor 119860 isin Q119898times119899 using a real structure-preserving algorithmwhere assignment numbers refer to the number of callingsubroutines or performing matrix operations In matrixoperations say 119861 = 119860119883 + 119884 we adopt the assignment119861 = 119860 lowast 119883 + 119884 to utilize vector pipelining arithmeticoperations rather than explicitly using triply nested for-end loops to speed up computations remarkably Thereforereal arithmetic numbers as well as assignment numbers areimportant measures See for example Chapter 1 of [20]
Toolbox for qSVDStructure-preservingReal SVD
0
10
20
30
40
50
60
70
CPU
tim
e (se
cond
)
10 20 30 40 500k (m = 10 lowast k)
Figure 2 Comparison of the SVD algorithms
We now provide a numerical example to compare theefficiencies of the three algorithms mentioned above Allthese computations are performed on an Intel Core i5220GHz8GB computer using Matlab R2013a
Example 7 For 119896 = 1 50 119898 = 10 lowast 119896 119860 isin Q119898times119898 we applythe above three different algorithms to compute the SVDWecompare the CPU times of three algorithms qSVD in toolboxfor 119860 [24] the structure-preserving SVD for 119860 and the realSVD for 119862
In [25 26] we have already shown that structure-preserving algorithm is superior to quaternion commandldquosvdrdquo in Matlab Figure 2 shows that the CPU time of the realSVD algorithm for 119862 is the smallest Particularly when thematrix is bigger its superiority is more obvious
From the above discussion and Figure 2 we see that thealgorithm performing the SVD for 119862 is the most efficient Inthe next section we will propose a new double color imagewatermarking algorithm based on the real SVD for 1198624 The Proposed Color Image
Watermarking Algorithm
In this section we describe our color image watermarkingalgorithm in which a color watermark image is embeddedas copyright message into a color host image
Assume that an original host image 119860 is a RGB colorimage of size 119873 times 119873 where 119873 = 2119899 and watermark image119882 is also a RGB color image of size 119872 times119872 where 119872 = 2119898and119873 ge 3119872
41 Watermark Embedding
Step 1 (color watermark preprocessing) We shuffle the colorwatermark image119882 119896 times by Arnold scrambling to obtain
Journal of Applied Mathematics 5
where 119896 can be used as the secret key for watermarkrecovery
Step 2 (partition) The original RGB host color image 119860 isdivided into nonoverlapping blocks 119860 119894119895 of size 119889 times 119889 pixelswhere 119894 119895 = 1 2 119872 119889 = 119873119872
Step 3 (pixels block rearranging) The R G and B colorcomponents of 119860 119894119895 are rearranged as rectangle matrices 119862119894119895of 3119889 times 119889Step 4 (performing the SVD) Perform the SVD for119862119894119895119862119894119895 =119880119894119895Σ119894119895119881119879119894119895 where Σ119894119895 = diag(120590(119894119895)1 120590(119894119895)
119889)
Step 5 (embedding) Form Σ119894119895 = diag(120590(119894119895)1 120590(119894119895)119889minus3
120572R119894119895
120572G119894119895 120572B
119894119895) where R119894119895 G
119894119895 and B119894119895 are red green and
blue values at (119894 119895) pixel of and 120572 is the scaling factor Form119894119895 = 119880119894119895Σ119894119895119881119879119894119895 and then obtain 119894119895 from 11989411989542 Watermark Extraction The watermark is extracted asfollows
Step 1 Both the original host image 119860 and the watermarkedimage are divided into nonoverlapping blocks 119860 119894119895 and 119894119895of size 119889 times 119889 pixels respectively where 119894 119895 = 1 2 119872 119889 =119873119872
Step 2 The R G and B color components of 119860 119894119895 and 119894119895are rearranged as rectangle matrices 119862119894119895 and 119894119895 of 3119889 times 119889respectively
Step 5 The inverse Arnold scrambling is applied to 119882 =(119882119894119895) to construct the watermark
5 Experimental Results and Analysis
To verify the effectiveness of the proposed algorithm a seriesof experiments were conducted where different host imagesare adopted We carried out our experiments in MatlabR2013a environment on a laptop with Intel i5 processor ratedat 25 GHz
In the first experiment color image Pepper of size 512 times512 is taken as the host image shown in Figure 3(a) A
color image Apple of size 64 times 64 is taken as watermarkimage shown in Figure 3(b) The detailed implementationprocedure of the proposed watermarking model is depictedin Figures 3(c)ndash3(f) The scrambled watermark after 5-foldArnold transformation is shown in Figure 3(c) Figure 3(d)shows the watermarked image based on the SVD and Arnoldtransformation with watermarking scaling factor 120572 = 014in which the embedded watermark is invisible Figure 3(e)shows the extracted scrambling watermark from the water-marked image Figure 3(f) shows the recovered watermarkby the inverse Arnold scrambling which is similar to theoriginal watermark
In the second experiment we use 256 times 256 color imageLena as host image and 64 times 64 color image Apple aswatermark After 5-fold Arnold scrambling the watermark isembedded with scaling factor 028The detailed implementa-tion procedure is depicted in Figures 4(a)ndash4(f)
In the third experiment the watermark is the same asin the above two experiments We use 1024 times 1024 colorimage Butterfly as host image and scaling factor 014 Thedetailed implementation procedure is depicted in Figures5(a)ndash5(f) We observe from the figures that visually extractedwatermarks are quite good compared with the origin ones
The visual fidelity can be measured by calculating aparameter known as peak signal-to-noise ratio (PSNR) andthe structured similarity index (MSSIM) between the originalhost image 119860 and the watermarked image PSNR isexpressed in decibel (dB) and is defined as
PSNR
= 10 lg 31198732 (max119860 (119909 119910 119896))2sum119873119909=1sum119873119910=1sum3119896=1 (119860 (119909 119910 119896) minus (119909 119910 119896))2
(16)
where max119860(119909 119910 119896) represents the maximum pixel value ofa color image and here it is 255 119860(119909 119910 119896) and (119909 119910 119896) arethe pixel values location at position (119909 119910 119896) in the originalhost image and the watermarked image respectively Ingeneral the larger the PSNR value is the more invisible thewatermark is
The mean structured similarity index (MSSIM) [28] is anobjective measure originally developed to assess perceptualimage quality It is superior to PSNR for image qualitycomparison and better at reflecting the overall similarityof two pictures in terms of appearance rather than sim-ple mathematical point-to-point difference In the practicalapplication the image can be divided into blocks by usingthe sliding window and the total number of blocks is 119871 andMSSIM is defined as
MSSIM (119860 ) = 1119871119871sum119896=1
SSIM (119909119896 119910119896) (17)
and the formula used for calculating SSIM is as follows
Figure 3 512 times 512 host image with scale factor 014
where 120583119909 120583119910 1205902119909 1205902119910 and 120590119909119910 are the average variance andcovariance of119909 and119910 respectively 1198881 and 1198882 are constantsThemore the MSSIM value gets close to one the more similar tothe original host image the watermarked image is
In order to measure the quality of the embedded andextracted watermark the Normalized Correlation (NC) iscalculated between the original watermark 119882 and extractedwatermark which is defined as
NC = sum119872119909=1sum119872119910=1sum3119896=1119882(119909 119910 119896) times (119909 119910 119896)radicsum119872119909=1sum119872119910=1sum3119896=11198822 (119909 119910 119896) times radicsum119872119909=1sum119872119910=1sum3119896=1 2 (119909 119910 119896)
(19)
A higher NC reveals that the extracted watermark resemblesthe original watermarkmore closely If a method has a higherNC value it is more robust
In Table 2 we list the PSNR values MSSIM values NCvalues and the CPU times in which N ST SF and CPUstand for size of original host image scrambling times ofArnold transformation scaling factor and execution timerespectively In all the experiments we observe the goodvisual quality of watermarked images and good similarityof extracted and original watermarks with the proposedalgorithm the execution time for preprocessing embeddingand extraction procedure was only 05110 seconds for 512 times512 host image when the number of Arnold iterations is 5Other examples also show that the execution times of this
algorithm are short and the algorithmhas low computationalcost and is easily implemented
6 Conclusions
In this paper we have proposed a new double RGB colorimage watermarking algorithm based on the real SVD andArnold scrambling First the color watermark image isscrambled by Arnold transformation to obtain a meaninglessimageThen the original host image is divided into nonover-lapping pixel blocks We form a real matrix with the redgreen and blue components in each pixel block and performthe SVD of the real matrices We then replace the threesmallest singular values of each real matrix by the red green
Journal of Applied Mathematics 7
(a) Original host image (b) Original watermark (c) Scrambling watermark
and blue values of corresponding pixel of the scrambledwatermark with scaling factor to form a new watermarkedhost image With the reserve procedure we can extract thewatermark from the watermarked host image
The experimental results show that the proposed algo-rithm achieves high PSNR high MSIIM of the watermarkedimage and high NC of the extracted watermark In additionin the process of the algorithm we only need to perform realnumber algebra operations which have very low computa-tional complexity and therefore are more effective than theone using the quaternion SVD of color image which costs alarge amount of quaternion operations
The idea described in this paper can also be appliedto other kinds of methods concerning double color imagewatermarking problems
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
This research is supported by the National Natural ScienceFoundation of China under Grants 11171226 and 11301247
8 Journal of Applied Mathematics
(a) Original host image (b) Original watermark (c) Scrambling watermark
Figure 5 1024 times 1024 host image with scale factor 014
the Natural Science Foundation of Shandong under GrantZR2012FQ005 Science and Technology Project of Depart-ment of Education Shandong Province (J15LI10) and theScience Foundation of Liaocheng University under Grants31805 and 318011318
References
[1] R G Schyndel A Z Tirkel and C F Osborne ldquoA digitalwatermarkrdquo in Proceedings of the IEEE International Conferenceon Image Processing vol 2 pp 86ndash91 1994
[2] K Bailey and K Curran ldquoAn evaluation of image basedsteganography methodsrdquo Multimedia Tools and Applicationsvol 30 no 1 pp 55ndash88 2006
[3] A Cheddad J Condell K Curran and P Mc Kevitt ldquoDigitalimage steganography survey and analysis of current methodsrdquoSignal Processing vol 90 no 3 pp 727ndash752 2010
[4] G C Langelaar I Setyawan and R L Lagendijk ldquoWater-marking digital image and video datardquo IEEE Signal ProcessingMagazine vol 17 no 5 pp 20ndash46 2000
[5] F A P Petitcolas and R J Anderson ldquoEvaluation of copyrightmarking systemsrdquo in Proceedings of the 6th International Con-ference onMultimedia Computing and Systems (ICMCS rsquo99) pp574ndash579 Florence Italy June 1999
[6] CDeng XGao X Li andDTao ldquoA local tchebichefmoments-based robust image watermarkingrdquo Signal Processing vol 89no 8 pp 1531ndash1539 2009
[7] C Deng X Gao X Li and D Tao ldquoLocal histogram basedgeometric invariant image watermarkingrdquo Signal Processingvol 90 no 12 pp 3256ndash3264 2010
[8] S Rawat and B Raman ldquoA new robust watermarking schemefor color imagesrdquo in Proceedings of the IEEE 2nd Interna-tional Advance Computing Conference (IACC rsquo10) pp 206ndash209Patiala India February 2010
[9] C G Thorat and B D Jadhav ldquoA blind digital watermarktechnique for color image based on integer wavelet transformand SIFTrdquo Procedia Computer Science vol 2 pp 236ndash241 2010
[10] Q Su Y Niu G Wang S Jia and J Yue ldquoColor image blindwatermarking scheme based on QR decompositionrdquo SignalProcessing vol 94 no 1 pp 219ndash235 2014
[11] C-H Chou and T-L Wu ldquoEmbedding color watermarksin color imagesrdquo EURASIP Journal on Advances in SignalProcessing vol 2003 Article ID 548941 pp 32ndash40 2003
[12] S Rastegar FNamazi K Yaghmaie andAAliabadian ldquoHybridwatermarking algorithm based on Singular Value Decompo-sition and Radon transformrdquo AEUmdashInternational Journal ofElectronics and Communications vol 65 no 7 pp 658ndash6632011
[13] Q Su Y Niu H Zou Y Zhao and T Yao ldquoA blind double colorimage watermarking algorithm based on QR decompositionrdquoMultimedia Tools and Applications vol 72 no 1 pp 987ndash10092014
[14] N E H Golea R Seghir and R Benzid ldquoA bind RGB colorimage watermarking based on singular value decompositionrdquoin Proceedings of the ACSIEEE International Conference on
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Computer Systems andApplications (AICCSA rsquo10) pp 1ndash5Ham-mamet Tunisia May 2010
[15] Y Xing and J Tan ldquoA color watermarking scheme based onblock-SVD and Arnold transformationrdquo in Proceedings of the2nd Workshop on Digital Media and Its Application in Museumand Heritage pp 3ndash8 December 2007
[16] M Kutter F D Jordan and F Bossen ldquoDigital signature of colorimages using amplitude modulationrdquo in Storage and Retrievalfor Image and Video Databases V vol 3022 of Proceedings ofSPIE pp 518ndash526 San Jose Calif USA February 1997
[17] E Ganic and A M Eskicioglu ldquoRobust DWT-SVD domainimage watermarking embedding data in all frequenciesrdquo inProceeding of the ACM Multimedia and Security Workshop pp166ndash174 Magdeburg Germany September 2004
[18] D Vaishnavi and T S Subashini ldquoRobust and invisible imagewatermarking in RGB color space using SVDrdquo Procedia Com-puter Science vol 46 pp 1770ndash1777 2015
[19] C-Q Yin L Li A-Q Lv and L Qu ldquoColor image watermark-ing algorithm based on DWT-SVDrdquo in Proceedings of the IEEEInternational Conference on Automation and Logistics (ICALrsquo07) pp 2607ndash2611 Jinan China August 2007
[20] G H Golub and C F Van Loan Matrix Computations TheJohns Hopkins University Press Baltimore Md USA 4thedition 2013
[21] H C Andrews and C L Patterson ldquoSingular value decom-positions and digital image processingrdquo IEEE Transactions onAcoustics Speech and Signal Processing vol 24 no 1 pp 26ndash531976
