Watermarking Polygonal Lines Using V-Descriptors

Xiaoyun Tie', Jiancheng Zoul, Wenqi Zhong', Dongxu Qi2IInstitute ofImage Processing and Pattern Recognition, North China University of Technology,

Beijing, Chinatiexy@ncut. edu. cn, jianchengzou@sohu. com, qingyunzhichi@tom. com

2Macau University ofScience and Technology, Macau, Chinadxqi@,must. edu.mo

Abstract

Based on the V- descriptors, a novel watermarkingalgorithm to protect the copyright of vector images isproposed in this paper. Watermarks are embedded inthe image's contours. Experiments show that thealgorithm is robust against some attacks such as

translation, scaling, rotation and local revision.

Keywords: Watermarking, Descriptor, informationsecurity

1. Introduction

The digital information revolution has brought aboutprofound changes in our society and our lives. Themany advantages of digital information have alsogenerated new challenges and new opportunities forinnovation. Digital products are very easy to copy,

reproduce, and maliciously process in a networkenvironment. Over the past decade, watermarking hasemerged as an important technology for protectingcopyright of multimedia products [1] . Currentwatermarking technology focuses on media types likestill image, and video and audio streams. However more

and more CAD -based 2D and 3D data are entering theWorld Wide Web, mostly as Virtual Reality ModelingLanguage scenes [2 5]. Many multimedia applicationsuse vector graphics image for example, digital maps,

geographic information systems (GIS) data, 2Dgraphics, and cartoon image. Classical watermarkingmethods embed a watermark by applying luminancealterations to the image. These methods are not suitablefor vector graphics images because in such imageswatermarks are perceivable and easily removed [6 7]

The most important information of vector graphics isstored in the contours. A watermarking method basedon Fourier descriptors is presented in [6]. Anotherwatermarking method based on complex wavelet isproposed in [7]. Based on the V- descriptors we propose

a novel watermarking algorithm to protect the copyrightof vector images in this paper. We embed watermarksin the image's contours. Experiments show that thealgorithm is robust against some attacks such as

translation, scaling, rotation and local revision.

2. V Systems and V Descriptors

2.1 V Systems

Orthogonal transformations are powerfulmathematical tools and have a lot of applications insignal processing. Orthogonal functions can be used inmany fields such as signal processing, engineeringcomputation, spectral analysis, etc.

The well known orthogonal functions are Fouriertriangular functions, polynomial orthogonal functions,Walsh functions, Haar functions, etc. Qi and Feng ([8-11]) explicitly constructed a kind of completeorthogonal function system - U systems. Song and Qi[12] constructed another kind of complete orthogonalfunction system - V systems, composed by a series ofpiecewise polynomials of degree k (k = 0, 1, 2, . . ),they are called the V-system of degree k. The V-systemconsists of not only smooth functions, but alsodiscontinuous functions at multi-levels. The Haarsystem is just a special case of the V-system with k = 0.

Definition [9]: If the k + 1 functions(x), i = 1,2,., k + 1} defined on the interval [0,1]

satisfies the following three properties

(1)fi(x) is a piecewise polynomial of degree

k with x=2as= a knot point, and f(x) is Ck-

continuous at X = , i =1,2,, kk+1;

(2)< t(x),f,(x) >= o5lj jEi,j 12 ,k }(3)< f(x),xi >= Oe{1,2, k +1},je {O,l,. ,k}.Where < ,@> denotes the inner product in L2 [0,1],

then we call {fi (x) , i = 1,2, , k + 1} generator

functions of degree k .

It's obvious that we can deduce the generatorfunctions of degree k from the coefficients of some

[10-12]equationsFor example, there is only one generator function ofzero degree

1-4244-0971-3/07/$25.00 ©)2007 IEEE.

1, xe[Oj).f(x) = 2 9

f -1, xe (2,1]There are two generator functions of degree one:

f1 (x)= (1-4x), O.<xlbV3 (4x-3), ±<x.l.

f2 (x) ={16x,O<x< 1

2 -X<12There are three generator functions of degree two:

V-5(16X2 _IOx+l), O<x<f1(x) = ( ) 2

V5[-16(1 -x)2 + 10(1 x) 1], I< x < 1L ~~~~~~~2

FX (30x2 -14x+1), <x<f2(x)= 1 2

[30(1 x)2 -14(1 -x)+1], <x<1L ~~~~~~~2

{40X2 -16x+1,f (y) =

0<x< 12

J 3 kA -]

40(1-x) +16(1-x)+1, 2 <x<1

The function values of the above functions at thediscontinuous point take the average value of the leftand right values.

