Improving the performance of spatial watermarking of images using channel coding

qThis work has been partially supported by ComisioH n Interministerial de Ciencia y TecnologmHa under project TIC-96-0500-C10-10and by Xunta de Galicia under project PGIDT99 PX132203B.

*Corresponding author. Tel.: #34-986-812-124; fax: 34-986-812-116.E-mail addresses: [email protected] (J.R. HernaH ndez), [email protected] (J.M. RodrmHguez), [email protected] (F. PeH rez-

GonzaH lez).

Signal Processing 80 (2000) 1261}1279

Improving the performance of spatial watermarking of imagesusing channel codingq

J.R. HernaH ndez, J.M. RodrmHguez, F. PeH rez-GonzaH lez*

Dept. Tecnologn&as de las Comunicaciones, ETSI Telecom., Universidad de Vigo, 36200 Vigo, Spain

Received 28 November 1997; received in revised form 2 September 1998

Abstract

In this paper we present a spatial-domain watermarking scheme for data hiding and copyright protection of stillimages and analyze the e!ect that the application of channel coding has on the performance of the watermark detectionand decoding tests. Detector structures are derived for both tests and analytical bounds and approximations are obtainedfor the bit error rate (BER) and the receiver operating characteristic (ROC) associated with the watermark decoding anddetection tests when block codes are employed. Finally, the analytical expressions are contrasted with experimentalresults in several cases of interest. ( 2000 Elsevier Science B.V. All rights reserved.

Zusammenfassung

In diesem Artikel praK sentieren wir ein Schema im Raumbereich, das Wasserzeichen erzeugt, um Daten zu versteckenund das Copyright an unbewegten Bildern zu schuK tzen. Au{erdem analysieren wir den E!ekt, den die Anwendung derKanalcodierung auf die LeistungsfaK higkeit der Wasserzeichendetektions- und -dekodierungstests hat. Detektorstruk-turen werden sowohl fuK r Testals auch fuK r analytische Grenzen abgeleitet. FuK r den Fall von Blockkodes werdenApproximationen der Bitfehlerrate (BER) und der Detektorcharakteristik (ROC) gegeben, die mit den Wasserzeichen-dekodierungs- und -detektionstests assoziiert sind. Zuletzt werden die analytischen AusdruK cke experimentellen Ergebnis-sen fuK r mehrere interessierende FaK lle gegenuK bergestellt. ( 2000 Elsevier Science B.V. All rights reserved.

Re2 sume2

Nous preH sentons dans ce papier une meH thode pour faire des "ligranes dans le domaine spatial a"n de cacher desdonneH es et proteH ger les droits de proprieH teH (copyright) d'images "xes, nous analysons les e!ets que l'application de codagecanal a sur les performances concernant la deH tection et deH codage des "ligranes. Nous proposons des structures dedeH tecteurs aussi bien pour les tests que pour deH terminer les bandes analytiques. Nous donnons des approximations dutaux d'erreur par bit (BER) et de la caracteH ristique du fonctionnement de reH cepteur (ROC) associeH s au deH codage du"ligrane et aux tests de deH tection quand des codes blocs sont utiliseH s. Finalement, les expressions analytiques sontcompareH es aux reH sultats expeH rimentaux dans plusierus cas inteH ressants. ( 2000 Elsevier Science B.V. All rights reserved.

0165-1684/00/$ - see front matter ( 2000 Elsevier Science B.V. All rights reserved.PII: S 0 1 6 5 - 1 6 8 4 ( 0 0 ) 0 0 0 3 4 - 7

1. Introduction

The representation of images in digital formatallows the use of communication networks andelectronic media to e$ciently store, distribute andmanipulate visual information. However, these re-sources can also be applied to illegally manipulateand redistribute digital information at low cost.Therefore, the use of communication networkssuch as Internet for commercial distribution of im-ages is not possible if there is no mechanismguaranteeing the protection of authorship rights.

Encryption is not a solution to this problem,since once the information is decrypted it can bemisused by the recipient. Watermarking techniquesconsist in adding an invisible and secret signal,called watermark, to an image. The watermark canidentify the issuer and the intended recipient of theimage, so it can be used to determine who is respon-sible for the illegal use. The watermark must beimperceptible since the watermarked image will bedelivered to intended recipients as a substitute forthe original image. For security purposes, thewatermark must be robust against both uninten-tional distortions and attacks aimed at corruptingor deleting the watermark.

Previous research on copyright protection of stillimages has resulted in the appearance of severalmethods based on watermarking. In all these tech-niques the contents of the original image are alteredin a fashion determined by a secret key and, option-ally, by a certain amount of information to behidden into the image. Some of these methodsperform the watermarking process in the spatialdomain using spread spectrum techniques [4,11].Other methods add the watermark in the frequencydomain by computing the DCT of the whole image[3] or in a block basis [2,4,11,12].

Even though di!erent proposals for solving thecopyright enforcement problem have been de-scribed and tested with diverse results, previousresearch in watermarking techniques has su!eredfrom the absence of a theoretical approach to thelimits in performance of these methods. In thispaper we present a spatial-domain watermarkingscheme which allows hiding a binary sequence in animage. There are two tests involved in the water-mark veri"cation process. The "rst one, called

watermark detection test, consists in decidingwhether the image has been watermarked with cer-tain secret key. In the second test an estimate of thehidden information is obtained. We assume thatthe original image is not available in any of thesetests. Hence, the watermark can be veri"ed withoutthe need to distribute the original image. Our maingoal in this paper is to analyze how the introduc-tion of channel coding a!ects the performance ofthe watermark detection and decoding tests.

In Section 2 we present the 2D-multipulse modu-lation scheme and derive an equivalent Gaussianvector channel. In Section 3 we analyze the perfor-mance of the hidden information decoding testwhen channel coding is used. In Section 4 we obtaindetector structures for the watermark detection testand analyze their performance in terms of the prob-ability of false alarm (P

F) and the probability of

detection (PD). In Section 5 we discuss di!erent

kinds of attacks and how they can be analyzedusing the model derived in previous sections.Finally, in Section 6 we present results from simula-tions.