[22] V J Arnold and A AvezTheMathematical Physics MonographSeriesergodic Problems of Classical Mechanics Benjamin NewYork NY USA 1968
[23] S J Sangwine ldquoFourier transforms of colour images usingquaternion or hypercomplex numbersrdquo Electronics Letters vol32 no 21 pp 1979ndash1980 1996
[24] S Sangwine and N L Bihan Quaternion toolbox for matlabhttpqtfmsourceforgenet
[25] Y Li MWei F Zhang and J Zhao ldquoA fast structure-preservingmethod for computing the singular value decomposition ofquaternion matricesrdquo Applied Mathematics and Computationvol 235 pp 157ndash167 2014
[26] Y Li M Wei F Zhang and J Zhao ldquoReal structure-preservingalgorithms of Householder based transformations for quater-nion matricesrdquo Journal of Computational and Applied Mathe-matics vol 305 pp 82ndash91 2016
[27] M Wang andW Ma ldquoA structure-preserving algorithm for thequaternion Cholesky decompositionrdquoAppliedMathematics andComputation vol 223 pp 354ndash361 2013
[28] ZWang A C Bovik H R Sheikh and E P Simoncelli ldquoImagequality assessment from error visibility to structural similarityrdquoIEEE Transactions on Image Processing vol 13 no 4 pp 600ndash612 2004
color channels and then treated independently [13] Whethersingle channel processing or multichannel synthesis itsessence is to treat greyscale image Therefore this approachcannot reflect the connection between the channels very well
(3) Algorithms Based on Quaternion A color image canbe considered as a quaternion matrix of pure imaginarynumbers In this way the color image can be processed ina holistic manner without losing color information Noticethat these kinds of algorithms require more quaternionoperations
An effective watermarking algorithm shouldmeet certainrequirements including transparency robustness adequateinformation capacity and low computational complexity
Watermarking algorithms utilizing the Singular ValueDecomposition (SVD) have become popular during the lastten years [14 17ndash19] because of the stability of the SVD SVD-based algorithms embed a watermark by modifying eitherthe singular values or the orthogonal matrices The workin [17] proposed a SVD-based watermarking scheme wherethe singular values of the watermark image were used tomodify the singular values of the subband images which wereobtained by using DWT to the host image The work in [18]proposed an algorithm to employ SVD on the blue channel ofthe host image to retrieve the singular values and embed thewatermark in these singular valuesThework in [19] proposeda SVD-based color imagewatermarking scheme inwhich thesingular values of original watermarkwere required to extractthe embedded singular values and then 119880 and 119881 orthogonalmatrices of the original watermark were utilized to recoverthe watermark The work in [14] proposed a block-SVD-based watermarking method to embed color watermark tocolor host image All these algorithms can preserve the visualquality of the image even for large singular value changes
In this paper we propose a new image watermarkingscheme based on the real SVD and Arnold scrambling toembed a color watermarking image into a color host imageBefore embedding thewatermark the color watermark image119882with size of119872times119872 is scrambled by Arnold transformationto obtain a meaningless image Then the color host image119860 with size of119873 times119873 is divided into nonoverlapping119873119872 times119873119872 pixel blocks In each (119894 119895) pixel block 119860 119894119895 we form areal matrix 119862119894119895 with the red green and blue components of119860 119894119895 and perform the SVD of 119862119894119895 We then replace the threesmallest singular values of 119862119894119895 by the red green and bluevalues of 119894119895 with scaling factor to form a new watermarkedhost image 119894119895 With the reserve procedure we can extractthe watermark from the watermarked host image
The remainder of this paper is organised as follows InSection 2 we present a brief overview regarding the SVDand image scrambling based on Arnold transformation InSection 3 we compare the SVD algorithms for color imagesIn Section 4 we propose embedding and extracting proce-dures of the new algorithm The experimental results andanalysis are presented in Section 5 Finally our conclusionsare stated in Section 6
2 Preliminaries
21The SVD The SVD [20] is one of themost powerful toolsand is widely applied to digital image processing [21]
The SVD of an119898 times 119899matrix 119860 is expressed as
119860 = 119880Σ119881119867 (1)
where both 119880119898times119898 and 119881119899times119899 are unitary matrices 119881119867 isconjugate transpose of 119881 and
Σ = [Σ1 00 0] (2)
where Σ1 = diag(1205901 1205902 120590119903) 1205901 ge 1205902 ge sdot sdot sdot ge 120590119903 gt 0 arepositive singular values of119860 and 119903 is the rank ofmatrix119860 If119860is a real matrix then both119880119898times119898 and119881119899times119899 are real orthogonalmatrices
119860 = 119880Σ119881119879 (3)
where 119881119879 is the transpose of 11988122 Arnold Scrambling Scrambling is a pretreatment stageof watermarking which transforms the meaningful imageinto another meaningless one and provides security in water-marking scheme Even if the aggressors obtain the embeddedwatermarking image they cannot extract the watermarkwithout the knowledge of the scrambling algorithm
In the proposed watermarking scheme the 2D Arnoldscrambling transformation [22] is used which shuffles thepixels positions of watermark image as follows
[] = 119860[119909119910] mod 119873[119909119910] = 119860minus1 [] mod 119873
119860 = [1 11 2]
119860minus1 = [ 2 minus1minus1 1 ]
(4)
where (119909 119910) is the old pixel coordinates of the originalimage and ( ) is the new pixel coordinates after iterativecomputation scrambling and 119873 is the size of the imageThe transformation is periodic and shows periodicity aftercompleting the period which is shown in Figure 1
3 Comparison of the SVD Algorithms forColor Images
31 Quaternion Representation of a Color Image A quater-nion 119902 isin 119876 is expressed as
119902 = 119886 + 119887119894 + 119888119895 + 119889119896 (5)where 119886 119887 119888 119889 isin 119877 and three imaginary units 119894 119895 and 119896satisfy
1198942 = 1198952 = 1198962 = 119894119895119896 = minus1 (6)When 119886 = 0 119902 is called pure quaternion
Journal of Applied Mathematics 3
(4) Scrambling times = 48(3) Scrambling times = 20
(2) Scrambling times = 1(1) Scrambling times = 0
Figure 1 Arnold scrambling
In [23] Sangwine proposed to encode the three channelcomponents of a RGB image on the three imaginary parts ofa pure quaternion that is
where 119903(119909 119910) 119892(119909 119910) and 119887(119909 119910) are the red green and bluevalues of the pixel (119909 119910) respectively Thus a color imagewith 119898 rows and 119899 columns can be represented by a pureimaginary quaternion matrix
Since then quaternion representation of a color imagehas attracted great attention Many researchers applied thequaternion matrix to study the problems of color imageprocessing due to the ability of quaternion matrices to treatthe three color channels holistically without losing colorinformation
32 Comparison of Real SVD and Quaternion SVD For aquaternion matrix 119860 isin Q119898times119899 of a color image we can per-form the quaternion SVD with different kinds of algorithmsFor example in [24] authors provided the function ldquosvdrdquoin Matlab Toolbox using quaternion arithmetics In [25 26]authors proposed a real structure-preserving algorithmbasedon the following results
For any quaternion matrix 119860 = 1198601 + 1198602119894 + 1198603119895 + 1198604119896where 1198601 1198602 1198603 1198604 isin R119898times119899 its real representation can bedefined as follows [25 26]
The properties of 119860119877 are as followsTheorem1 (see [27]) Let119860 119861 isin Q119898times119899 119862 isin Q119899times119904 and 119886 isin RThen one has the following
(1) (119860 + 119861)119877 = 119860119877 + 119861119877 (119886119860)119877 = 119886119860119877 (119860119862)119877 = 119860119877119862119877(2) (119860119867)119877 = (119860119877)119879(3) 119860 isin Q119898times119898 is a unitary matrix if and only if 119860119877 is an
orthogonal matrix
From (9) andTheorem 1 for the matrix119860119877 we only needto store the first column block of 119860119877 denoted as
119860119877119888 =[[[[[
1198601119860211986031198604
]]]]] (10)
From this notation andTheorem 1 we have the followingresult
Theorem2 Let119860 119861 isin Q119898times119899 119862 isin Q119899times119904 119902 isin Q119898 and 119886 isin RThen one has the following
(1) (119860 + 119861)119877119888 = 119860119877119888 + 119861119877119888 (119886119860)119877119888 = 119886119860119877119888 (119860119862)119877119888 = 119860119877119862119877119888 (2) (119860119867)119877119888 = [(119860119877)119879]119888(3) 119860119865 = 119860119877119888119865 1199022 = 119902119877119888 2In addition we have the following results about the SVD
of a quaternion matrix and its real representation
Theorem 3 Let 119860 isin Q119898times119899 Then the singular values of 119860119877appear in fours
Therefore if we want to compute the SVD of an 119898 times 119899quaternion matrix 119860 then we can deal with the SVD ofthe 4119898 times 4119899 real representation matrix 119860119877 In fact underthe orthogonal transformations the real representation of aquaternion matrix has the standard form stated in the nexttheorem
Theorem 4 (see [25]) Suppose that 119860 isin Q119898times119899 and 119860119877 is thereal representation of 119860 Then there exist orthogonal matrices119880 isin R4119898times4119898 and 119881 isin R4119899times4119899 such that
119880119879119860119877119881 = [[[[[
119863 0 0 00 119863 0 00 0 119863 00 0 0 119863
]]]]] (11)
where119863 isin R119898times119899 is a bidiagonal matrix
4 Journal of Applied Mathematics
Table 1 Computation amounts and assignment numbers for realSVD and quaternion SVD
FromTheorems 3 and 4 the SVD of 119860119877 can be obtainedby computing the SVD of bidiagonal matrix 119863 So weshould first get bidiagonalmatrix119863 usingHouseholder basedtransformation In [26] we listed three forms of Householderbased transformations that appeared in the literatures andproposed a new formof quaternionHouseholder based trans-formation And then we gave the real structure-preservingalgorithms for these quaternion Householder based trans-formations By comparison on computation amounts andassignment numbers we obtained the most flexible andefficient one which is described as follows
Theorem 5 Suppose that 0 = 119910 isin Q119899 is not a multiple of 1198901denote 119906 = (119910 minus 1205721198901)119910 minus 1205721198901 where
120572 = minus 1199101100381610038161003816100381611991011003816100381610038161003816
120572119872 = diag(120572|120572| 119868119899minus1) and 119867 = 120572119872(119868 minus 2119906119906119867) and then 119867maps 119910 to |120572|1198901
This kind of Householder based transformation is 1198674introduced in [26] For its specific real structure-preservingalgorithm refer to Algorithms 41 42 and 48 in [26]
After the bidiagonal matrix 119863 is calculated we canperform a sequence of iterations using Givens rotations onit and then the SVD of quaternion matrix is obtained
On the other hand we can extract the three imaginaryparts R G and B of matrix 119860 to rearrange as a new realrectangle matrix
119862 = [[[
RGB
]]] (13)
and then directly use the function ldquosvdrdquo on the real matrix119862Remark 6 Not only is the computation time related tothe number of floating points arithmetics but also it hasa lot to do with the assignment number In Table 1 welist the numbers of real flops and assignment numbers forcomputing real SVD for 119862 isin R3119898times119899 and quaternion SVDfor 119860 isin Q119898times119899 using a real structure-preserving algorithmwhere assignment numbers refer to the number of callingsubroutines or performing matrix operations In matrixoperations say 119861 = 119860119883 + 119884 we adopt the assignment119861 = 119860 lowast 119883 + 119884 to utilize vector pipelining arithmeticoperations rather than explicitly using triply nested for-end loops to speed up computations remarkably Thereforereal arithmetic numbers as well as assignment numbers areimportant measures See for example Chapter 1 of [20]
Toolbox for qSVDStructure-preservingReal SVD
0
10
20
30
40
50
60
70
CPU
tim
e (se
cond
)
10 20 30 40 500k (m = 10 lowast k)
Figure 2 Comparison of the SVD algorithms
We now provide a numerical example to compare theefficiencies of the three algorithms mentioned above Allthese computations are performed on an Intel Core i5220GHz8GB computer using Matlab R2013a
Example 7 For 119896 = 1 50 119898 = 10 lowast 119896 119860 isin Q119898times119898 we applythe above three different algorithms to compute the SVDWecompare the CPU times of three algorithms qSVD in toolboxfor 119860 [24] the structure-preserving SVD for 119860 and the realSVD for 119862
In [25 26] we have already shown that structure-preserving algorithm is superior to quaternion commandldquosvdrdquo in Matlab Figure 2 shows that the CPU time of the realSVD algorithm for 119862 is the smallest Particularly when thematrix is bigger its superiority is more obvious
From the above discussion and Figure 2 we see that thealgorithm performing the SVD for 119862 is the most efficient Inthe next section we will propose a new double color imagewatermarking algorithm based on the real SVD for 1198624 The Proposed Color Image
Watermarking Algorithm
In this section we describe our color image watermarkingalgorithm in which a color watermark image is embeddedas copyright message into a color host image
Assume that an original host image 119860 is a RGB colorimage of size 119873 times 119873 where 119873 = 2119899 and watermark image119882 is also a RGB color image of size 119872 times119872 where 119872 = 2119898and119873 ge 3119872
41 Watermark Embedding
Step 1 (color watermark preprocessing) We shuffle the colorwatermark image119882 119896 times by Arnold scrambling to obtain
Journal of Applied Mathematics 5
where 119896 can be used as the secret key for watermarkrecovery
Step 2 (partition) The original RGB host color image 119860 isdivided into nonoverlapping blocks 119860 119894119895 of size 119889 times 119889 pixelswhere 119894 119895 = 1 2 119872 119889 = 119873119872
Step 3 (pixels block rearranging) The R G and B colorcomponents of 119860 119894119895 are rearranged as rectangle matrices 119862119894119895of 3119889 times 119889Step 4 (performing the SVD) Perform the SVD for119862119894119895119862119894119895 =119880119894119895Σ119894119895119881119879119894119895 where Σ119894119895 = diag(120590(119894119895)1 120590(119894119895)