Now let V,1 (x),V,71(x), , Vk', (x) are the former

k +1 Legendre polynomials in the interval [0,1 ], thenwe can construct generators with order k .

{&,2 (X) , i = 1,2., k + 1}, which are orthogonalwith each other, and they are also orthogonalwith VA'l (x), Vk2l (x), * , (x) . Then the V- systemsofdere redfieahVfllwnof degree n are defined as the following

V2-,-2V i -2 2x 2)1'V17I(x) =k

,V' 2[2 2n(X 2 )]

O,

-1 1

2 ' 2n-2else

i=12..,k+1, Ij=1,2, ,2 325,

2.2 Piecewise Linear V- systems

The general formula of the V - System of degree oneare the following

V2 1 2 [2 (x n2j10, else

0 [4j 3-2 x) x E ( 2n-2 2 n-I

V3(2n2 n

x-4j + 1), x E (2j-1 J )

V2'n(x) e 2

22(X_2x0, else

F 22(61j5-3.2"x) XE(n

1 2] -1

2`2(6j-1 32x)3 xe(21 )

O, else

I=1,2,",2 2;n- = 3,4,***' ~~~~~~~~~~~~~~~~~~~L S _a T,~~~~~~~~~~~~~~~~~~~~~~~~~2!| | ~~~~~~~~~~~~~~~_ LI!2~~~~~~~~~~I :-,3~

Nr~Nr~

Figure 1: the former 16 base functions of V Systems o1

(11 j .n

Where~~~~~~~~N

(ii)VSytem ofdegreek arecopeeiL2[,]

(iv) FourePr-Vperties Reproduction:_IfFnisraVpiecewis

piTsefonmer-qe(qiandriearie insyteges) ithane bhe

serloies. rpetisTheLcliyk-dgresuprVssetem ,hasmuhandt

2n-2~ ~ ~ ~ v1 2

discontinuity 12,,,,inoraio. Eseial using.

reproducSytionmw candoge somre forequienyoanalysisao

geometricin formation2.VDescripto

<~~ Vkn, , k m

Vvlheredegeeon

2.3i TheSystertisofdegeewisae mLeteain Systems

The fourier-VmerisRpoution:ed ipiecewiseVsseshv h

folloingprpetis

di)sLontinity. Thesupormtiston. Eseial usins

reproduction,tem candoge ksare norequien analsisofoageometricn infoL2ation]2.4 ~ ~ <kVZ'Descriptor

Given a graph set of the planar curve

P(t) = x(t) + iy(t) with 2n pieces,where i = -1 , x(t), y(t) are piecewise

0, else

polynomials of degree k. x(t), y(t) can be exactlyexpressed by the V- systems of degree k. By dividinguniformly the interval [0,1] into 2n subintervals, wecan map X(t), y(t) to the subintervals respectively:

X(t) = Xj (t), tE [j2,j')},j=0,1,.*.,2 1,

y(t) = {yj (t), t El [ j )} j = 0,1, .,2 1,Where xj (t), yj (t) are the polynomials of degree

over [ j2 -j)nl By the reproducibility of the V- series,

we have(k+1)2 -1 (k+1)2n1

P(t) =x(t) + iy(t) = IE (jvIt+ )jv t/=0 /=0

Where Ai$ fx(t)vj(t)dt i = fy(t)vj(t)dtj = 0 ,1,2, -, (k + 1)2 n -1I

Denote i(j) = ,x + i,%{, then A(j) is called as the

j - th descriptor ofP(t), namelyu(i =) x(t)vj (t)dt + i I y(t)vj (t)dt JIP(t)vj (t)dt

j = 0,1,2, -.., (k + 1)2n -1. (2)

then P(t) (1)21k+l=o,(j)v (t)

2.5 The Properties of V- descriptor

set d(j)= AU) , j=

Then d(j) is the j-th normalized V- descriptor

ofP(t), j = 1,2, .

Theorem ([9]): () By translation zo, scaling a and

rotation 0, the descriptor A(j) is the following

i'(j) = ae'o [A(j)+zo 9(j)]. j = 0,1,2,**

where(j) j

(ii) The normalized descriptor d(j) (j = 1,2,are invariant under translation, scaling and rotationtransformations.

3. The Watermarking Algorithm

In the following we give a watermarking embeddingalgorithm for commercial bill based on one degree Vsystem, by according to the sign of matrix coefficientafter orthogonal transform.