2. 2D-Multipulse amplitude modulation

2.1. Dexnitions

In a 2D-multipulse amplitude modulation water-marking scheme [5,6], the watermark w[m,n] canbe expressed as a linear combination of N ortho-gonal functions Mp

i[m, n]N, i3M1,2, NN:

w[m,n]"N+i/1

bipi[m, n], (1)

where the coe$cients b1,2, b

Nencode a hidden

message. The watermark w[m, n] is added to theoriginal image x[m, n] to obtain the watermarkedversion y[m, n]"x[m,n]#w[m, n]. In the schemewe are considering [5,6], the pulses p

i[m, n] are

de"ned as

pi[m, n]OG

a[m, n]s[m, n] if (m, n)3Si,

0 otherwise,(2)

where s[m, n] is the output of a pseudorandomsequence generator initialized to a state which

1262 J.R. Herna& ndez et al. / Signal Processing 80 (2000) 1261}1279

depends on the value of a secret key K, only knownto the copyright owner. The sets MS

iN indicate the

spatial shape of the pulses. We will assume in thesequel non-overlapping pulses, i.e. S

iWS

j"0,

∀iOj, so they will always be orthogonal. We alsopropose the use of key-dependent sparse sets MS

iN

scattered over the whole image to add spatial un-certainty about the location of each hidden in-formation bit and to increase the resilience tocropping attacks. The sequence a[m, n] is calledperceptual mask and shapes the output from thepseudorandom sequence generator so that invisi-bility is guaranteed. We have assumed that s[m, n]has unit variance for every (m, n), so a[m, n] indi-cates the maximum standard deviation that w[m, n]is allowed to have at each pixel.

The sequence a[m, n] is obtained via a perceptualanalysis of the image to be watermarked. First, theMasking Function M[m, n], described, in [7,9] isobtained using the expression proposed in [1],

M[m, n]"m`P+

k1/m~P

n`P+

k2/n~P

0.35J(k1~m)2`(k2~n)2

](Dx[k1#1, k

2]!x[k

1!1, k

2]D

#Dx[k1, k

2#1]!x[k

1, k

2!1]D), (3)

computed in a (2P#1)](2P#1) neighborhood of[m, n]. From here, the value of a2[m, n] is obtainedas

a2[m, n]"n7*4*"

ebM*m,n+, (4)

where n7*4*"

controls the noise visibility level andb is a settable real parameter.

The modulation technique presented in this sec-tion is similar to a direct-sequence spread spectrum(SS) scheme. However, the main di!erence withrespect to classical SS systems is that in our contextthe jammer is not limited to additive noise attacks.It can in fact play the role of a worst-case channelespecially designed to attack the hidden signalwithout perceptually degrading the image.

2.2. Equivalent vector channel

Due to the lack of good statistical models forimages, we will reduce the observation space to theprojection of the image onto the pulses Mp

iN. This

simpli"cation will lead to suboptimum detection

tests since the information in the subspace ortho-gonal to the pulses p

iis ignored. Nevertheless, the

information contained in the subspace spanned bythese pulses is enough to achieve fairly good perfor-mance in the detection tests. We assume that theoriginal image x[m, n] is not available in the detec-tion process and that the watermarked image is"ltered with a space-variant linear "lter h

kl[m, n],

which may be used to model a linear "ltering attackor a preprocessing step used to improve the per-formance of the detection process. The output ofthe linear "lter will be denoted by z[m, n]. Let usde"ne xk,l[m, n]Ox[m!k, n!l] and pk,l

i[m, n]O

pi[m!k, n!l]. Then,

riOSz, p

iT"

N+j/1

bj+k,l

Shk,l

pk,lj

, piT

#+k,l

Shk,l

xk,l, piT. (5)

Unfortunately, there are no satisfactory spatial-domain statistical models for common images. Asa consequence, the analysis of performancemeasures averaged over the statistical space formedby real-world images for a "xed K is infeasible.Alternatively, we can consider a given image asa "xed deterministic signal and base our analysison a statistical model of the secret key space. Fromthis new point of view, a random experiment consistsin taking a secret key at random, watermarking theimage under study with that key, passing the water-marked image through a jamming channel, and"nally testing the resulting image. We have followedthis approach in the analysis exposed in this paper.

Assume K is the only random variable in thewatermarking system. We will model the sequences[m, n] as a zero-mean i.i.d. random process withmarginal pdf f

s(s) and variance E[s2]"1. For

a "xed marginal pdf, the independence assumptionmaximizes the entropy, i.e. the amount of uncer-tainty when the secret key is not known. The en-tropy of the watermark can also be controlledthrough the choice of f

s(s).

As we have previously stated, we propose the useof sparse pulses randomly spread out over thewhole image. We will assume that the probabilityof a pixel (m, n) being assigned to any set S

iis 1/N

and that the assignment is done independently for

J.R. Herna& ndez et al. / Signal Processing 80 (2000) 1261}1279 1263

each pixel. Using central limit theorem arguments,it turns out that the vector r"(r

1,2, r

N) can be

accurately approximated as the output of an addi-tive Gaussian channel [5,6]

r"Ab#n, (6)

where b"(b1,2, b

N) is the vector of coe$cients

used to modulate the pulses, A is a deterministicdiagonal matrix and n is a zero-mean Gaussianrandom vector. Let C be the covariance matrix of n.The elements of A"[a

ij] and C"[c

ij] are [5,6]:

aij"d

ij

1

N+m,n

h0,0

[m, n]a2[m, n], (7)

cii"

1

N+m,n

a2[m, n]x2&[m, n]

#b2i

1

N+m,n

h20,0

[m, n]a4[m, n](E[s4]!1)

#b2i

1

N+

(k,l)E(0,0)

+m,n

h2k,l

[m, n]

]a2[m, n]a2[m!k, n!l]

#b2i

N!1

N2+m,n

h20,0

[m, n]a4[m, n], (8)

cij"!b

ibj

1

N2+m,n

h20,0

[m, n]a4[m, n], iOj, (9)

where dij

is the Kronecker delta function andx&[m, n] is the image "ltered by h

kl[m, n]. Even

though C is non-diagonal, the cross-covarianceterms are in practice small compared to the termsin the diagonal if N is large enough. Therefore,r can be accurately modeled as the output fromN uses of an AWGN (Additive white Gaussian noise)channel. We will assume in the sequel thatbi3M!1,1N, ∀i (binary antipodal constellation), so

the elements in the diagonal of A and C are inde-pendent of vector b. For simplicity, we will denotein the sequel aOa

ii, p2Oc

iifor any i3M1,2, NN.