119889)
Step 5 (embedding) Form Σ119894119895 = diag(120590(119894119895)1 120590(119894119895)119889minus3
120572R119894119895
120572G119894119895 120572B
119894119895) where R119894119895 G
119894119895 and B119894119895 are red green and
blue values at (119894 119895) pixel of and 120572 is the scaling factor Form119894119895 = 119880119894119895Σ119894119895119881119879119894119895 and then obtain 119894119895 from 11989411989542 Watermark Extraction The watermark is extracted asfollows
Step 1 Both the original host image 119860 and the watermarkedimage are divided into nonoverlapping blocks 119860 119894119895 and 119894119895of size 119889 times 119889 pixels respectively where 119894 119895 = 1 2 119872 119889 =119873119872
Step 2 The R G and B color components of 119860 119894119895 and 119894119895are rearranged as rectangle matrices 119862119894119895 and 119894119895 of 3119889 times 119889respectively
Step 5 The inverse Arnold scrambling is applied to 119882 =(119882119894119895) to construct the watermark
5 Experimental Results and Analysis
To verify the effectiveness of the proposed algorithm a seriesof experiments were conducted where different host imagesare adopted We carried out our experiments in MatlabR2013a environment on a laptop with Intel i5 processor ratedat 25 GHz
In the first experiment color image Pepper of size 512 times512 is taken as the host image shown in Figure 3(a) A
color image Apple of size 64 times 64 is taken as watermarkimage shown in Figure 3(b) The detailed implementationprocedure of the proposed watermarking model is depictedin Figures 3(c)ndash3(f) The scrambled watermark after 5-foldArnold transformation is shown in Figure 3(c) Figure 3(d)shows the watermarked image based on the SVD and Arnoldtransformation with watermarking scaling factor 120572 = 014in which the embedded watermark is invisible Figure 3(e)shows the extracted scrambling watermark from the water-marked image Figure 3(f) shows the recovered watermarkby the inverse Arnold scrambling which is similar to theoriginal watermark
In the second experiment we use 256 times 256 color imageLena as host image and 64 times 64 color image Apple aswatermark After 5-fold Arnold scrambling the watermark isembedded with scaling factor 028The detailed implementa-tion procedure is depicted in Figures 4(a)ndash4(f)
In the third experiment the watermark is the same asin the above two experiments We use 1024 times 1024 colorimage Butterfly as host image and scaling factor 014 Thedetailed implementation procedure is depicted in Figures5(a)ndash5(f) We observe from the figures that visually extractedwatermarks are quite good compared with the origin ones
The visual fidelity can be measured by calculating aparameter known as peak signal-to-noise ratio (PSNR) andthe structured similarity index (MSSIM) between the originalhost image 119860 and the watermarked image PSNR isexpressed in decibel (dB) and is defined as
PSNR
= 10 lg 31198732 (max119860 (119909 119910 119896))2sum119873119909=1sum119873119910=1sum3119896=1 (119860 (119909 119910 119896) minus (119909 119910 119896))2
(16)
where max119860(119909 119910 119896) represents the maximum pixel value ofa color image and here it is 255 119860(119909 119910 119896) and (119909 119910 119896) arethe pixel values location at position (119909 119910 119896) in the originalhost image and the watermarked image respectively Ingeneral the larger the PSNR value is the more invisible thewatermark is
The mean structured similarity index (MSSIM) [28] is anobjective measure originally developed to assess perceptualimage quality It is superior to PSNR for image qualitycomparison and better at reflecting the overall similarityof two pictures in terms of appearance rather than sim-ple mathematical point-to-point difference In the practicalapplication the image can be divided into blocks by usingthe sliding window and the total number of blocks is 119871 andMSSIM is defined as
MSSIM (119860 ) = 1119871119871sum119896=1
SSIM (119909119896 119910119896) (17)
and the formula used for calculating SSIM is as follows
Figure 3 512 times 512 host image with scale factor 014
where 120583119909 120583119910 1205902119909 1205902119910 and 120590119909119910 are the average variance andcovariance of119909 and119910 respectively 1198881 and 1198882 are constantsThemore the MSSIM value gets close to one the more similar tothe original host image the watermarked image is
In order to measure the quality of the embedded andextracted watermark the Normalized Correlation (NC) iscalculated between the original watermark 119882 and extractedwatermark which is defined as
NC = sum119872119909=1sum119872119910=1sum3119896=1119882(119909 119910 119896) times (119909 119910 119896)radicsum119872119909=1sum119872119910=1sum3119896=11198822 (119909 119910 119896) times radicsum119872119909=1sum119872119910=1sum3119896=1 2 (119909 119910 119896)
(19)
A higher NC reveals that the extracted watermark resemblesthe original watermarkmore closely If a method has a higherNC value it is more robust
In Table 2 we list the PSNR values MSSIM values NCvalues and the CPU times in which N ST SF and CPUstand for size of original host image scrambling times ofArnold transformation scaling factor and execution timerespectively In all the experiments we observe the goodvisual quality of watermarked images and good similarityof extracted and original watermarks with the proposedalgorithm the execution time for preprocessing embeddingand extraction procedure was only 05110 seconds for 512 times512 host image when the number of Arnold iterations is 5Other examples also show that the execution times of this
algorithm are short and the algorithmhas low computationalcost and is easily implemented
6 Conclusions
In this paper we have proposed a new double RGB colorimage watermarking algorithm based on the real SVD andArnold scrambling First the color watermark image isscrambled by Arnold transformation to obtain a meaninglessimageThen the original host image is divided into nonover-lapping pixel blocks We form a real matrix with the redgreen and blue components in each pixel block and performthe SVD of the real matrices We then replace the threesmallest singular values of each real matrix by the red green
Journal of Applied Mathematics 7
(a) Original host image (b) Original watermark (c) Scrambling watermark
and blue values of corresponding pixel of the scrambledwatermark with scaling factor to form a new watermarkedhost image With the reserve procedure we can extract thewatermark from the watermarked host image
The experimental results show that the proposed algo-rithm achieves high PSNR high MSIIM of the watermarkedimage and high NC of the extracted watermark In additionin the process of the algorithm we only need to perform realnumber algebra operations which have very low computa-tional complexity and therefore are more effective than theone using the quaternion SVD of color image which costs alarge amount of quaternion operations
The idea described in this paper can also be appliedto other kinds of methods concerning double color imagewatermarking problems
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
This research is supported by the National Natural ScienceFoundation of China under Grants 11171226 and 11301247
8 Journal of Applied Mathematics
(a) Original host image (b) Original watermark (c) Scrambling watermark
Figure 5 1024 times 1024 host image with scale factor 014
the Natural Science Foundation of Shandong under GrantZR2012FQ005 Science and Technology Project of Depart-ment of Education Shandong Province (J15LI10) and theScience Foundation of Liaocheng University under Grants31805 and 318011318
References
[1] R G Schyndel A Z Tirkel and C F Osborne ldquoA digitalwatermarkrdquo in Proceedings of the IEEE International Conferenceon Image Processing vol 2 pp 86ndash91 1994
[2] K Bailey and K Curran ldquoAn evaluation of image basedsteganography methodsrdquo Multimedia Tools and Applicationsvol 30 no 1 pp 55ndash88 2006
[3] A Cheddad J Condell K Curran and P Mc Kevitt ldquoDigitalimage steganography survey and analysis of current methodsrdquoSignal Processing vol 90 no 3 pp 727ndash752 2010
[4] G C Langelaar I Setyawan and R L Lagendijk ldquoWater-marking digital image and video datardquo IEEE Signal ProcessingMagazine vol 17 no 5 pp 20ndash46 2000
[5] F A P Petitcolas and R J Anderson ldquoEvaluation of copyrightmarking systemsrdquo in Proceedings of the 6th International Con-ference onMultimedia Computing and Systems (ICMCS rsquo99) pp574ndash579 Florence Italy June 1999
[6] CDeng XGao X Li andDTao ldquoA local tchebichefmoments-based robust image watermarkingrdquo Signal Processing vol 89no 8 pp 1531ndash1539 2009
[7] C Deng X Gao X Li and D Tao ldquoLocal histogram basedgeometric invariant image watermarkingrdquo Signal Processingvol 90 no 12 pp 3256ndash3264 2010
[8] S Rawat and B Raman ldquoA new robust watermarking schemefor color imagesrdquo in Proceedings of the IEEE 2nd Interna-tional Advance Computing Conference (IACC rsquo10) pp 206ndash209Patiala India February 2010
[9] C G Thorat and B D Jadhav ldquoA blind digital watermarktechnique for color image based on integer wavelet transformand SIFTrdquo Procedia Computer Science vol 2 pp 236ndash241 2010
[10] Q Su Y Niu G Wang S Jia and J Yue ldquoColor image blindwatermarking scheme based on QR decompositionrdquo SignalProcessing vol 94 no 1 pp 219ndash235 2014
[11] C-H Chou and T-L Wu ldquoEmbedding color watermarksin color imagesrdquo EURASIP Journal on Advances in SignalProcessing vol 2003 Article ID 548941 pp 32ndash40 2003
[12] S Rastegar FNamazi K Yaghmaie andAAliabadian ldquoHybridwatermarking algorithm based on Singular Value Decompo-sition and Radon transformrdquo AEUmdashInternational Journal ofElectronics and Communications vol 65 no 7 pp 658ndash6632011
[13] Q Su Y Niu H Zou Y Zhao and T Yao ldquoA blind double colorimage watermarking algorithm based on QR decompositionrdquoMultimedia Tools and Applications vol 72 no 1 pp 987ndash10092014
[14] N E H Golea R Seghir and R Benzid ldquoA bind RGB colorimage watermarking based on singular value decompositionrdquoin Proceedings of the ACSIEEE International Conference on
Journal of Applied Mathematics 9
Computer Systems andApplications (AICCSA rsquo10) pp 1ndash5Ham-mamet Tunisia May 2010
[15] Y Xing and J Tan ldquoA color watermarking scheme based onblock-SVD and Arnold transformationrdquo in Proceedings of the2nd Workshop on Digital Media and Its Application in Museumand Heritage pp 3ndash8 December 2007
[16] M Kutter F D Jordan and F Bossen ldquoDigital signature of colorimages using amplitude modulationrdquo in Storage and Retrievalfor Image and Video Databases V vol 3022 of Proceedings ofSPIE pp 518ndash526 San Jose Calif USA February 1997
[17] E Ganic and A M Eskicioglu ldquoRobust DWT-SVD domainimage watermarking embedding data in all frequenciesrdquo inProceeding of the ACM Multimedia and Security Workshop pp166ndash174 Magdeburg Germany September 2004
[18] D Vaishnavi and T S Subashini ldquoRobust and invisible imagewatermarking in RGB color space using SVDrdquo Procedia Com-puter Science vol 46 pp 1770ndash1777 2015
[19] C-Q Yin L Li A-Q Lv and L Qu ldquoColor image watermark-ing algorithm based on DWT-SVDrdquo in Proceedings of the IEEEInternational Conference on Automation and Logistics (ICALrsquo07) pp 2607ndash2611 Jinan China August 2007
[20] G H Golub and C F Van Loan Matrix Computations TheJohns Hopkins University Press Baltimore Md USA 4thedition 2013
[21] H C Andrews and C L Patterson ldquoSingular value decom-positions and digital image processingrdquo IEEE Transactions onAcoustics Speech and Signal Processing vol 24 no 1 pp 26ndash531976
[22] V J Arnold and A AvezTheMathematical Physics MonographSeriesergodic Problems of Classical Mechanics Benjamin NewYork NY USA 1968
[23] S J Sangwine ldquoFourier transforms of colour images usingquaternion or hypercomplex numbersrdquo Electronics Letters vol32 no 21 pp 1979ndash1980 1996
[24] S Sangwine and N L Bihan Quaternion toolbox for matlabhttpqtfmsourceforgenet
[25] Y Li MWei F Zhang and J Zhao ldquoA fast structure-preservingmethod for computing the singular value decomposition ofquaternion matricesrdquo Applied Mathematics and Computationvol 235 pp 157ndash167 2014
[26] Y Li M Wei F Zhang and J Zhao ldquoReal structure-preservingalgorithms of Householder based transformations for quater-nion matricesrdquo Journal of Computational and Applied Mathe-matics vol 305 pp 82ndash91 2016
[27] M Wang andW Ma ldquoA structure-preserving algorithm for thequaternion Cholesky decompositionrdquoAppliedMathematics andComputation vol 223 pp 354ndash361 2013
[28] ZWang A C Bovik H R Sheikh and E P Simoncelli ldquoImagequality assessment from error visibility to structural similarityrdquoIEEE Transactions on Image Processing vol 13 no 4 pp 600ndash612 2004
where 119903(119909 119910) 119892(119909 119910) and 119887(119909 119910) are the red green and bluevalues of the pixel (119909 119910) respectively Thus a color imagewith 119898 rows and 119899 columns can be represented by a pureimaginary quaternion matrix
Since then quaternion representation of a color imagehas attracted great attention Many researchers applied thequaternion matrix to study the problems of color imageprocessing due to the ability of quaternion matrices to treatthe three color channels holistically without losing colorinformation
32 Comparison of Real SVD and Quaternion SVD For aquaternion matrix 119860 isin Q119898times119899 of a color image we can per-form the quaternion SVD with different kinds of algorithmsFor example in [24] authors provided the function ldquosvdrdquoin Matlab Toolbox using quaternion arithmetics In [25 26]authors proposed a real structure-preserving algorithmbasedon the following results
For any quaternion matrix 119860 = 1198601 + 1198602119894 + 1198603119895 + 1198604119896where 1198601 1198602 1198603 1198604 isin R119898times119899 its real representation can bedefined as follows [25 26]
The properties of 119860119877 are as followsTheorem1 (see [27]) Let119860 119861 isin Q119898times119899 119862 isin Q119899times119904 and 119886 isin RThen one has the following
(1) (119860 + 119861)119877 = 119860119877 + 119861119877 (119886119860)119877 = 119886119860119877 (119860119862)119877 = 119860119877119862119877(2) (119860119867)119877 = (119860119877)119879(3) 119860 isin Q119898times119898 is a unitary matrix if and only if 119860119877 is an
orthogonal matrix
From (9) andTheorem 1 for the matrix119860119877 we only needto store the first column block of 119860119877 denoted as
119860119877119888 =[[[[[
1198601119860211986031198604
]]]]] (10)
From this notation andTheorem 1 we have the followingresult