3.1 The flow chart of the watermarkingalgorithm

Feature po i nts0 Contour __ 2D graphextract d t

V transform

Watermark embedding

Ve s

Watrmark Random number keydescriptors + ~signals j generator

Inverse V transform

WatermarkedWaemkendntnl (, 2D graph

Figure 2 the flow chart ofwatermarking embedding

Feature Xpointsextract

I CDntourextraction

V tansform- Wate-mark

sigals

Water-nard-Extract

CalcMlatecorrelation

A Water=narked2D graph

R-ndomn numbergenerator I key

Judge if there iswatemiark or not

Figure 3 the flow chart ofwatermark detection

3.2 Watermarking Algorithm

3.2.1 Extracting the contour of the graphics image

For a given image, the contour is a closed polygonalline that consists of N vertices. By the polygonalapproximation algorithm, we can also reconstruct theimage based on the feature points of the polygonal.

3.2.2 Calculate the V- descriptors

Based on the above polygonal P(t), suppose P1 (t)be the polygonal line between two adjacent featurepoints(Xj (t), yjy (t)), (xj,l (t), yj,l (t)), j = 0,1, ..* N - I,then we can calculate the V -descriptors ofP1 (t)= xj (t) + iyj (t) according to formula (2),

where i = 1 , x(t), y(t) are piecewisepolynomials.

3.2.3 Watermark Construction and embedding

Using a pseudorandom generator, we create a bi-valued ± 1 random sequence and use it to produce awatermark. A bi-valued ± 1 random sequence has zeromean value and unit variance. By changing thecoefficients of the V- descriptors, we can embed awatermark into the polygonal.

V'(k) = V(k) ' p x W(k) (3)

Where p is a factor that determines the watermarkpower, w is the watermark signal, which is a bi-valued

± 1 random sequence controlled by a secret key, V' isthe watermarked V-descriptors ofthe polygonal lines.

3.2.4 Watermark Detection

The correlation coefficient c between V' and Wcan be used to detect the watermark's presence.. Thedetection formula is the following:

Figure 5 watermarked polygonalwith

Figure 6 anti clock rotation

90 degree to figure 5

v = V'-v W=W-W

V*WC = IV~:7 -*-, W-~* (4)v Iw*w Figure 7 X-direction

translation to figure 5Figure 8 y-directiontranslation to figure 5

Where V and W are the mean values of V andW respectively, V' is the V- descriptor of thecandidate. According to the value ofc, we can detect ifthere is watermark or not. We can think watermark existif c is greater than a theoretical mean value. Otherwiseno watermark or watermark destroyed. In this paper wetake = 0.7.

4. Simulation Results

Experiments show that the algorithms presented hereis robust to the general geometrical transformation suchas rotation, translation, scaling and local revision.

(a) is the original image with size 128X128, (b) is thecontour polygonal of image (a), which has 41 featurepoints, (c) is the watermarked polygonal. Thewatermark is a bi-valued ± 1 random sequence that haszero mean value and unit variance.

(a) Original image (b) original polygonal (c) watermarked polygonalFigure 4

In the following we test the robustness of thewatermarking algorithm by some attacks to thewatermarked polygonal, such as translation, rotation,scaling and local revision. Figure 5 is the watermarkedcontour of the original image. Figure 6 is the image byanti clock rotation 90 degree to figure 5, figure 7 andfigure 8 are the images by X-direction and Y- directiontranslating to figure 5 respectively. Figure 9 is thescaled image by 0.75times to figure 5. Figure 10 is theimage revised 8 feature points to figure 5. Diagram 1 isthe results of robustness tests (d is 0.7) described bythe correlation. Experiments show that the algorithm isrobust against some attacks such as translation, scaling,rotation and local revision.

Figure 9 scale 0.75times Figure 10 local revisionof figure 5 to figure 5

Diagram 1 the results of robustness tests ((5 is 0.7)uge is O Vmc tle 6 piLs m aiskt 6 pls m Re 9 ScaleLct 1

chanes m.g e d > a x-direLion y-decLimn de gee O0.75txs revisn

blmltian 0.0000 0.7069 0.7034 0.7057 0.7048 0.7068 0OfO68

5. ConclusionBased on the V-descriptor we introduce a novel

watermarking algorithm in this paper. Experimentsshow that the algorithm is robust against some attackssuch as translation, scaling, rotation and local revision.So this algorithm can be used to protect the copyright ofvector images. In the future we should compare theknown watermarking algorithms with ours and find theadvantages and the disadvantages of each algorithm tospecific application backgrounds.

6. Acknowledgements

This work was supported by the Development Fundof Science and Technology of Macau (045/2006/A),National 973 program of China (2002CB312104), NSFof China (No. 10671002), NSF of Beijing (1062006),PHR(IHLB) and the grant for those retuned fromoverseas.

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