3. Channel coding

3.1. Binary antipodal signaling

Suppose that codewords b(k), bi(k)3M1,!1N,

i3M1,2,NN, k3M1,2,2NN, in a binary antipodal

constellation are used to encode hidden messages.The bit-by-bit hard decoder is close to the optimalML detector since the cross-covariance terms in thenoise covariance matrix are negligible if N is largeenough. The probability of bit error averaged overall the keys for a given image is:

P""QA

a

pB. (10)

where Q(x)O1/J2n:=xe~t

2@2dt. Channel codes canbe used to improve the performance of the datahiding system in terms of the bit error probability.From Eqs. (7) and (8) we infer that the SNR ofthe equivalent channel decreases as we increase thelength of the encoded message. Hence, two are themain factors that determine the performance ofa code when applied to the Gaussian channel de-rived in Section 2.2: the minimum distance and theredundancy of the code. The best code for a givenminimum distance is the one with minimum re-dundancy.

3.2. Coding

If we use a bit-by-bit hard decoder, the result canbe modeled as the output of a BSC (binary symmet-ric channel) with transition probability P

", which

can be obtained from (10).The Bhattacharyya upper bound for the bit error

probability P#

of a linear (n, k) block code withminimum distance d

.*/when applied to this BSC is

P#)

M

2(M!1)

M+l/2

[4P"(1!P

")]wl @2, (11)

where M"2k is the number of codewords, wlis the

Hamming weight of the lth codeword, and l"1corresponds to the all-zeros codeword. In our ex-periments (Section 6) we have obtained the bit errorprobabilities using the approximation [10]

P#K

1

n

n+

m/t`1

m(nm)Pm

"(1!P

")n~m, (12)

where t is the maximum number of bit errors thatthe code can correct. Using (12) it is possible todetermine approximately the minimum number ofpixels per pulse for which P

#)P

"or, in other

words, the maximum value of N below which


coding is pro"table. Note that with P#"P

"in (12),

a polynomial equation results. The solution PH"

tothis equation can be used to invert (10), thus result-ing in a value of a/p from which N can be derived.

Given a certain desired message length, the op-timization of the channel coding scheme can lead todi!erent results depending on the characteristics ofthe image. This fact might lead us to think thatwatermark extraction would be impossible withouta priori knowledge of the code used. However,channel parameters a and p2 can be estimated fromthe watermarked image. If we assume that thehidden message length is a previously "xed systemparameter, then we can "nd the best code for thosechannel parameters and that message length. Alter-natively, we could perform a watermark detectiontest with all the possible coding schemes, since onlythe code actually used would lead to a positivedecision and in practice the number of choices willbe reasonably small.

Bounds similar to (11) can be found for convolu-tional codes as a function of the parameter d

&3%%.

The rate of the convolutional code also plays animportant role, since the addition of redundancyproduces a degradation of the SNR in the equiva-lent channel. The optimum ML detector is theminimum Euclidean distance sequence detector,since the channel is approximately Gaussian andwhite. Therefore, a Viterbi algorithm implementa-tion is applicable.

4. Sync recovery and watermark detection

So far we have assumed that the exact location ofthe pulses was known. However, attacks such ascropping and a$ne transforms may change thespatial location of the watermark. Assume that thewatermarked image z[m, n] may have su!ereda geometrical transformation ¹( ) ) with unknownparameters n for which we do not suppose anya priori distribution. Under this kind of attacks, thepulses generated during veri"cation and those ac-tually present in the geometrically transformedwatermark do not match, and as a result the statis-tics of the correlation coe$cients r

iare severely

degraded due to the white nature of the pseudoran-dom sequence. To solve this problem we can use

a synchronization algorithm to obtain an estimatenK of the unknown parameters so that we can applythe same transformation to the locally generatedmodulation pulses.

The watermark detection test, in which we decidewhether z[m, n] corresponds to an image water-marked with certain key K, can be formulated asthe binary hypothesis test

H1: z[m, n]"¹(x[m, n]#w[m, n], n),

H0: z[m, n]"¹(x[m, n], n).

(13)

A uniformly most powerful (UMP) test does notexist in general. However, we can exploit the factthat the synchronization recovery algorithm andthe watermark detection test are actually intimatelyrelated. When the former succeeds/fails to acquiresynchronization, we can infer that the image iswatermarked/not watermarked. Therefore, in thefollowing discussion we will refer to the watermarkdetection test and the synchronization algorithm asequivalent processes.

As we did in the decoder, we will use the correla-tion coe$cients r

i"Sz, p

iT as the inputs to the

watermark detection test. We will limit our analysisto geometrical transformations consisting in inte-ger spatial shifts. Suppose also that N

4out of the

N pulses are reserved for synchronization purposesand are thus modulated by known coe$cients (as-sume #1).