Theorem2 Let119860 119861 isin Q119898times119899 119862 isin Q119899times119904 119902 isin Q119898 and 119886 isin RThen one has the following
(1) (119860 + 119861)119877119888 = 119860119877119888 + 119861119877119888 (119886119860)119877119888 = 119886119860119877119888 (119860119862)119877119888 = 119860119877119862119877119888 (2) (119860119867)119877119888 = [(119860119877)119879]119888(3) 119860119865 = 119860119877119888119865 1199022 = 119902119877119888 2In addition we have the following results about the SVD
of a quaternion matrix and its real representation
Theorem 3 Let 119860 isin Q119898times119899 Then the singular values of 119860119877appear in fours
Therefore if we want to compute the SVD of an 119898 times 119899quaternion matrix 119860 then we can deal with the SVD ofthe 4119898 times 4119899 real representation matrix 119860119877 In fact underthe orthogonal transformations the real representation of aquaternion matrix has the standard form stated in the nexttheorem
Theorem 4 (see [25]) Suppose that 119860 isin Q119898times119899 and 119860119877 is thereal representation of 119860 Then there exist orthogonal matrices119880 isin R4119898times4119898 and 119881 isin R4119899times4119899 such that
119880119879119860119877119881 = [[[[[
119863 0 0 00 119863 0 00 0 119863 00 0 0 119863
]]]]] (11)
where119863 isin R119898times119899 is a bidiagonal matrix
4 Journal of Applied Mathematics
Table 1 Computation amounts and assignment numbers for realSVD and quaternion SVD
FromTheorems 3 and 4 the SVD of 119860119877 can be obtainedby computing the SVD of bidiagonal matrix 119863 So weshould first get bidiagonalmatrix119863 usingHouseholder basedtransformation In [26] we listed three forms of Householderbased transformations that appeared in the literatures andproposed a new formof quaternionHouseholder based trans-formation And then we gave the real structure-preservingalgorithms for these quaternion Householder based trans-formations By comparison on computation amounts andassignment numbers we obtained the most flexible andefficient one which is described as follows
Theorem 5 Suppose that 0 = 119910 isin Q119899 is not a multiple of 1198901denote 119906 = (119910 minus 1205721198901)119910 minus 1205721198901 where
120572 = minus 1199101100381610038161003816100381611991011003816100381610038161003816
120572119872 = diag(120572|120572| 119868119899minus1) and 119867 = 120572119872(119868 minus 2119906119906119867) and then 119867maps 119910 to |120572|1198901
This kind of Householder based transformation is 1198674introduced in [26] For its specific real structure-preservingalgorithm refer to Algorithms 41 42 and 48 in [26]
After the bidiagonal matrix 119863 is calculated we canperform a sequence of iterations using Givens rotations onit and then the SVD of quaternion matrix is obtained
On the other hand we can extract the three imaginaryparts R G and B of matrix 119860 to rearrange as a new realrectangle matrix
119862 = [[[
RGB
]]] (13)
and then directly use the function ldquosvdrdquo on the real matrix119862Remark 6 Not only is the computation time related tothe number of floating points arithmetics but also it hasa lot to do with the assignment number In Table 1 welist the numbers of real flops and assignment numbers forcomputing real SVD for 119862 isin R3119898times119899 and quaternion SVDfor 119860 isin Q119898times119899 using a real structure-preserving algorithmwhere assignment numbers refer to the number of callingsubroutines or performing matrix operations In matrixoperations say 119861 = 119860119883 + 119884 we adopt the assignment119861 = 119860 lowast 119883 + 119884 to utilize vector pipelining arithmeticoperations rather than explicitly using triply nested for-end loops to speed up computations remarkably Thereforereal arithmetic numbers as well as assignment numbers areimportant measures See for example Chapter 1 of [20]
Toolbox for qSVDStructure-preservingReal SVD
0
10
20
30
40
50
60
70
CPU
tim
e (se
cond
)
10 20 30 40 500k (m = 10 lowast k)
Figure 2 Comparison of the SVD algorithms
We now provide a numerical example to compare theefficiencies of the three algorithms mentioned above Allthese computations are performed on an Intel Core i5220GHz8GB computer using Matlab R2013a
Example 7 For 119896 = 1 50 119898 = 10 lowast 119896 119860 isin Q119898times119898 we applythe above three different algorithms to compute the SVDWecompare the CPU times of three algorithms qSVD in toolboxfor 119860 [24] the structure-preserving SVD for 119860 and the realSVD for 119862
In [25 26] we have already shown that structure-preserving algorithm is superior to quaternion commandldquosvdrdquo in Matlab Figure 2 shows that the CPU time of the realSVD algorithm for 119862 is the smallest Particularly when thematrix is bigger its superiority is more obvious
From the above discussion and Figure 2 we see that thealgorithm performing the SVD for 119862 is the most efficient Inthe next section we will propose a new double color imagewatermarking algorithm based on the real SVD for 1198624 The Proposed Color Image
Watermarking Algorithm
In this section we describe our color image watermarkingalgorithm in which a color watermark image is embeddedas copyright message into a color host image
Assume that an original host image 119860 is a RGB colorimage of size 119873 times 119873 where 119873 = 2119899 and watermark image119882 is also a RGB color image of size 119872 times119872 where 119872 = 2119898and119873 ge 3119872
41 Watermark Embedding
Step 1 (color watermark preprocessing) We shuffle the colorwatermark image119882 119896 times by Arnold scrambling to obtain
Journal of Applied Mathematics 5
where 119896 can be used as the secret key for watermarkrecovery
Step 2 (partition) The original RGB host color image 119860 isdivided into nonoverlapping blocks 119860 119894119895 of size 119889 times 119889 pixelswhere 119894 119895 = 1 2 119872 119889 = 119873119872
Step 3 (pixels block rearranging) The R G and B colorcomponents of 119860 119894119895 are rearranged as rectangle matrices 119862119894119895of 3119889 times 119889Step 4 (performing the SVD) Perform the SVD for119862119894119895119862119894119895 =119880119894119895Σ119894119895119881119879119894119895 where Σ119894119895 = diag(120590(119894119895)1 120590(119894119895)
119889)
Step 5 (embedding) Form Σ119894119895 = diag(120590(119894119895)1 120590(119894119895)119889minus3
120572R119894119895
120572G119894119895 120572B
119894119895) where R119894119895 G
119894119895 and B119894119895 are red green and
blue values at (119894 119895) pixel of and 120572 is the scaling factor Form119894119895 = 119880119894119895Σ119894119895119881119879119894119895 and then obtain 119894119895 from 11989411989542 Watermark Extraction The watermark is extracted asfollows
Step 1 Both the original host image 119860 and the watermarkedimage are divided into nonoverlapping blocks 119860 119894119895 and 119894119895of size 119889 times 119889 pixels respectively where 119894 119895 = 1 2 119872 119889 =119873119872
Step 2 The R G and B color components of 119860 119894119895 and 119894119895are rearranged as rectangle matrices 119862119894119895 and 119894119895 of 3119889 times 119889respectively
Step 5 The inverse Arnold scrambling is applied to 119882 =(119882119894119895) to construct the watermark
5 Experimental Results and Analysis
To verify the effectiveness of the proposed algorithm a seriesof experiments were conducted where different host imagesare adopted We carried out our experiments in MatlabR2013a environment on a laptop with Intel i5 processor ratedat 25 GHz
In the first experiment color image Pepper of size 512 times512 is taken as the host image shown in Figure 3(a) A
color image Apple of size 64 times 64 is taken as watermarkimage shown in Figure 3(b) The detailed implementationprocedure of the proposed watermarking model is depictedin Figures 3(c)ndash3(f) The scrambled watermark after 5-foldArnold transformation is shown in Figure 3(c) Figure 3(d)shows the watermarked image based on the SVD and Arnoldtransformation with watermarking scaling factor 120572 = 014in which the embedded watermark is invisible Figure 3(e)shows the extracted scrambling watermark from the water-marked image Figure 3(f) shows the recovered watermarkby the inverse Arnold scrambling which is similar to theoriginal watermark
In the second experiment we use 256 times 256 color imageLena as host image and 64 times 64 color image Apple aswatermark After 5-fold Arnold scrambling the watermark isembedded with scaling factor 028The detailed implementa-tion procedure is depicted in Figures 4(a)ndash4(f)
In the third experiment the watermark is the same asin the above two experiments We use 1024 times 1024 colorimage Butterfly as host image and scaling factor 014 Thedetailed implementation procedure is depicted in Figures5(a)ndash5(f) We observe from the figures that visually extractedwatermarks are quite good compared with the origin ones
The visual fidelity can be measured by calculating aparameter known as peak signal-to-noise ratio (PSNR) andthe structured similarity index (MSSIM) between the originalhost image 119860 and the watermarked image PSNR isexpressed in decibel (dB) and is defined as
PSNR
= 10 lg 31198732 (max119860 (119909 119910 119896))2sum119873119909=1sum119873119910=1sum3119896=1 (119860 (119909 119910 119896) minus (119909 119910 119896))2
(16)
where max119860(119909 119910 119896) represents the maximum pixel value ofa color image and here it is 255 119860(119909 119910 119896) and (119909 119910 119896) arethe pixel values location at position (119909 119910 119896) in the originalhost image and the watermarked image respectively Ingeneral the larger the PSNR value is the more invisible thewatermark is
The mean structured similarity index (MSSIM) [28] is anobjective measure originally developed to assess perceptualimage quality It is superior to PSNR for image qualitycomparison and better at reflecting the overall similarityof two pictures in terms of appearance rather than sim-ple mathematical point-to-point difference In the practicalapplication the image can be divided into blocks by usingthe sliding window and the total number of blocks is 119871 andMSSIM is defined as
MSSIM (119860 ) = 1119871119871sum119896=1
SSIM (119909119896 119910119896) (17)
and the formula used for calculating SSIM is as follows
Figure 3 512 times 512 host image with scale factor 014
where 120583119909 120583119910 1205902119909 1205902119910 and 120590119909119910 are the average variance andcovariance of119909 and119910 respectively 1198881 and 1198882 are constantsThemore the MSSIM value gets close to one the more similar tothe original host image the watermarked image is
In order to measure the quality of the embedded andextracted watermark the Normalized Correlation (NC) iscalculated between the original watermark 119882 and extractedwatermark which is defined as
NC = sum119872119909=1sum119872119910=1sum3119896=1119882(119909 119910 119896) times (119909 119910 119896)radicsum119872119909=1sum119872119910=1sum3119896=11198822 (119909 119910 119896) times radicsum119872119909=1sum119872119910=1sum3119896=1 2 (119909 119910 119896)
(19)
A higher NC reveals that the extracted watermark resemblesthe original watermarkmore closely If a method has a higherNC value it is more robust
In Table 2 we list the PSNR values MSSIM values NCvalues and the CPU times in which N ST SF and CPUstand for size of original host image scrambling times ofArnold transformation scaling factor and execution timerespectively In all the experiments we observe the goodvisual quality of watermarked images and good similarityof extracted and original watermarks with the proposedalgorithm the execution time for preprocessing embeddingand extraction procedure was only 05110 seconds for 512 times512 host image when the number of Arnold iterations is 5Other examples also show that the execution times of this
algorithm are short and the algorithmhas low computationalcost and is easily implemented
6 Conclusions
In this paper we have proposed a new double RGB colorimage watermarking algorithm based on the real SVD andArnold scrambling First the color watermark image isscrambled by Arnold transformation to obtain a meaninglessimageThen the original host image is divided into nonover-lapping pixel blocks We form a real matrix with the redgreen and blue components in each pixel block and performthe SVD of the real matrices We then replace the threesmallest singular values of each real matrix by the red green
Journal of Applied Mathematics 7
(a) Original host image (b) Original watermark (c) Scrambling watermark
and blue values of corresponding pixel of the scrambledwatermark with scaling factor to form a new watermarkedhost image With the reserve procedure we can extract thewatermark from the watermarked host image
The experimental results show that the proposed algo-rithm achieves high PSNR high MSIIM of the watermarkedimage and high NC of the extracted watermark In additionin the process of the algorithm we only need to perform realnumber algebra operations which have very low computa-tional complexity and therefore are more effective than theone using the quaternion SVD of color image which costs alarge amount of quaternion operations
The idea described in this paper can also be appliedto other kinds of methods concerning double color imagewatermarking problems
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
This research is supported by the National Natural ScienceFoundation of China under Grants 11171226 and 11301247
8 Journal of Applied Mathematics
(a) Original host image (b) Original watermark (c) Scrambling watermark
Figure 5 1024 times 1024 host image with scale factor 014
the Natural Science Foundation of Shandong under GrantZR2012FQ005 Science and Technology Project of Depart-ment of Education Shandong Province (J15LI10) and theScience Foundation of Liaocheng University under Grants31805 and 318011318
References
[1] R G Schyndel A Z Tirkel and C F Osborne ldquoA digitalwatermarkrdquo in Proceedings of the IEEE International Conferenceon Image Processing vol 2 pp 86ndash91 1994
[2] K Bailey and K Curran ldquoAn evaluation of image basedsteganography methodsrdquo Multimedia Tools and Applicationsvol 30 no 1 pp 55ndash88 2006
[3] A Cheddad J Condell K Curran and P Mc Kevitt ldquoDigitalimage steganography survey and analysis of current methodsrdquoSignal Processing vol 90 no 3 pp 727ndash752 2010
[4] G C Langelaar I Setyawan and R L Lagendijk ldquoWater-marking digital image and video datardquo IEEE Signal ProcessingMagazine vol 17 no 5 pp 20ndash46 2000
[5] F A P Petitcolas and R J Anderson ldquoEvaluation of copyrightmarking systemsrdquo in Proceedings of the 6th International Con-ference onMultimedia Computing and Systems (ICMCS rsquo99) pp574ndash579 Florence Italy June 1999
[6] CDeng XGao X Li andDTao ldquoA local tchebichefmoments-based robust image watermarkingrdquo Signal Processing vol 89no 8 pp 1531ndash1539 2009
[7] C Deng X Gao X Li and D Tao ldquoLocal histogram basedgeometric invariant image watermarkingrdquo Signal Processingvol 90 no 12 pp 3256ndash3264 2010
[8] S Rawat and B Raman ldquoA new robust watermarking schemefor color imagesrdquo in Proceedings of the IEEE 2nd Interna-tional Advance Computing Conference (IACC rsquo10) pp 206ndash209Patiala India February 2010
[9] C G Thorat and B D Jadhav ldquoA blind digital watermarktechnique for color image based on integer wavelet transformand SIFTrdquo Procedia Computer Science vol 2 pp 236ndash241 2010
[10] Q Su Y Niu G Wang S Jia and J Yue ldquoColor image blindwatermarking scheme based on QR decompositionrdquo SignalProcessing vol 94 no 1 pp 219ndash235 2014
[11] C-H Chou and T-L Wu ldquoEmbedding color watermarksin color imagesrdquo EURASIP Journal on Advances in SignalProcessing vol 2003 Article ID 548941 pp 32ndash40 2003
[12] S Rastegar FNamazi K Yaghmaie andAAliabadian ldquoHybridwatermarking algorithm based on Singular Value Decompo-sition and Radon transformrdquo AEUmdashInternational Journal ofElectronics and Communications vol 65 no 7 pp 658ndash6632011
[13] Q Su Y Niu H Zou Y Zhao and T Yao ldquoA blind double colorimage watermarking algorithm based on QR decompositionrdquoMultimedia Tools and Applications vol 72 no 1 pp 987ndash10092014
[14] N E H Golea R Seghir and R Benzid ldquoA bind RGB colorimage watermarking based on singular value decompositionrdquoin Proceedings of the ACSIEEE International Conference on
Journal of Applied Mathematics 9
Computer Systems andApplications (AICCSA rsquo10) pp 1ndash5Ham-mamet Tunisia May 2010
[15] Y Xing and J Tan ldquoA color watermarking scheme based onblock-SVD and Arnold transformationrdquo in Proceedings of the2nd Workshop on Digital Media and Its Application in Museumand Heritage pp 3ndash8 December 2007
[16] M Kutter F D Jordan and F Bossen ldquoDigital signature of colorimages using amplitude modulationrdquo in Storage and Retrievalfor Image and Video Databases V vol 3022 of Proceedings ofSPIE pp 518ndash526 San Jose Calif USA February 1997
[17] E Ganic and A M Eskicioglu ldquoRobust DWT-SVD domainimage watermarking embedding data in all frequenciesrdquo inProceeding of the ACM Multimedia and Security Workshop pp166ndash174 Magdeburg Germany September 2004
[18] D Vaishnavi and T S Subashini ldquoRobust and invisible imagewatermarking in RGB color space using SVDrdquo Procedia Com-puter Science vol 46 pp 1770ndash1777 2015
[19] C-Q Yin L Li A-Q Lv and L Qu ldquoColor image watermark-ing algorithm based on DWT-SVDrdquo in Proceedings of the IEEEInternational Conference on Automation and Logistics (ICALrsquo07) pp 2607ndash2611 Jinan China August 2007
[20] G H Golub and C F Van Loan Matrix Computations TheJohns Hopkins University Press Baltimore Md USA 4thedition 2013
[21] H C Andrews and C L Patterson ldquoSingular value decom-positions and digital image processingrdquo IEEE Transactions onAcoustics Speech and Signal Processing vol 24 no 1 pp 26ndash531976
[22] V J Arnold and A AvezTheMathematical Physics MonographSeriesergodic Problems of Classical Mechanics Benjamin NewYork NY USA 1968
[23] S J Sangwine ldquoFourier transforms of colour images usingquaternion or hypercomplex numbersrdquo Electronics Letters vol32 no 21 pp 1979ndash1980 1996
[24] S Sangwine and N L Bihan Quaternion toolbox for matlabhttpqtfmsourceforgenet
[25] Y Li MWei F Zhang and J Zhao ldquoA fast structure-preservingmethod for computing the singular value decomposition ofquaternion matricesrdquo Applied Mathematics and Computationvol 235 pp 157ndash167 2014
[26] Y Li M Wei F Zhang and J Zhao ldquoReal structure-preservingalgorithms of Householder based transformations for quater-nion matricesrdquo Journal of Computational and Applied Mathe-matics vol 305 pp 82ndash91 2016
[27] M Wang andW Ma ldquoA structure-preserving algorithm for thequaternion Cholesky decompositionrdquoAppliedMathematics andComputation vol 223 pp 354ndash361 2013
[28] ZWang A C Bovik H R Sheikh and E P Simoncelli ldquoImagequality assessment from error visibility to structural similarityrdquoIEEE Transactions on Image Processing vol 13 no 4 pp 600ndash612 2004
FromTheorems 3 and 4 the SVD of 119860119877 can be obtainedby computing the SVD of bidiagonal matrix 119863 So weshould first get bidiagonalmatrix119863 usingHouseholder basedtransformation In [26] we listed three forms of Householderbased transformations that appeared in the literatures andproposed a new formof quaternionHouseholder based trans-formation And then we gave the real structure-preservingalgorithms for these quaternion Householder based trans-formations By comparison on computation amounts andassignment numbers we obtained the most flexible andefficient one which is described as follows
Theorem 5 Suppose that 0 = 119910 isin Q119899 is not a multiple of 1198901denote 119906 = (119910 minus 1205721198901)119910 minus 1205721198901 where
120572 = minus 1199101100381610038161003816100381611991011003816100381610038161003816
120572119872 = diag(120572|120572| 119868119899minus1) and 119867 = 120572119872(119868 minus 2119906119906119867) and then 119867maps 119910 to |120572|1198901
This kind of Householder based transformation is 1198674introduced in [26] For its specific real structure-preservingalgorithm refer to Algorithms 41 42 and 48 in [26]
After the bidiagonal matrix 119863 is calculated we canperform a sequence of iterations using Givens rotations onit and then the SVD of quaternion matrix is obtained
On the other hand we can extract the three imaginaryparts R G and B of matrix 119860 to rearrange as a new realrectangle matrix
119862 = [[[
RGB
]]] (13)
and then directly use the function ldquosvdrdquo on the real matrix119862Remark 6 Not only is the computation time related tothe number of floating points arithmetics but also it hasa lot to do with the assignment number In Table 1 welist the numbers of real flops and assignment numbers forcomputing real SVD for 119862 isin R3119898times119899 and quaternion SVDfor 119860 isin Q119898times119899 using a real structure-preserving algorithmwhere assignment numbers refer to the number of callingsubroutines or performing matrix operations In matrixoperations say 119861 = 119860119883 + 119884 we adopt the assignment119861 = 119860 lowast 119883 + 119884 to utilize vector pipelining arithmeticoperations rather than explicitly using triply nested for-end loops to speed up computations remarkably Thereforereal arithmetic numbers as well as assignment numbers areimportant measures See for example Chapter 1 of [20]
Toolbox for qSVDStructure-preservingReal SVD
0
10
20
30
40
50
60
70
CPU
tim
e (se
cond
)
10 20 30 40 500k (m = 10 lowast k)
Figure 2 Comparison of the SVD algorithms
We now provide a numerical example to compare theefficiencies of the three algorithms mentioned above Allthese computations are performed on an Intel Core i5220GHz8GB computer using Matlab R2013a
Example 7 For 119896 = 1 50 119898 = 10 lowast 119896 119860 isin Q119898times119898 we applythe above three different algorithms to compute the SVDWecompare the CPU times of three algorithms qSVD in toolboxfor 119860 [24] the structure-preserving SVD for 119860 and the realSVD for 119862
In [25 26] we have already shown that structure-preserving algorithm is superior to quaternion commandldquosvdrdquo in Matlab Figure 2 shows that the CPU time of the realSVD algorithm for 119862 is the smallest Particularly when thematrix is bigger its superiority is more obvious
From the above discussion and Figure 2 we see that thealgorithm performing the SVD for 119862 is the most efficient Inthe next section we will propose a new double color imagewatermarking algorithm based on the real SVD for 1198624 The Proposed Color Image
Watermarking Algorithm
In this section we describe our color image watermarkingalgorithm in which a color watermark image is embeddedas copyright message into a color host image
Assume that an original host image 119860 is a RGB colorimage of size 119873 times 119873 where 119873 = 2119899 and watermark image119882 is also a RGB color image of size 119872 times119872 where 119872 = 2119898and119873 ge 3119872
41 Watermark Embedding
Step 1 (color watermark preprocessing) We shuffle the colorwatermark image119882 119896 times by Arnold scrambling to obtain
Journal of Applied Mathematics 5
where 119896 can be used as the secret key for watermarkrecovery
Step 2 (partition) The original RGB host color image 119860 isdivided into nonoverlapping blocks 119860 119894119895 of size 119889 times 119889 pixelswhere 119894 119895 = 1 2 119872 119889 = 119873119872
Step 3 (pixels block rearranging) The R G and B colorcomponents of 119860 119894119895 are rearranged as rectangle matrices 119862119894119895of 3119889 times 119889Step 4 (performing the SVD) Perform the SVD for119862119894119895119862119894119895 =119880119894119895Σ119894119895119881119879119894119895 where Σ119894119895 = diag(120590(119894119895)1 120590(119894119895)
119889)
Step 5 (embedding) Form Σ119894119895 = diag(120590(119894119895)1 120590(119894119895)119889minus3
120572R119894119895
120572G119894119895 120572B
119894119895) where R119894119895 G
119894119895 and B119894119895 are red green and
blue values at (119894 119895) pixel of and 120572 is the scaling factor Form119894119895 = 119880119894119895Σ119894119895119881119879119894119895 and then obtain 119894119895 from 11989411989542 Watermark Extraction The watermark is extracted asfollows
Step 1 Both the original host image 119860 and the watermarkedimage are divided into nonoverlapping blocks 119860 119894119895 and 119894119895of size 119889 times 119889 pixels respectively where 119894 119895 = 1 2 119872 119889 =119873119872
Step 2 The R G and B color components of 119860 119894119895 and 119894119895are rearranged as rectangle matrices 119862119894119895 and 119894119895 of 3119889 times 119889respectively
Step 5 The inverse Arnold scrambling is applied to 119882 =(119882119894119895) to construct the watermark
5 Experimental Results and Analysis
To verify the effectiveness of the proposed algorithm a seriesof experiments were conducted where different host imagesare adopted We carried out our experiments in MatlabR2013a environment on a laptop with Intel i5 processor ratedat 25 GHz
In the first experiment color image Pepper of size 512 times512 is taken as the host image shown in Figure 3(a) A
color image Apple of size 64 times 64 is taken as watermarkimage shown in Figure 3(b) The detailed implementationprocedure of the proposed watermarking model is depictedin Figures 3(c)ndash3(f) The scrambled watermark after 5-foldArnold transformation is shown in Figure 3(c) Figure 3(d)shows the watermarked image based on the SVD and Arnoldtransformation with watermarking scaling factor 120572 = 014in which the embedded watermark is invisible Figure 3(e)shows the extracted scrambling watermark from the water-marked image Figure 3(f) shows the recovered watermarkby the inverse Arnold scrambling which is similar to theoriginal watermark
In the second experiment we use 256 times 256 color imageLena as host image and 64 times 64 color image Apple aswatermark After 5-fold Arnold scrambling the watermark isembedded with scaling factor 028The detailed implementa-tion procedure is depicted in Figures 4(a)ndash4(f)
In the third experiment the watermark is the same asin the above two experiments We use 1024 times 1024 colorimage Butterfly as host image and scaling factor 014 Thedetailed implementation procedure is depicted in Figures5(a)ndash5(f) We observe from the figures that visually extractedwatermarks are quite good compared with the origin ones
The visual fidelity can be measured by calculating aparameter known as peak signal-to-noise ratio (PSNR) andthe structured similarity index (MSSIM) between the originalhost image 119860 and the watermarked image PSNR isexpressed in decibel (dB) and is defined as
PSNR
= 10 lg 31198732 (max119860 (119909 119910 119896))2sum119873119909=1sum119873119910=1sum3119896=1 (119860 (119909 119910 119896) minus (119909 119910 119896))2
(16)
where max119860(119909 119910 119896) represents the maximum pixel value ofa color image and here it is 255 119860(119909 119910 119896) and (119909 119910 119896) arethe pixel values location at position (119909 119910 119896) in the originalhost image and the watermarked image respectively Ingeneral the larger the PSNR value is the more invisible thewatermark is
The mean structured similarity index (MSSIM) [28] is anobjective measure originally developed to assess perceptualimage quality It is superior to PSNR for image qualitycomparison and better at reflecting the overall similarityof two pictures in terms of appearance rather than sim-ple mathematical point-to-point difference In the practicalapplication the image can be divided into blocks by usingthe sliding window and the total number of blocks is 119871 andMSSIM is defined as
MSSIM (119860 ) = 1119871119871sum119896=1
SSIM (119909119896 119910119896) (17)
and the formula used for calculating SSIM is as follows
Figure 3 512 times 512 host image with scale factor 014
where 120583119909 120583119910 1205902119909 1205902119910 and 120590119909119910 are the average variance andcovariance of119909 and119910 respectively 1198881 and 1198882 are constantsThemore the MSSIM value gets close to one the more similar tothe original host image the watermarked image is
In order to measure the quality of the embedded andextracted watermark the Normalized Correlation (NC) iscalculated between the original watermark 119882 and extractedwatermark which is defined as
NC = sum119872119909=1sum119872119910=1sum3119896=1119882(119909 119910 119896) times (119909 119910 119896)radicsum119872119909=1sum119872119910=1sum3119896=11198822 (119909 119910 119896) times radicsum119872119909=1sum119872119910=1sum3119896=1 2 (119909 119910 119896)
(19)
A higher NC reveals that the extracted watermark resemblesthe original watermarkmore closely If a method has a higherNC value it is more robust
In Table 2 we list the PSNR values MSSIM values NCvalues and the CPU times in which N ST SF and CPUstand for size of original host image scrambling times ofArnold transformation scaling factor and execution timerespectively In all the experiments we observe the goodvisual quality of watermarked images and good similarityof extracted and original watermarks with the proposedalgorithm the execution time for preprocessing embeddingand extraction procedure was only 05110 seconds for 512 times512 host image when the number of Arnold iterations is 5Other examples also show that the execution times of this