We propose the following synchronization algo-rithm. First, we derive the maximum-likelihood(ML) test assuming that an arbitrary tentative vec-tor of parameters n is correct. Then we compute theresulting-likelihood function at every possiblen and "nd its maximum. If this maximum is greaterthan a certain threshold then we infer that thealgorithm has acquired synchronization for thatn and, therefore, we decide H

1. This procedure is

equivalent to the test [5]

l(z)"maxn

M+i/1

f (r D b(i), n,H1)

f (r DH0,n)

H1

jH0

g, (14)

where b(i)"(b0(i),2, b

N(i)), i3M1,2, MN is an

antipodal vector used to encode one of M pos-sible messages. We will assume that b

j(i)3M!1,1N,

∀j3M1,2, NN, i3M1,2, MN. For every n the pdf


f (r D b(i), n,H1) can be approximated as a Gaussian

pdf with mean Ab(i) and covariance matrix C [5]. Ifthe image is not watermarked, then r"n, wheren is zero-mean white Gaussian noise with variance

p20"

1

N+m,n

a2[m, n]x2&[m, n]. (15)

The maximization of the likelihood function can bedi$cult to implement when general a$ne trans-forms are possible, since the maximum is verynarrow. Work is in progress with di!erent pulsegeneration techniques to allow the use of combinedbrute force search and gradient optimization algo-rithms. In the following sections we derive the like-lihood tests conditioned to n for di!erent channelcoding schemes.

4.1. Binary antipodal signaling

If we neglect the cross-covariance terms in C, thetest de"ned in (14) can be expressed as [5]

l(z)"maxn

N lnp0p

!

a2N

2p2#

1

2A1

p20

!

1

p2BN+i/1

r2i(n)

#

a

p2

N4

+i/1

biri(n)

#

N+

i/N4`1

ln coshAar

i(n)

p2 BH1

jH0

g, (16)

where ri(n)"Sz,¹(p

i,n)T. The coe$cients b

1,2,

bN4

are reserved for synchronization purposes andare therefore assumed to be known in the detector.

4.2. Coding

In this case the ML test conditioned to a certainn is not practical due to its high computationalcomplexity. Instead, we can de"ne the followingsuboptimal test: "rst, obtain an estimate bK of theencoded message using a hard decisor and a min-imum Hamming distance decoder; then, decide be-tween the two hypotheses

H1: r is watermarked with bK .

H0: r is not watermarked.

(17)

Then, the resulting watermark detection test is

l(z)"maxn

N lnp0p

!

a2N

2p2#

1

2A1

p20

!

1

p2BN+i/1

r2i(n)

#

a

p2

N4

+i/1

biri(n)

#

a

p2

N+

i/N4`1

ri(n)bK

i(n)

H1

jH0

g, (18)

where ri(n)"Sz,¹(p

i,n)T. The coe$cients b

1,2,

bN4

are reserved for synchronization purposes andare therefore assumed to be known in the detector.The suboptimal algorithm proposed for blockcodes is also applicable to convolutional codes andeither hard or soft decoding. In this case, bK isobtained using a Viterbi sequence detector and thewatermark detection test is still given by (18).

4.3. Performance evaluation

In this section we derive bounds for the probabil-ities of false alarm (P

F) and detection (P

D) condi-

tioned to a certain arbitrary vector n of geometricaltransformation parameters (Section 4) for thewatermark detector structures presented in pre-vious sections. The actual value of P

Fcan be

upper-bounded by the product of the number ofpoints n for which the likelihood function is evalu-ated during the maximization process and thePF

conditioned to any of these points n. If theconditioned P

Fis very small, then the actual value

of PD

is approximately equal to the PD

conditionedto any n. Therefore, the bounds that we will providefor the probabilities P

Fand P

Dconditioned to an

arbitrary n are of practical interest.

4.3.1. Binary antipodal signalingLet k(s)OlnE[esl(r)]. The probabilities of false

alarm (PF) and detection (P

D) conditioned to an

arbitrary n (Section 4) can be bounded using theCherno! bounds [13]

PF)ek(s)~sk5 (s), s'0,

PD*1!ek(s)`(1~s)k5 (s), s(1,

(19)


1k5 (s)O(d/ds)k(s).

where k5 (s)"g.1 If we use the expression in Eq. (16)to derive k(s), we get [5]

k(s)"AN lnp0p

!

a2N

2p2 Bs# (N!N

4) lnECe~s(r2@2)((1@p2)~(1@p2

0 ))coshsAar

p2BD#N

4lnE[e~s(r2@2)((1@p2)~(1@p2

0))es(ar@p2)], (20)

where the expectations are evaluated with respectto r&N(a,p2

0). Note that we have assumed

bi"1, ∀i3M1,2,N

4N.

4.3.2. Channel codes and hard decodingAssume that a (n, k) block code with minimum

distance d.*/

is employed. Let n#"(N!N

4)/n be

the number of codewords carried by the watermarkfor that code. The following theorem providesbounds for the P

Fand P

Dconditioned to an arbit-

rary n (Section 4) in such a situation.

Theorem 1. Let r be a real-valued N-dimensionalobservation vector. Let us dexne the binary hypothe-sis test

H1: r"abK #n1 ,

H0: r"n0 ,

(21)

where n1 and n0 are zero-mean Gaussian randomvectors with covariance matrices p2I

Nand p2

0IN,

respectively, and bK is a binary antipodal vector, i.e.bKi3M!1,1N ∀i3M1,2,NN, whose components can

be expressed in terms of an N-dimensional binarysequence x( consisting of n

#n-bit codewords from

a (n, k) block code with minimum distance d.*/

, andN

4bits reserved for synchronization, as bK

i"2x(

i!1,

i3M1,2,NN. The binary vector x( is obtained fromr using a bit-by-bit hard decisor followed by a min-imum Hamming distance binary decoder which ob-tains the closest encoded message. The probabilitiesof false alarm (P

F) and detection (P

D) can be bounded

as

PF)ek(s)~sk5 (s), s'0, (22)

PD*(1!P

%)n#(1!ek(s)`(1~s)k5 (s)), s(1, (23)

where

P%O1!q(1), (24)

k(s)On#k#(s)#N

4g(s), (25)

k#(s)O ln q(s)!s ln q(1)!(1!s) ln q(0)#ng(s),

(26)

q(s)Oxd.*/@2

y

+d/0

(nd)p(s)d,(1!p(s))n~d, (27)

p(s)OQAsap

0/p

Jsp20#(1!s)p2B, (28)

g(s)O1

2lnA

p2(1~s)p2s0

sp20#(1!s)p2B#

1

2

a2s(s!1)

sp20#(1!s)p2

(29)

and k5 (s)"g#n#ln(q(0)/q(1)).