algorithm are short and the algorithmhas low computationalcost and is easily implemented
6 Conclusions
In this paper we have proposed a new double RGB colorimage watermarking algorithm based on the real SVD andArnold scrambling First the color watermark image isscrambled by Arnold transformation to obtain a meaninglessimageThen the original host image is divided into nonover-lapping pixel blocks We form a real matrix with the redgreen and blue components in each pixel block and performthe SVD of the real matrices We then replace the threesmallest singular values of each real matrix by the red green
Journal of Applied Mathematics 7
(a) Original host image (b) Original watermark (c) Scrambling watermark
and blue values of corresponding pixel of the scrambledwatermark with scaling factor to form a new watermarkedhost image With the reserve procedure we can extract thewatermark from the watermarked host image
The experimental results show that the proposed algo-rithm achieves high PSNR high MSIIM of the watermarkedimage and high NC of the extracted watermark In additionin the process of the algorithm we only need to perform realnumber algebra operations which have very low computa-tional complexity and therefore are more effective than theone using the quaternion SVD of color image which costs alarge amount of quaternion operations
The idea described in this paper can also be appliedto other kinds of methods concerning double color imagewatermarking problems
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
This research is supported by the National Natural ScienceFoundation of China under Grants 11171226 and 11301247
8 Journal of Applied Mathematics
(a) Original host image (b) Original watermark (c) Scrambling watermark
Figure 5 1024 times 1024 host image with scale factor 014
the Natural Science Foundation of Shandong under GrantZR2012FQ005 Science and Technology Project of Depart-ment of Education Shandong Province (J15LI10) and theScience Foundation of Liaocheng University under Grants31805 and 318011318
References
[1] R G Schyndel A Z Tirkel and C F Osborne ldquoA digitalwatermarkrdquo in Proceedings of the IEEE International Conferenceon Image Processing vol 2 pp 86ndash91 1994
[2] K Bailey and K Curran ldquoAn evaluation of image basedsteganography methodsrdquo Multimedia Tools and Applicationsvol 30 no 1 pp 55ndash88 2006
[3] A Cheddad J Condell K Curran and P Mc Kevitt ldquoDigitalimage steganography survey and analysis of current methodsrdquoSignal Processing vol 90 no 3 pp 727ndash752 2010
[4] G C Langelaar I Setyawan and R L Lagendijk ldquoWater-marking digital image and video datardquo IEEE Signal ProcessingMagazine vol 17 no 5 pp 20ndash46 2000
[5] F A P Petitcolas and R J Anderson ldquoEvaluation of copyrightmarking systemsrdquo in Proceedings of the 6th International Con-ference onMultimedia Computing and Systems (ICMCS rsquo99) pp574ndash579 Florence Italy June 1999
[6] CDeng XGao X Li andDTao ldquoA local tchebichefmoments-based robust image watermarkingrdquo Signal Processing vol 89no 8 pp 1531ndash1539 2009
[7] C Deng X Gao X Li and D Tao ldquoLocal histogram basedgeometric invariant image watermarkingrdquo Signal Processingvol 90 no 12 pp 3256ndash3264 2010
[8] S Rawat and B Raman ldquoA new robust watermarking schemefor color imagesrdquo in Proceedings of the IEEE 2nd Interna-tional Advance Computing Conference (IACC rsquo10) pp 206ndash209Patiala India February 2010
[9] C G Thorat and B D Jadhav ldquoA blind digital watermarktechnique for color image based on integer wavelet transformand SIFTrdquo Procedia Computer Science vol 2 pp 236ndash241 2010
[10] Q Su Y Niu G Wang S Jia and J Yue ldquoColor image blindwatermarking scheme based on QR decompositionrdquo SignalProcessing vol 94 no 1 pp 219ndash235 2014
[11] C-H Chou and T-L Wu ldquoEmbedding color watermarksin color imagesrdquo EURASIP Journal on Advances in SignalProcessing vol 2003 Article ID 548941 pp 32ndash40 2003
[12] S Rastegar FNamazi K Yaghmaie andAAliabadian ldquoHybridwatermarking algorithm based on Singular Value Decompo-sition and Radon transformrdquo AEUmdashInternational Journal ofElectronics and Communications vol 65 no 7 pp 658ndash6632011
[13] Q Su Y Niu H Zou Y Zhao and T Yao ldquoA blind double colorimage watermarking algorithm based on QR decompositionrdquoMultimedia Tools and Applications vol 72 no 1 pp 987ndash10092014
[14] N E H Golea R Seghir and R Benzid ldquoA bind RGB colorimage watermarking based on singular value decompositionrdquoin Proceedings of the ACSIEEE International Conference on
Journal of Applied Mathematics 9
Computer Systems andApplications (AICCSA rsquo10) pp 1ndash5Ham-mamet Tunisia May 2010
[15] Y Xing and J Tan ldquoA color watermarking scheme based onblock-SVD and Arnold transformationrdquo in Proceedings of the2nd Workshop on Digital Media and Its Application in Museumand Heritage pp 3ndash8 December 2007
[16] M Kutter F D Jordan and F Bossen ldquoDigital signature of colorimages using amplitude modulationrdquo in Storage and Retrievalfor Image and Video Databases V vol 3022 of Proceedings ofSPIE pp 518ndash526 San Jose Calif USA February 1997
[17] E Ganic and A M Eskicioglu ldquoRobust DWT-SVD domainimage watermarking embedding data in all frequenciesrdquo inProceeding of the ACM Multimedia and Security Workshop pp166ndash174 Magdeburg Germany September 2004
[18] D Vaishnavi and T S Subashini ldquoRobust and invisible imagewatermarking in RGB color space using SVDrdquo Procedia Com-puter Science vol 46 pp 1770ndash1777 2015
[19] C-Q Yin L Li A-Q Lv and L Qu ldquoColor image watermark-ing algorithm based on DWT-SVDrdquo in Proceedings of the IEEEInternational Conference on Automation and Logistics (ICALrsquo07) pp 2607ndash2611 Jinan China August 2007
[20] G H Golub and C F Van Loan Matrix Computations TheJohns Hopkins University Press Baltimore Md USA 4thedition 2013
[21] H C Andrews and C L Patterson ldquoSingular value decom-positions and digital image processingrdquo IEEE Transactions onAcoustics Speech and Signal Processing vol 24 no 1 pp 26ndash531976
[22] V J Arnold and A AvezTheMathematical Physics MonographSeriesergodic Problems of Classical Mechanics Benjamin NewYork NY USA 1968
[23] S J Sangwine ldquoFourier transforms of colour images usingquaternion or hypercomplex numbersrdquo Electronics Letters vol32 no 21 pp 1979ndash1980 1996
[24] S Sangwine and N L Bihan Quaternion toolbox for matlabhttpqtfmsourceforgenet
[25] Y Li MWei F Zhang and J Zhao ldquoA fast structure-preservingmethod for computing the singular value decomposition ofquaternion matricesrdquo Applied Mathematics and Computationvol 235 pp 157ndash167 2014
[26] Y Li M Wei F Zhang and J Zhao ldquoReal structure-preservingalgorithms of Householder based transformations for quater-nion matricesrdquo Journal of Computational and Applied Mathe-matics vol 305 pp 82ndash91 2016
[27] M Wang andW Ma ldquoA structure-preserving algorithm for thequaternion Cholesky decompositionrdquoAppliedMathematics andComputation vol 223 pp 354ndash361 2013
[28] ZWang A C Bovik H R Sheikh and E P Simoncelli ldquoImagequality assessment from error visibility to structural similarityrdquoIEEE Transactions on Image Processing vol 13 no 4 pp 600ndash612 2004
where 119896 can be used as the secret key for watermarkrecovery
Step 2 (partition) The original RGB host color image 119860 isdivided into nonoverlapping blocks 119860 119894119895 of size 119889 times 119889 pixelswhere 119894 119895 = 1 2 119872 119889 = 119873119872
Step 3 (pixels block rearranging) The R G and B colorcomponents of 119860 119894119895 are rearranged as rectangle matrices 119862119894119895of 3119889 times 119889Step 4 (performing the SVD) Perform the SVD for119862119894119895119862119894119895 =119880119894119895Σ119894119895119881119879119894119895 where Σ119894119895 = diag(120590(119894119895)1 120590(119894119895)
119889)
Step 5 (embedding) Form Σ119894119895 = diag(120590(119894119895)1 120590(119894119895)119889minus3
120572R119894119895
120572G119894119895 120572B
119894119895) where R119894119895 G
119894119895 and B119894119895 are red green and
blue values at (119894 119895) pixel of and 120572 is the scaling factor Form119894119895 = 119880119894119895Σ119894119895119881119879119894119895 and then obtain 119894119895 from 11989411989542 Watermark Extraction The watermark is extracted asfollows
Step 1 Both the original host image 119860 and the watermarkedimage are divided into nonoverlapping blocks 119860 119894119895 and 119894119895of size 119889 times 119889 pixels respectively where 119894 119895 = 1 2 119872 119889 =119873119872
Step 2 The R G and B color components of 119860 119894119895 and 119894119895are rearranged as rectangle matrices 119862119894119895 and 119894119895 of 3119889 times 119889respectively
Step 5 The inverse Arnold scrambling is applied to 119882 =(119882119894119895) to construct the watermark
5 Experimental Results and Analysis
To verify the effectiveness of the proposed algorithm a seriesof experiments were conducted where different host imagesare adopted We carried out our experiments in MatlabR2013a environment on a laptop with Intel i5 processor ratedat 25 GHz
In the first experiment color image Pepper of size 512 times512 is taken as the host image shown in Figure 3(a) A
color image Apple of size 64 times 64 is taken as watermarkimage shown in Figure 3(b) The detailed implementationprocedure of the proposed watermarking model is depictedin Figures 3(c)ndash3(f) The scrambled watermark after 5-foldArnold transformation is shown in Figure 3(c) Figure 3(d)shows the watermarked image based on the SVD and Arnoldtransformation with watermarking scaling factor 120572 = 014in which the embedded watermark is invisible Figure 3(e)shows the extracted scrambling watermark from the water-marked image Figure 3(f) shows the recovered watermarkby the inverse Arnold scrambling which is similar to theoriginal watermark
In the second experiment we use 256 times 256 color imageLena as host image and 64 times 64 color image Apple aswatermark After 5-fold Arnold scrambling the watermark isembedded with scaling factor 028The detailed implementa-tion procedure is depicted in Figures 4(a)ndash4(f)
In the third experiment the watermark is the same asin the above two experiments We use 1024 times 1024 colorimage Butterfly as host image and scaling factor 014 Thedetailed implementation procedure is depicted in Figures5(a)ndash5(f) We observe from the figures that visually extractedwatermarks are quite good compared with the origin ones
The visual fidelity can be measured by calculating aparameter known as peak signal-to-noise ratio (PSNR) andthe structured similarity index (MSSIM) between the originalhost image 119860 and the watermarked image PSNR isexpressed in decibel (dB) and is defined as
PSNR
= 10 lg 31198732 (max119860 (119909 119910 119896))2sum119873119909=1sum119873119910=1sum3119896=1 (119860 (119909 119910 119896) minus (119909 119910 119896))2
(16)
where max119860(119909 119910 119896) represents the maximum pixel value ofa color image and here it is 255 119860(119909 119910 119896) and (119909 119910 119896) arethe pixel values location at position (119909 119910 119896) in the originalhost image and the watermarked image respectively Ingeneral the larger the PSNR value is the more invisible thewatermark is
The mean structured similarity index (MSSIM) [28] is anobjective measure originally developed to assess perceptualimage quality It is superior to PSNR for image qualitycomparison and better at reflecting the overall similarityof two pictures in terms of appearance rather than sim-ple mathematical point-to-point difference In the practicalapplication the image can be divided into blocks by usingthe sliding window and the total number of blocks is 119871 andMSSIM is defined as
MSSIM (119860 ) = 1119871119871sum119896=1
SSIM (119909119896 119910119896) (17)
and the formula used for calculating SSIM is as follows
Figure 3 512 times 512 host image with scale factor 014
where 120583119909 120583119910 1205902119909 1205902119910 and 120590119909119910 are the average variance andcovariance of119909 and119910 respectively 1198881 and 1198882 are constantsThemore the MSSIM value gets close to one the more similar tothe original host image the watermarked image is
In order to measure the quality of the embedded andextracted watermark the Normalized Correlation (NC) iscalculated between the original watermark 119882 and extractedwatermark which is defined as
NC = sum119872119909=1sum119872119910=1sum3119896=1119882(119909 119910 119896) times (119909 119910 119896)radicsum119872119909=1sum119872119910=1sum3119896=11198822 (119909 119910 119896) times radicsum119872119909=1sum119872119910=1sum3119896=1 2 (119909 119910 119896)
(19)
A higher NC reveals that the extracted watermark resemblesthe original watermarkmore closely If a method has a higherNC value it is more robust
In Table 2 we list the PSNR values MSSIM values NCvalues and the CPU times in which N ST SF and CPUstand for size of original host image scrambling times ofArnold transformation scaling factor and execution timerespectively In all the experiments we observe the goodvisual quality of watermarked images and good similarityof extracted and original watermarks with the proposedalgorithm the execution time for preprocessing embeddingand extraction procedure was only 05110 seconds for 512 times512 host image when the number of Arnold iterations is 5Other examples also show that the execution times of this
algorithm are short and the algorithmhas low computationalcost and is easily implemented
6 Conclusions
In this paper we have proposed a new double RGB colorimage watermarking algorithm based on the real SVD andArnold scrambling First the color watermark image isscrambled by Arnold transformation to obtain a meaninglessimageThen the original host image is divided into nonover-lapping pixel blocks We form a real matrix with the redgreen and blue components in each pixel block and performthe SVD of the real matrices We then replace the threesmallest singular values of each real matrix by the red green
Journal of Applied Mathematics 7
(a) Original host image (b) Original watermark (c) Scrambling watermark
and blue values of corresponding pixel of the scrambledwatermark with scaling factor to form a new watermarkedhost image With the reserve procedure we can extract thewatermark from the watermarked host image