Proof. See Appendix A. h

In these bounds we have neglected the probabil-ity of detection conditioned to an error in thedecoding step. In fact, the lower bound for P

Dtends

to (1!P%)n# as the threshold decreases towards

!R. A tighter lower bound for PD

can be ob-tained if the decoding errors are considered in thederivations.

Theorem 2. Let r be a real-valued N-dimensionalobservation vector. Let us dexne the binary hypo-thesis test

H1: r"abK #n

1,

H0: r"n

0,

(30)

where n1 and n0 are zero-mean Gaussian randomvectors with covariance matrices p2I

Nand p2

0IN,

respectively, and bK is a binary antipodal vector, i.e.bKi3M!1,1N ∀i3M1,2,NN, whose components can

be expressed in terms of an N-dimensional binarysequence x( consisting of n

#n-bit codewords from

a (n, k) block code with minimum distance d.*/

,and N

4bits reserved for synchronization, as

bKi"2x(

i!1, i3M1,2, NN. The binary vector x( is

obtained from r using a bit-by-bit hard decisor fol-lowed by a minimum Hamming distance binary de-coder which obtains the closest encoded message. The


probability of detection (PD) can be bounded as

PD*A

n+

w/0

n(w)A(0,w)Bn#!ek1 (s)~sk5 1 (s), s(0,

(31)

where k51(s)"g and

k1(s)On

#kc1

(s)#N4g(s#1), (32)

kc1

(s)O lnn+

w/0

n(w)A(s,w)eh(s,w)#ng(s#1), (33)

where n(w) is the number of codewords with Hammingweight w and

A(s,w)Oxd.*/@2

y

+d/0

.*/(d,w)+k/0

(wk)(n~wd~k

)p0(s)w~k(1!p

0(s))k

]p1(s)d~k(1!p

1(s))n~w~d`k, (34)

h(s,w)O!

2sa2wp20/p2

(s#1)p20!sp2

, (35)

p1(s)OQA

a(1#s)p0/p

J(1#s)p20!sp2B, (36)

p0(s)OQA

a(1!s)p0/p

J(1#s)p20!sp2B (37)

and g(s) is dexned in (29).

Proof. See Appendix B. h

This lower bound for PD

is tighter than thebound given in Theorem 1 for values of P

Dclose to

one and is clearly superior when P%

is not negli-gible. However, the bound in Theorem 1 is com-putationally simpler and is a good approximationwhen P

%is very small.

5. Attacks

The watermarked image may su!er attacksaimed at deleting or corrupting the watermark.To delete the watermark, the attacker transformsthe image into a perceptually similar version witha high probability of yielding a negative result inthe watermark detection test. To corrupt the water-mark, the attacker will transform the image with

the aim of increasing the probability of error in thedecoding process. In both cases the attacker will tryto degrade the resulting equivalent vector channelas much as possible, decreasing the probability ofdetection P

Dor increasing the decoding probability

of error P%. The equivalent channel we have derived

in Section 2.2 can be used to model the e!ect ofattacks such as linear "ltering, cropping and addi-tive noise. The "rst kind of attack is explicitlyrepresented by the space-variant linear "lterhkl[m, n]. An example of the application of linear

"ltering to watermark deletion attacks is the use ofa Wiener "lter to estimate the original image. Wehave studied the e!ect that an attack based on anadaptive Wiener "lter [7] has on the performanceof the decoder (Section 6).

The e!ect of cropping can be studied as a reduc-tion in the pulse size and, as a consequence, a de-crease in the SNR of the equivalent vector channel.The cropping attack a!ects all the hidden bits inapproximately the same amount, since the modula-tion pulses are sparse and randomly spread overthe whole image. This property facilitates the de-sign of error protection codes, which usually rely onindependence among noise components.

The watermarked image could also be attackedwith zero-mean additive white noise. If the noisevariance at pixel (m, n) is p2

n[m, n], then we can

analyze the e!ect of this attack simply adding theterm (+

m,na2[m, n]p2

n[m, n])/N to the variance de-

rived in Eq. (8).

6. Experimental results and comparisons

In Figs. 1}4 we can see the images used in theexperiments and the corresponding perceptualmasks. In all cases the empirical values have beenobtained by watermarking the image under testwith 100 keys and then averaging out the results.The two-dimensional pseudorandom sequences[m, n] used to generate the watermark is white andits marginal distribution is zero mean, discrete, with

four equiprobable levels M!J85,!J2

5,J2

5,J8

5N.

Figs. 5}7 show plots of the empirical BER (biterror rate) as a function of the number of pixels permessage information bit for the uncoded case


Fig. 1. Lena (256]256). (a) Original image, (b) perceptual mask a[m, n].

Fig. 2. Brick (160]200). (a) Original image, (b) perceptual mask a[m, n].

2The numbers (n, k) are the codeword length and the messagelength, respectively, of the block code.

(binary antipodal signaling) and Bose}Chaudhuri}Hocquenghem (BCH) block codes [8] BCH(63,10),BCH(63,36), BCH(63,57),2 whose minimum distan-ces d

.*/are 27, 11 and 1, respectively. The approxi-

mation to P#

given by (12) and based on theempirical P

"for the uncoded case is also shown for

these block codes. In Fig. 5 we show the BER forLena when Wiener "ltering is performed as a pre-processing step before detection to eliminate part ofthe noise due to the original image and assumingthat no attack is su!ered by the watermarked im-

age. Fig. 6 shows the BER when the image isattacked with additive Gaussian noise with vari-ance at each pixel shaped by the perceptual maskfor n

7*4*""8. Fig. 7 shows a similar plot for

a Wiener "ltering attack consisting in the subtrac-tion from the watermarked image of a linear esti-mate of the watermark. Two factors determine theperformance of a block code: the minimum distanceand the redundancy. Obviously, the greater theminimum distance, the better the performance.However, redundancy introduced by the codeto achieve that minimum distance results ina degradation of the SNR in the equivalent chan-nel. Therefore, the best code is the one with the


Fig. 3. b32 (160]200). (a) Original image, (b) perceptual mask a[m, n].

Fig. 4. Fabric (160]200). (a) Original image, (b) perceptual mask a[m, n].

Fig. 5. Bit error rate with Wiener "lter preprocessing prior todetection and no attacks, for Lena (256]256).

Fig. 6. Bit error rate with additive Gaussian noise attack andWiener "lter preprocessing prior to detection for Lena(256]256).


Fig. 7. Bit error rate with Wiener "ltering attack and Wiener"lter preprocessing prior to detection for Lena (256]256).

Fig. 8. Bit error rate with Wiener "lter preprocessing prior todetection and no attacks, for Lena (256]256).

Fig. 9. Bit error rate with additive Gaussian noise attack andWiener "lter preprocessing prior to detection for Lena(256]256).

Fig. 10. Bit error rate with Wiener "ltering attack and Wiener"lter preprocessing prior to detection for Lena (256]256).

most e$cient combination of minimum distanceand redundancy. In fact, we can see in the plots thatthe best code is BCH (63,36), which is neither theone with the largest minimum distance nor the onewith the smallest redundancy. We can also observethat coding results in an improvement in perfor-mance only when the number of pixels per informa-tion bit is greater than a certain minimum amount.

In Figs. 8}10 we show the empirical BER and thetheoretical approximations to P

"and P

#using ex-

pressions (7) and (8) to estimate the parametersa and p in Eq. (10) for Lena and the same channelcoding schemes as in previous plots. The slightdi!erence between the empirical and the theoreticalBER curves is due to small errors in the estimationof a and p2. Speci"cally, these errors are due to thefact that the coe$cients of Wiener "lter used asa preprocessing step are key-dependent, while thespace variant "lter considered in the theoreticalmodel was assumed to be independent of the key.


Fig. 11. Bit error rate with Wiener "lter preprocessing prior todetection and no attacks for brick (160]200), b32 (160]200)and fabric (160]200).

Fig. 12. Bit error rate with Wiener "lter preprocessing prior todetection and no attacks for brick (160]200), b32 (160]200)and fabric (160]200).

Fig. 13. Cherno! bound for the ROC of the watermark detec-tion test performed to Lena (256]256) with Wiener "lteringbefore detection and no attacks.

Similar experiments, not shown here, have beenperformed with key-independent preprocessing "l-ters, producing more accurate results.

In Fig. 11 we show the plots of the empiricalP"

for the uncoded case and the empirical P#

fora BCH(63,36) block code that resulted from testsperformed on the images brick, b32 and fabric (seeFigs. 2}4). We also show an approximation toP#

obtained applying the empirical P"

to Eq. (12).In Fig. 12 the same empirical curves are contrastedwith theoretical approximations to P

"and P

#based on the estimation of the equivalent channelparameters a and p using Eqs. (7) and (8). We canobserve in these plots that performance in terms ofBER is highly dependent on the characteristics ofthe image to be watermarked. In fact, we can seehigh di!erences between the pair of images brick,b32 and the pair Lena, fabric. The reason for thesedi!erences is that the "rst two images have noisytextures, while the second two images have many#at regions, resulting in higher values of the SNR atthe output of the Wiener preprocessing step. In-deed, we have con"rmed through experimentationthe striking fact that even though in image regionswhere there is high contrast in luminance thewatermark has a larger power level, the variance ofthe original image luminance causes the SNR inthose regions to be much worse than the SNR inareas with #at luminance.

Figs. 13}16 show the plots of the Cherno!bounds for the probability of false alarm (P

F) and

the probability of detection (PD) given in Theorem

2 computed for all the images under test. Curves areprovided for the uncoded case (binary antipodalsignaling) and a Golay (23,12) block code [8] formessage lengths of 6 0 and 240 bits. In all cases wehave not reserved any pulses for synchronization


Fig. 14. Cherno! bound for the ROC of the watermark detec-tion test performed to brick (160]200) with Wiener "lteringbefore detection and no attacks.

Fig. 15. Cherno! bound for the ROC of the watermark detec-tion test performed to b32 (160]200) with Wiener "lteringbefore detection and no attacks.

Fig. 16. Cherno! bound for the ROC of the watermark detec-tion test performed to fabric (160]200) with Wiener "lteringbefore detection and no attacks.

purposes. We have also plotted curves representingthe empirical P

Dversus the Cherno! bound for P

F(note that P

Fis extremely low and cannot be esti-

mated through simulation). The empirical valueshave been obtained by averaging over 100 keys.For a fair comparison between all the images,watermark detection tests have been performed toa 160]200 portion of Lena right in the center of

the image. We can observe how the ROC degradesas we increase the message length due to the de-crease in SNR of the equivalent channel. We canalso see how coding degrades the ROC for a "xedmessage length, due to the decrease in SNR asso-ciated with the redundancy introduced by the code.The degradation is more apparent for large mess-age lengths.

Again, discrepancies between the theoretical andempirical curves are due to small errors in theestimation of the equivalent channel parameters.Note that there are errors in the estimation of a, p2

and p20, required for the computation of the max-

imum likelihood function, from the watermarkedimages. Even in some cases (see Figs. 14 and 15)estimation errors cause the empirical curves to fallbelow the theoretical Cherno! bounds, whichshould be thus considered as approximationsrather than bounds. In this sense, note the di!erentbehavior of the theoretical ROC in images brick,b32 and in images Lena, fabric.