The experimental results show that the proposed algo-rithm achieves high PSNR high MSIIM of the watermarkedimage and high NC of the extracted watermark In additionin the process of the algorithm we only need to perform realnumber algebra operations which have very low computa-tional complexity and therefore are more effective than theone using the quaternion SVD of color image which costs alarge amount of quaternion operations
The idea described in this paper can also be appliedto other kinds of methods concerning double color imagewatermarking problems
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
This research is supported by the National Natural ScienceFoundation of China under Grants 11171226 and 11301247
8 Journal of Applied Mathematics
(a) Original host image (b) Original watermark (c) Scrambling watermark
Figure 5 1024 times 1024 host image with scale factor 014
the Natural Science Foundation of Shandong under GrantZR2012FQ005 Science and Technology Project of Depart-ment of Education Shandong Province (J15LI10) and theScience Foundation of Liaocheng University under Grants31805 and 318011318
References
[1] R G Schyndel A Z Tirkel and C F Osborne ldquoA digitalwatermarkrdquo in Proceedings of the IEEE International Conferenceon Image Processing vol 2 pp 86ndash91 1994
[2] K Bailey and K Curran ldquoAn evaluation of image basedsteganography methodsrdquo Multimedia Tools and Applicationsvol 30 no 1 pp 55ndash88 2006
[3] A Cheddad J Condell K Curran and P Mc Kevitt ldquoDigitalimage steganography survey and analysis of current methodsrdquoSignal Processing vol 90 no 3 pp 727ndash752 2010
[4] G C Langelaar I Setyawan and R L Lagendijk ldquoWater-marking digital image and video datardquo IEEE Signal ProcessingMagazine vol 17 no 5 pp 20ndash46 2000
[5] F A P Petitcolas and R J Anderson ldquoEvaluation of copyrightmarking systemsrdquo in Proceedings of the 6th International Con-ference onMultimedia Computing and Systems (ICMCS rsquo99) pp574ndash579 Florence Italy June 1999
[6] CDeng XGao X Li andDTao ldquoA local tchebichefmoments-based robust image watermarkingrdquo Signal Processing vol 89no 8 pp 1531ndash1539 2009
[7] C Deng X Gao X Li and D Tao ldquoLocal histogram basedgeometric invariant image watermarkingrdquo Signal Processingvol 90 no 12 pp 3256ndash3264 2010
[8] S Rawat and B Raman ldquoA new robust watermarking schemefor color imagesrdquo in Proceedings of the IEEE 2nd Interna-tional Advance Computing Conference (IACC rsquo10) pp 206ndash209Patiala India February 2010
[9] C G Thorat and B D Jadhav ldquoA blind digital watermarktechnique for color image based on integer wavelet transformand SIFTrdquo Procedia Computer Science vol 2 pp 236ndash241 2010
[10] Q Su Y Niu G Wang S Jia and J Yue ldquoColor image blindwatermarking scheme based on QR decompositionrdquo SignalProcessing vol 94 no 1 pp 219ndash235 2014
[11] C-H Chou and T-L Wu ldquoEmbedding color watermarksin color imagesrdquo EURASIP Journal on Advances in SignalProcessing vol 2003 Article ID 548941 pp 32ndash40 2003
[12] S Rastegar FNamazi K Yaghmaie andAAliabadian ldquoHybridwatermarking algorithm based on Singular Value Decompo-sition and Radon transformrdquo AEUmdashInternational Journal ofElectronics and Communications vol 65 no 7 pp 658ndash6632011
[13] Q Su Y Niu H Zou Y Zhao and T Yao ldquoA blind double colorimage watermarking algorithm based on QR decompositionrdquoMultimedia Tools and Applications vol 72 no 1 pp 987ndash10092014
[14] N E H Golea R Seghir and R Benzid ldquoA bind RGB colorimage watermarking based on singular value decompositionrdquoin Proceedings of the ACSIEEE International Conference on
Journal of Applied Mathematics 9
Computer Systems andApplications (AICCSA rsquo10) pp 1ndash5Ham-mamet Tunisia May 2010
[15] Y Xing and J Tan ldquoA color watermarking scheme based onblock-SVD and Arnold transformationrdquo in Proceedings of the2nd Workshop on Digital Media and Its Application in Museumand Heritage pp 3ndash8 December 2007
[16] M Kutter F D Jordan and F Bossen ldquoDigital signature of colorimages using amplitude modulationrdquo in Storage and Retrievalfor Image and Video Databases V vol 3022 of Proceedings ofSPIE pp 518ndash526 San Jose Calif USA February 1997
[17] E Ganic and A M Eskicioglu ldquoRobust DWT-SVD domainimage watermarking embedding data in all frequenciesrdquo inProceeding of the ACM Multimedia and Security Workshop pp166ndash174 Magdeburg Germany September 2004
[18] D Vaishnavi and T S Subashini ldquoRobust and invisible imagewatermarking in RGB color space using SVDrdquo Procedia Com-puter Science vol 46 pp 1770ndash1777 2015
[19] C-Q Yin L Li A-Q Lv and L Qu ldquoColor image watermark-ing algorithm based on DWT-SVDrdquo in Proceedings of the IEEEInternational Conference on Automation and Logistics (ICALrsquo07) pp 2607ndash2611 Jinan China August 2007
[20] G H Golub and C F Van Loan Matrix Computations TheJohns Hopkins University Press Baltimore Md USA 4thedition 2013
[21] H C Andrews and C L Patterson ldquoSingular value decom-positions and digital image processingrdquo IEEE Transactions onAcoustics Speech and Signal Processing vol 24 no 1 pp 26ndash531976
[22] V J Arnold and A AvezTheMathematical Physics MonographSeriesergodic Problems of Classical Mechanics Benjamin NewYork NY USA 1968
[23] S J Sangwine ldquoFourier transforms of colour images usingquaternion or hypercomplex numbersrdquo Electronics Letters vol32 no 21 pp 1979ndash1980 1996
[24] S Sangwine and N L Bihan Quaternion toolbox for matlabhttpqtfmsourceforgenet
[25] Y Li MWei F Zhang and J Zhao ldquoA fast structure-preservingmethod for computing the singular value decomposition ofquaternion matricesrdquo Applied Mathematics and Computationvol 235 pp 157ndash167 2014
[26] Y Li M Wei F Zhang and J Zhao ldquoReal structure-preservingalgorithms of Householder based transformations for quater-nion matricesrdquo Journal of Computational and Applied Mathe-matics vol 305 pp 82ndash91 2016
[27] M Wang andW Ma ldquoA structure-preserving algorithm for thequaternion Cholesky decompositionrdquoAppliedMathematics andComputation vol 223 pp 354ndash361 2013
[28] ZWang A C Bovik H R Sheikh and E P Simoncelli ldquoImagequality assessment from error visibility to structural similarityrdquoIEEE Transactions on Image Processing vol 13 no 4 pp 600ndash612 2004
Figure 3 512 times 512 host image with scale factor 014
where 120583119909 120583119910 1205902119909 1205902119910 and 120590119909119910 are the average variance andcovariance of119909 and119910 respectively 1198881 and 1198882 are constantsThemore the MSSIM value gets close to one the more similar tothe original host image the watermarked image is
In order to measure the quality of the embedded andextracted watermark the Normalized Correlation (NC) iscalculated between the original watermark 119882 and extractedwatermark which is defined as
NC = sum119872119909=1sum119872119910=1sum3119896=1119882(119909 119910 119896) times (119909 119910 119896)radicsum119872119909=1sum119872119910=1sum3119896=11198822 (119909 119910 119896) times radicsum119872119909=1sum119872119910=1sum3119896=1 2 (119909 119910 119896)
(19)
A higher NC reveals that the extracted watermark resemblesthe original watermarkmore closely If a method has a higherNC value it is more robust
In Table 2 we list the PSNR values MSSIM values NCvalues and the CPU times in which N ST SF and CPUstand for size of original host image scrambling times ofArnold transformation scaling factor and execution timerespectively In all the experiments we observe the goodvisual quality of watermarked images and good similarityof extracted and original watermarks with the proposedalgorithm the execution time for preprocessing embeddingand extraction procedure was only 05110 seconds for 512 times512 host image when the number of Arnold iterations is 5Other examples also show that the execution times of this
algorithm are short and the algorithmhas low computationalcost and is easily implemented
6 Conclusions
In this paper we have proposed a new double RGB colorimage watermarking algorithm based on the real SVD andArnold scrambling First the color watermark image isscrambled by Arnold transformation to obtain a meaninglessimageThen the original host image is divided into nonover-lapping pixel blocks We form a real matrix with the redgreen and blue components in each pixel block and performthe SVD of the real matrices We then replace the threesmallest singular values of each real matrix by the red green
Journal of Applied Mathematics 7
(a) Original host image (b) Original watermark (c) Scrambling watermark
and blue values of corresponding pixel of the scrambledwatermark with scaling factor to form a new watermarkedhost image With the reserve procedure we can extract thewatermark from the watermarked host image
The experimental results show that the proposed algo-rithm achieves high PSNR high MSIIM of the watermarkedimage and high NC of the extracted watermark In additionin the process of the algorithm we only need to perform realnumber algebra operations which have very low computa-tional complexity and therefore are more effective than theone using the quaternion SVD of color image which costs alarge amount of quaternion operations
The idea described in this paper can also be appliedto other kinds of methods concerning double color imagewatermarking problems
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
This research is supported by the National Natural ScienceFoundation of China under Grants 11171226 and 11301247
8 Journal of Applied Mathematics
(a) Original host image (b) Original watermark (c) Scrambling watermark
Figure 5 1024 times 1024 host image with scale factor 014
the Natural Science Foundation of Shandong under GrantZR2012FQ005 Science and Technology Project of Depart-ment of Education Shandong Province (J15LI10) and theScience Foundation of Liaocheng University under Grants31805 and 318011318
References
[1] R G Schyndel A Z Tirkel and C F Osborne ldquoA digitalwatermarkrdquo in Proceedings of the IEEE International Conferenceon Image Processing vol 2 pp 86ndash91 1994
[2] K Bailey and K Curran ldquoAn evaluation of image basedsteganography methodsrdquo Multimedia Tools and Applicationsvol 30 no 1 pp 55ndash88 2006
[3] A Cheddad J Condell K Curran and P Mc Kevitt ldquoDigitalimage steganography survey and analysis of current methodsrdquoSignal Processing vol 90 no 3 pp 727ndash752 2010
[4] G C Langelaar I Setyawan and R L Lagendijk ldquoWater-marking digital image and video datardquo IEEE Signal ProcessingMagazine vol 17 no 5 pp 20ndash46 2000
[5] F A P Petitcolas and R J Anderson ldquoEvaluation of copyrightmarking systemsrdquo in Proceedings of the 6th International Con-ference onMultimedia Computing and Systems (ICMCS rsquo99) pp574ndash579 Florence Italy June 1999
[6] CDeng XGao X Li andDTao ldquoA local tchebichefmoments-based robust image watermarkingrdquo Signal Processing vol 89no 8 pp 1531ndash1539 2009
[7] C Deng X Gao X Li and D Tao ldquoLocal histogram basedgeometric invariant image watermarkingrdquo Signal Processingvol 90 no 12 pp 3256ndash3264 2010
[8] S Rawat and B Raman ldquoA new robust watermarking schemefor color imagesrdquo in Proceedings of the IEEE 2nd Interna-tional Advance Computing Conference (IACC rsquo10) pp 206ndash209Patiala India February 2010
[9] C G Thorat and B D Jadhav ldquoA blind digital watermarktechnique for color image based on integer wavelet transformand SIFTrdquo Procedia Computer Science vol 2 pp 236ndash241 2010
[10] Q Su Y Niu G Wang S Jia and J Yue ldquoColor image blindwatermarking scheme based on QR decompositionrdquo SignalProcessing vol 94 no 1 pp 219ndash235 2014
[11] C-H Chou and T-L Wu ldquoEmbedding color watermarksin color imagesrdquo EURASIP Journal on Advances in SignalProcessing vol 2003 Article ID 548941 pp 32ndash40 2003
[12] S Rastegar FNamazi K Yaghmaie andAAliabadian ldquoHybridwatermarking algorithm based on Singular Value Decompo-sition and Radon transformrdquo AEUmdashInternational Journal ofElectronics and Communications vol 65 no 7 pp 658ndash6632011
[13] Q Su Y Niu H Zou Y Zhao and T Yao ldquoA blind double colorimage watermarking algorithm based on QR decompositionrdquoMultimedia Tools and Applications vol 72 no 1 pp 987ndash10092014
[14] N E H Golea R Seghir and R Benzid ldquoA bind RGB colorimage watermarking based on singular value decompositionrdquoin Proceedings of the ACSIEEE International Conference on
Journal of Applied Mathematics 9
Computer Systems andApplications (AICCSA rsquo10) pp 1ndash5Ham-mamet Tunisia May 2010
[15] Y Xing and J Tan ldquoA color watermarking scheme based onblock-SVD and Arnold transformationrdquo in Proceedings of the2nd Workshop on Digital Media and Its Application in Museumand Heritage pp 3ndash8 December 2007
[16] M Kutter F D Jordan and F Bossen ldquoDigital signature of colorimages using amplitude modulationrdquo in Storage and Retrievalfor Image and Video Databases V vol 3022 of Proceedings ofSPIE pp 518ndash526 San Jose Calif USA February 1997
[17] E Ganic and A M Eskicioglu ldquoRobust DWT-SVD domainimage watermarking embedding data in all frequenciesrdquo inProceeding of the ACM Multimedia and Security Workshop pp166ndash174 Magdeburg Germany September 2004
[18] D Vaishnavi and T S Subashini ldquoRobust and invisible imagewatermarking in RGB color space using SVDrdquo Procedia Com-puter Science vol 46 pp 1770ndash1777 2015
[19] C-Q Yin L Li A-Q Lv and L Qu ldquoColor image watermark-ing algorithm based on DWT-SVDrdquo in Proceedings of the IEEEInternational Conference on Automation and Logistics (ICALrsquo07) pp 2607ndash2611 Jinan China August 2007
[20] G H Golub and C F Van Loan Matrix Computations TheJohns Hopkins University Press Baltimore Md USA 4thedition 2013
[21] H C Andrews and C L Patterson ldquoSingular value decom-positions and digital image processingrdquo IEEE Transactions onAcoustics Speech and Signal Processing vol 24 no 1 pp 26ndash531976
[22] V J Arnold and A AvezTheMathematical Physics MonographSeriesergodic Problems of Classical Mechanics Benjamin NewYork NY USA 1968
[23] S J Sangwine ldquoFourier transforms of colour images usingquaternion or hypercomplex numbersrdquo Electronics Letters vol32 no 21 pp 1979ndash1980 1996
[24] S Sangwine and N L Bihan Quaternion toolbox for matlabhttpqtfmsourceforgenet
[25] Y Li MWei F Zhang and J Zhao ldquoA fast structure-preservingmethod for computing the singular value decomposition ofquaternion matricesrdquo Applied Mathematics and Computationvol 235 pp 157ndash167 2014
[26] Y Li M Wei F Zhang and J Zhao ldquoReal structure-preservingalgorithms of Householder based transformations for quater-nion matricesrdquo Journal of Computational and Applied Mathe-matics vol 305 pp 82ndash91 2016
[27] M Wang andW Ma ldquoA structure-preserving algorithm for thequaternion Cholesky decompositionrdquoAppliedMathematics andComputation vol 223 pp 354ndash361 2013