7. Conclusions

In this paper we have studied the application ofchannel coding schemes in a spatial-domain water-marking system for copyright protection of images.


We have obtained detector structures and analyti-cal approximations to the BER and the ROC whichcan be used to know the achievable performancefor a given image. These bounds have been contras-ted with simulation results performed to severalimages with several block codes. We have observedthat the performance of the codes depends on thecharacteristics of the image and that coding gain isobtained only above certain minimum pulse length.We have also observed that for a given bit-rate,coding results in a degraded ROC. These con-clusions can be easily extended to other codingschemes such as convolutional codes.

In our theoretical model we have also coveredattacks consisting in additive noise, cropping andspace-variant linear "ltering processes. Thus, thismodel allows us to study the impact of codingschemes on the performance when di!erent kinds ofattacks are su!ered by the watermarked image.Performance under JPEG compression can be ana-lyzed through a combination of block "ltering andquantization noise. However, only rough estimatescan be expected since JPEG is a transform-domaintechnique.

Results from experimentation have shown thatthe Gaussian assumption is accurate. Di!erencesbetween theoretical and empirical performancemeasures still are present due to errors in the es-timation of the equivalent channel parameters.Work is in progress to improve the accurateness ofthe estimates from both the original and the water-marked images. Speci"cally, better estimates wouldbring about theoretical ROC curves actually be-having as bounds for all the images under study.

We are also currently working on the extensionof the theoretical framework presented in this pa-per to transform-domain watermarking techniquesand the design of watermark generation schemesmore resilient against synchronization attacks suchas geometrical transformations.

Appendix A. Proof of Theorem 1

Assume that a (n, k) block code is used for errorprotection purposes. Let x

i"(x1

i,2, xn

i) be the ith

binary codeword in this code, where i3M1,2,2kN,and D

iLRn the decoding region associated with

the codeword xi. Let b

i, bj

i"2xj

i!1, j"1,2, n

be the real-valued antipodal n-dimensional vectorwhich is actually introduced into the equivalentGaussian vector channel. Assume "rst that onlyone codeword is carried by the watermark.

In the bit-by-bit hard decision step, the vector ris transformed into a binary word y

j3M1,0Nn,

j3M1,2,2nN. The decoding region associated withyj

is an orthant Oj"Mr3Rn D sgn(r

i)"2yi

j!1,

i"1,2, nN. Finally, yjis processed by a minimum

Hamming distance decoder and a binary codewordx(3Mx

1,2, x

2kN in the (n, k) code is decided. Thelikelihood function used to decide whether theimage is watermarked is

¸(r)"lnf (r D H

1, x( )

f (r DH0)

. (A.1)

Due to the hard decision step, each decoding regionD

iconsists of the union of a set of orthants in Rn.

The probabilities of detection (PD) and false alarm

(PF) can be expressed as

PD"

M+i/1

p(xi)M+j/1

PrMr3Dj,¸(r)'g D H

1, x

iN, (A.2)

PF"

M+i/1

PrMr3Di,¸(r)'g DH

0N, (A.3)

where M"2k and g is the decision threshold. If theprobability of error in the decoding process is su$-ciently small, we can neglect the terms for whichiOj in the summation in Eq. (A.2). With this sim-pli"cation, P

Dis lower bounded as

PD*

M+i/1

p(xi)PrMr3D

i,¸(r)'g DH

1, x

iN. (A.4)

Now we can obtain the Cherno! bounds [13]

PrM¸(r)'g D H0,D

iN)e~sgU

0(s,D

i), ∀s'0,

(A.5)

PrM¸(r)'g D H1, x

i, D

iN*1!e~sgU

1(s,D

i),

∀s(0, (A.6)

where U0(s, D

i)OE[esL D H

0,D

i] and U

1(s,D

i)O

E[esL DH1, x

i,D

i] are the moment generating


3dH(x

j, x

i) is the Hamming distance between binary words

xj

and xi.

functions conditioned to Di.

U0(s,D

i)"P

DiAf (r DH

1, x

i)

f (r DH0) B

sf (r DH

0,D

i) dr

"

1

P(DiD H

0)P

Di

f (r D H1, x

i)sf (r DH

0)1~sdr.

(A.7)

Let p0OP(D

iDH

0). We can exploit the fact that

f (r D H1, x

i) is a Gaussian pdf with covariance

matrix p2Inand centered at codeword x

i, f (r DH

0) is

a zero-mean Gaussian pdf with covariance matrixp20

In, and D

iis the union of a set of n-dimensional

orthants to get

U0(s,D

i)"

p~10

+Oj$Di

POj

expM!s/2p2DDr!abiDD2!(1!s)/2p2

0DDrDD2N

(2p)n@2psnp(1~s)n0

dr

(A.8)

which can be transformed after some manipula-tions into

U0(s,D

i)"p~1

0eng(s) +

Oj$DiPOj

f (r, s) dr, (A.9)

where f (r, s) is a Gaussian pdf N((sap20/(sp2

0#

(1!s)p2))bi,(p2p2

0/(sp2

0#(1!s)p2))I

n) and

g(s)O1

2lnA

p2(1~s)p2s0

sp20#(1!s)p2B#

1

2

a2s(s!1)

sp20#(1!s)p2

.