[28] ZWang A C Bovik H R Sheikh and E P Simoncelli ldquoImagequality assessment from error visibility to structural similarityrdquoIEEE Transactions on Image Processing vol 13 no 4 pp 600ndash612 2004
and blue values of corresponding pixel of the scrambledwatermark with scaling factor to form a new watermarkedhost image With the reserve procedure we can extract thewatermark from the watermarked host image
The experimental results show that the proposed algo-rithm achieves high PSNR high MSIIM of the watermarkedimage and high NC of the extracted watermark In additionin the process of the algorithm we only need to perform realnumber algebra operations which have very low computa-tional complexity and therefore are more effective than theone using the quaternion SVD of color image which costs alarge amount of quaternion operations
The idea described in this paper can also be appliedto other kinds of methods concerning double color imagewatermarking problems
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
This research is supported by the National Natural ScienceFoundation of China under Grants 11171226 and 11301247
8 Journal of Applied Mathematics
(a) Original host image (b) Original watermark (c) Scrambling watermark
Figure 5 1024 times 1024 host image with scale factor 014
the Natural Science Foundation of Shandong under GrantZR2012FQ005 Science and Technology Project of Depart-ment of Education Shandong Province (J15LI10) and theScience Foundation of Liaocheng University under Grants31805 and 318011318
References
[1] R G Schyndel A Z Tirkel and C F Osborne ldquoA digitalwatermarkrdquo in Proceedings of the IEEE International Conferenceon Image Processing vol 2 pp 86ndash91 1994
[2] K Bailey and K Curran ldquoAn evaluation of image basedsteganography methodsrdquo Multimedia Tools and Applicationsvol 30 no 1 pp 55ndash88 2006
[3] A Cheddad J Condell K Curran and P Mc Kevitt ldquoDigitalimage steganography survey and analysis of current methodsrdquoSignal Processing vol 90 no 3 pp 727ndash752 2010
[4] G C Langelaar I Setyawan and R L Lagendijk ldquoWater-marking digital image and video datardquo IEEE Signal ProcessingMagazine vol 17 no 5 pp 20ndash46 2000
[5] F A P Petitcolas and R J Anderson ldquoEvaluation of copyrightmarking systemsrdquo in Proceedings of the 6th International Con-ference onMultimedia Computing and Systems (ICMCS rsquo99) pp574ndash579 Florence Italy June 1999
[6] CDeng XGao X Li andDTao ldquoA local tchebichefmoments-based robust image watermarkingrdquo Signal Processing vol 89no 8 pp 1531ndash1539 2009
[7] C Deng X Gao X Li and D Tao ldquoLocal histogram basedgeometric invariant image watermarkingrdquo Signal Processingvol 90 no 12 pp 3256ndash3264 2010
[8] S Rawat and B Raman ldquoA new robust watermarking schemefor color imagesrdquo in Proceedings of the IEEE 2nd Interna-tional Advance Computing Conference (IACC rsquo10) pp 206ndash209Patiala India February 2010
[9] C G Thorat and B D Jadhav ldquoA blind digital watermarktechnique for color image based on integer wavelet transformand SIFTrdquo Procedia Computer Science vol 2 pp 236ndash241 2010
[10] Q Su Y Niu G Wang S Jia and J Yue ldquoColor image blindwatermarking scheme based on QR decompositionrdquo SignalProcessing vol 94 no 1 pp 219ndash235 2014
[11] C-H Chou and T-L Wu ldquoEmbedding color watermarksin color imagesrdquo EURASIP Journal on Advances in SignalProcessing vol 2003 Article ID 548941 pp 32ndash40 2003
[12] S Rastegar FNamazi K Yaghmaie andAAliabadian ldquoHybridwatermarking algorithm based on Singular Value Decompo-sition and Radon transformrdquo AEUmdashInternational Journal ofElectronics and Communications vol 65 no 7 pp 658ndash6632011
[13] Q Su Y Niu H Zou Y Zhao and T Yao ldquoA blind double colorimage watermarking algorithm based on QR decompositionrdquoMultimedia Tools and Applications vol 72 no 1 pp 987ndash10092014
[14] N E H Golea R Seghir and R Benzid ldquoA bind RGB colorimage watermarking based on singular value decompositionrdquoin Proceedings of the ACSIEEE International Conference on
Journal of Applied Mathematics 9
Computer Systems andApplications (AICCSA rsquo10) pp 1ndash5Ham-mamet Tunisia May 2010
[15] Y Xing and J Tan ldquoA color watermarking scheme based onblock-SVD and Arnold transformationrdquo in Proceedings of the2nd Workshop on Digital Media and Its Application in Museumand Heritage pp 3ndash8 December 2007
[16] M Kutter F D Jordan and F Bossen ldquoDigital signature of colorimages using amplitude modulationrdquo in Storage and Retrievalfor Image and Video Databases V vol 3022 of Proceedings ofSPIE pp 518ndash526 San Jose Calif USA February 1997
[17] E Ganic and A M Eskicioglu ldquoRobust DWT-SVD domainimage watermarking embedding data in all frequenciesrdquo inProceeding of the ACM Multimedia and Security Workshop pp166ndash174 Magdeburg Germany September 2004
[18] D Vaishnavi and T S Subashini ldquoRobust and invisible imagewatermarking in RGB color space using SVDrdquo Procedia Com-puter Science vol 46 pp 1770ndash1777 2015
[19] C-Q Yin L Li A-Q Lv and L Qu ldquoColor image watermark-ing algorithm based on DWT-SVDrdquo in Proceedings of the IEEEInternational Conference on Automation and Logistics (ICALrsquo07) pp 2607ndash2611 Jinan China August 2007
[20] G H Golub and C F Van Loan Matrix Computations TheJohns Hopkins University Press Baltimore Md USA 4thedition 2013
[21] H C Andrews and C L Patterson ldquoSingular value decom-positions and digital image processingrdquo IEEE Transactions onAcoustics Speech and Signal Processing vol 24 no 1 pp 26ndash531976
[22] V J Arnold and A AvezTheMathematical Physics MonographSeriesergodic Problems of Classical Mechanics Benjamin NewYork NY USA 1968
[23] S J Sangwine ldquoFourier transforms of colour images usingquaternion or hypercomplex numbersrdquo Electronics Letters vol32 no 21 pp 1979ndash1980 1996
[24] S Sangwine and N L Bihan Quaternion toolbox for matlabhttpqtfmsourceforgenet
[25] Y Li MWei F Zhang and J Zhao ldquoA fast structure-preservingmethod for computing the singular value decomposition ofquaternion matricesrdquo Applied Mathematics and Computationvol 235 pp 157ndash167 2014
[26] Y Li M Wei F Zhang and J Zhao ldquoReal structure-preservingalgorithms of Householder based transformations for quater-nion matricesrdquo Journal of Computational and Applied Mathe-matics vol 305 pp 82ndash91 2016
[27] M Wang andW Ma ldquoA structure-preserving algorithm for thequaternion Cholesky decompositionrdquoAppliedMathematics andComputation vol 223 pp 354ndash361 2013
[28] ZWang A C Bovik H R Sheikh and E P Simoncelli ldquoImagequality assessment from error visibility to structural similarityrdquoIEEE Transactions on Image Processing vol 13 no 4 pp 600ndash612 2004
Figure 5 1024 times 1024 host image with scale factor 014
the Natural Science Foundation of Shandong under GrantZR2012FQ005 Science and Technology Project of Depart-ment of Education Shandong Province (J15LI10) and theScience Foundation of Liaocheng University under Grants31805 and 318011318
References
[1] R G Schyndel A Z Tirkel and C F Osborne ldquoA digitalwatermarkrdquo in Proceedings of the IEEE International Conferenceon Image Processing vol 2 pp 86ndash91 1994
[2] K Bailey and K Curran ldquoAn evaluation of image basedsteganography methodsrdquo Multimedia Tools and Applicationsvol 30 no 1 pp 55ndash88 2006
[3] A Cheddad J Condell K Curran and P Mc Kevitt ldquoDigitalimage steganography survey and analysis of current methodsrdquoSignal Processing vol 90 no 3 pp 727ndash752 2010
[4] G C Langelaar I Setyawan and R L Lagendijk ldquoWater-marking digital image and video datardquo IEEE Signal ProcessingMagazine vol 17 no 5 pp 20ndash46 2000
[5] F A P Petitcolas and R J Anderson ldquoEvaluation of copyrightmarking systemsrdquo in Proceedings of the 6th International Con-ference onMultimedia Computing and Systems (ICMCS rsquo99) pp574ndash579 Florence Italy June 1999
[6] CDeng XGao X Li andDTao ldquoA local tchebichefmoments-based robust image watermarkingrdquo Signal Processing vol 89no 8 pp 1531ndash1539 2009
[7] C Deng X Gao X Li and D Tao ldquoLocal histogram basedgeometric invariant image watermarkingrdquo Signal Processingvol 90 no 12 pp 3256ndash3264 2010
[8] S Rawat and B Raman ldquoA new robust watermarking schemefor color imagesrdquo in Proceedings of the IEEE 2nd Interna-tional Advance Computing Conference (IACC rsquo10) pp 206ndash209Patiala India February 2010
[9] C G Thorat and B D Jadhav ldquoA blind digital watermarktechnique for color image based on integer wavelet transformand SIFTrdquo Procedia Computer Science vol 2 pp 236ndash241 2010
[10] Q Su Y Niu G Wang S Jia and J Yue ldquoColor image blindwatermarking scheme based on QR decompositionrdquo SignalProcessing vol 94 no 1 pp 219ndash235 2014
[11] C-H Chou and T-L Wu ldquoEmbedding color watermarksin color imagesrdquo EURASIP Journal on Advances in SignalProcessing vol 2003 Article ID 548941 pp 32ndash40 2003
[12] S Rastegar FNamazi K Yaghmaie andAAliabadian ldquoHybridwatermarking algorithm based on Singular Value Decompo-sition and Radon transformrdquo AEUmdashInternational Journal ofElectronics and Communications vol 65 no 7 pp 658ndash6632011
[13] Q Su Y Niu H Zou Y Zhao and T Yao ldquoA blind double colorimage watermarking algorithm based on QR decompositionrdquoMultimedia Tools and Applications vol 72 no 1 pp 987ndash10092014
[14] N E H Golea R Seghir and R Benzid ldquoA bind RGB colorimage watermarking based on singular value decompositionrdquoin Proceedings of the ACSIEEE International Conference on
Journal of Applied Mathematics 9
Computer Systems andApplications (AICCSA rsquo10) pp 1ndash5Ham-mamet Tunisia May 2010
[15] Y Xing and J Tan ldquoA color watermarking scheme based onblock-SVD and Arnold transformationrdquo in Proceedings of the2nd Workshop on Digital Media and Its Application in Museumand Heritage pp 3ndash8 December 2007
[16] M Kutter F D Jordan and F Bossen ldquoDigital signature of colorimages using amplitude modulationrdquo in Storage and Retrievalfor Image and Video Databases V vol 3022 of Proceedings ofSPIE pp 518ndash526 San Jose Calif USA February 1997
[17] E Ganic and A M Eskicioglu ldquoRobust DWT-SVD domainimage watermarking embedding data in all frequenciesrdquo inProceeding of the ACM Multimedia and Security Workshop pp166ndash174 Magdeburg Germany September 2004
[18] D Vaishnavi and T S Subashini ldquoRobust and invisible imagewatermarking in RGB color space using SVDrdquo Procedia Com-puter Science vol 46 pp 1770ndash1777 2015
[19] C-Q Yin L Li A-Q Lv and L Qu ldquoColor image watermark-ing algorithm based on DWT-SVDrdquo in Proceedings of the IEEEInternational Conference on Automation and Logistics (ICALrsquo07) pp 2607ndash2611 Jinan China August 2007
[20] G H Golub and C F Van Loan Matrix Computations TheJohns Hopkins University Press Baltimore Md USA 4thedition 2013
[21] H C Andrews and C L Patterson ldquoSingular value decom-positions and digital image processingrdquo IEEE Transactions onAcoustics Speech and Signal Processing vol 24 no 1 pp 26ndash531976
[22] V J Arnold and A AvezTheMathematical Physics MonographSeriesergodic Problems of Classical Mechanics Benjamin NewYork NY USA 1968
[23] S J Sangwine ldquoFourier transforms of colour images usingquaternion or hypercomplex numbersrdquo Electronics Letters vol32 no 21 pp 1979ndash1980 1996
[24] S Sangwine and N L Bihan Quaternion toolbox for matlabhttpqtfmsourceforgenet
[25] Y Li MWei F Zhang and J Zhao ldquoA fast structure-preservingmethod for computing the singular value decomposition ofquaternion matricesrdquo Applied Mathematics and Computationvol 235 pp 157ndash167 2014
[26] Y Li M Wei F Zhang and J Zhao ldquoReal structure-preservingalgorithms of Householder based transformations for quater-nion matricesrdquo Journal of Computational and Applied Mathe-matics vol 305 pp 82ndash91 2016
[27] M Wang andW Ma ldquoA structure-preserving algorithm for thequaternion Cholesky decompositionrdquoAppliedMathematics andComputation vol 223 pp 354ndash361 2013
[28] ZWang A C Bovik H R Sheikh and E P Simoncelli ldquoImagequality assessment from error visibility to structural similarityrdquoIEEE Transactions on Image Processing vol 13 no 4 pp 600ndash612 2004
Computer Systems andApplications (AICCSA rsquo10) pp 1ndash5Ham-mamet Tunisia May 2010
[15] Y Xing and J Tan ldquoA color watermarking scheme based onblock-SVD and Arnold transformationrdquo in Proceedings of the2nd Workshop on Digital Media and Its Application in Museumand Heritage pp 3ndash8 December 2007
[16] M Kutter F D Jordan and F Bossen ldquoDigital signature of colorimages using amplitude modulationrdquo in Storage and Retrievalfor Image and Video Databases V vol 3022 of Proceedings ofSPIE pp 518ndash526 San Jose Calif USA February 1997
[17] E Ganic and A M Eskicioglu ldquoRobust DWT-SVD domainimage watermarking embedding data in all frequenciesrdquo inProceeding of the ACM Multimedia and Security Workshop pp166ndash174 Magdeburg Germany September 2004
[18] D Vaishnavi and T S Subashini ldquoRobust and invisible imagewatermarking in RGB color space using SVDrdquo Procedia Com-puter Science vol 46 pp 1770ndash1777 2015
[19] C-Q Yin L Li A-Q Lv and L Qu ldquoColor image watermark-ing algorithm based on DWT-SVDrdquo in Proceedings of the IEEEInternational Conference on Automation and Logistics (ICALrsquo07) pp 2607ndash2611 Jinan China August 2007
[20] G H Golub and C F Van Loan Matrix Computations TheJohns Hopkins University Press Baltimore Md USA 4thedition 2013
[21] H C Andrews and C L Patterson ldquoSingular value decom-positions and digital image processingrdquo IEEE Transactions onAcoustics Speech and Signal Processing vol 24 no 1 pp 26ndash531976
[22] V J Arnold and A AvezTheMathematical Physics MonographSeriesergodic Problems of Classical Mechanics Benjamin NewYork NY USA 1968
[23] S J Sangwine ldquoFourier transforms of colour images usingquaternion or hypercomplex numbersrdquo Electronics Letters vol32 no 21 pp 1979ndash1980 1996
[24] S Sangwine and N L Bihan Quaternion toolbox for matlabhttpqtfmsourceforgenet
[25] Y Li MWei F Zhang and J Zhao ldquoA fast structure-preservingmethod for computing the singular value decomposition ofquaternion matricesrdquo Applied Mathematics and Computationvol 235 pp 157ndash167 2014
[26] Y Li M Wei F Zhang and J Zhao ldquoReal structure-preservingalgorithms of Householder based transformations for quater-nion matricesrdquo Journal of Computational and Applied Mathe-matics vol 305 pp 82ndash91 2016
[27] M Wang andW Ma ldquoA structure-preserving algorithm for thequaternion Cholesky decompositionrdquoAppliedMathematics andComputation vol 223 pp 354ndash361 2013
[28] ZWang A C Bovik H R Sheikh and E P Simoncelli ldquoImagequality assessment from error visibility to structural similarityrdquoIEEE Transactions on Image Processing vol 13 no 4 pp 600ndash612 2004