(A.10)

Let q(s,Di) be the summation in Eq. (A.9). Then,

q(s,Di)O +

Oj$DiPOj

f (r, s) dr

" +Oj$Di

p(s)dji (1!p(s))n~dji , (A.11)

where djiOd

H(x

j, x

i)3 and

p(s)OQAsap

0/p

Jsp20#(1!s)p2B. (A.12)

Note that p0"q(0,D

i). If the code is not per-

fect, the decoding region Di

can be intricate andthe moment generating functions U

0(s, D

i) and

U1(s,D

i) may be di$cult to evaluate. We can, how-

ever, obtain bounds of practical interest assumingsimpler decoding regions. We will consider regionsD@

iLD

iconsisting of all the orthants O

kLRn as-

sociated to n-bit words at Hamming distance atmost xd

.*//2y from the codeword x

i. All the points

r in D@ifor which ¸(r)'g, also lie inside D

i. Hence,

PrMr3Di, ¸(r)'g D H

1, x

iN

*PrMr3D@i, ¸(r)'g DH

1, x

iN. (A.13)

We can, therefore, substitute Diby D@

iin Eqs. (A.4)

and (A.6) to obtain a lower bound for PD. This kind

of argument is not applicable to PF, since in this

case we are interested in "nding an upper bound.However, as we show in the following lemma, wecan upper-bound the moment generating functionin Eq. (A.5) by conditioning the expectation to D@

iinstead of D

i.

Lemma 3. If Di

and D@i

are such that D@iLD

iand

dH( y

j, y

i))d

H(y

l, y

i), ∀O

jLD@

i, O

lLD

iCD@

i, then

U0(s,D

i))U

0(s, D@

i), ∀s'0. (A.14)

Proof. From Eqs. (A.9) and (A.11) we can infer that

U0(s,D

i)"

q(s,Di)

q(0,Di)eng(s)"

q(s,D@i)#q(s,D

iCD@

i)

q(0,D@i)#q(0,D

iCD@

i)eng(s)

"

1#(q(s,DiCD@

i)/q(s,D@

i))

1#(q(0,DiCD@

i)/q(0,D@

i))

q(s,D@i)

q(0,D@i)eng(s)

"

1#(q(s,DiCD@

i)/q(s,D@

i))

1#(q(0,DiCD@

i)/q(0,D@

i))U(s,D@

i). (A.15)

Now, using Eq. (A.11), we have

q(s,DiCD@

i)

q(s,D@i)

"

+Ol$Di

CD{ip(s)dli(1!p(s))n~dli

+Oj$D{i

p(s)dji(1!p(s))n~dji

)

p(s)d1(1!p(s))n~d1 DDiCD@

iD

p(s)d2(1!p(s))n~d2 DD@iD

"Ap(s)

1!p(s)Bd1~d2 DD

iCD@

iD

DD@iD

, (A.16)


where

d1" min

Ol$DiCD{i

dli, d

2" max

Oj$D{i

dji

(A.17)

and DDiCD@

iD, DD@

iD are the number of orthants in

DiCD@

iand D@

i, respectively. The inequality is due

to the fact that p(s)d(1!p(s))n~d is non-increasing ind for all s'0 since p(s))1

2, ∀s'0. If d

ji)d

li,

∀OjLD@

i, O

lLD

iCD@

ithen d

1*d

2. Since

p(s))12, then

q(s,DiCD@

i)

q(s,D@i)

)

DDiCD@

iD

DD@iD

"

q(0,DiCD@

i)

q(0,D@i)

, ∀s'0.

(A.18)

Applying this inequality in Eq. (A.15) we infer that

U0(s,D

i))U

0(s, D@

i), ∀s'0, (A.19)

which completes the proof. h

Therefore, we can transform Eq. (A.5) into

PrM¸(r)'g D H0,D

iN)e~sgU

0(s,D@

i), ∀s'0,

(A.20)

which is nothing but the Cherno! bound condi-tioned to D@

iinstead of D

i. In addition, this new

bound can be computed even for non-perfect blockcodes. Let us now proceed with U

1(s,D

i):

U1(s,D

i)"P

DiAf (r DH

1, x

i)

f (r DH0) B

sf (r DH

1, x

i,D

i) dr

"

1

P(DiD H

1, x

i)

] +Oj$Di

POj

f (r D H1, x

i)(s`1)f(r DH

0)~sdr.

(A.21)

If we examine the de"nition of q(s,Di) in (A.11) we

can see that P(DiD H

1, x

i)"q(1,D

i). Furthermore,

comparing (A.21) with (A.7) we can infer that

U1(s,D

i)"

q(0,Di)

q(1,Di)U

0(s#1,D

i)

"

q(s#1,Di)

q(1,Di)

eng(s`1). (A.22)

Now, applying the inequalities in (A.13) and (A.6)we have that

PrMr3Di,¸(r)'g D H

1, x

iN

*P(D@iD H

1, x

i)(1!e~gsU

1(s, D@

i))

" q(1,D@i)A1!e~gs

q(0,D@i)

q(1,D@i)U

0(s#1,D@

i)B,

∀s(0. (A.23)

After a change of variable, this expression is equiva-lent to

PrMr3Di,¸(r)'g D H

1, x

iN

*q(1,D@i)!q(0,D@

i)eg(1~s)U

0(s,D@

i),

∀s(1. (A.24)

Note that the expressions in Eqs. (A.20) and (A.24)are independent of i, since the regions D@

ihave

the same structure for all i3M1,2,2kN. Hence,we can de"ne the functions q(s)Oq(s,D@

i) and

U0(s)OU

0(s,D@

i) for any i3M1,2,2kN. Now, let us

de"ne the function

k#(s)OlnU

0(s)#s ln

q(0)

q(1). (A.25)

Then, the Cherno! bounds for PF

and PD

are

PF)e~gs`k# (s)~s -/(q(0)@q(1)), ∀s'0 (A.26)

PD*q(1)!q(0)e(1~s)g`k# (s)~s -/(q(0)@q(1)),

∀s(1, (A.27)

where we have used the fact that +2k

i/1P(D

iDH

0)

"1 and +2k

i/1p(x

i)"1.

The best bounds within the allowed values ofs correspond to the minimum of the right-hand sideof Eq. (A.26) and the maximum of the right-handside of Eq. (A.27). It can be easily veri"ed that bothextrema occur at the value of s which satis"es

g"k5#(s)!ln

q(0)

q(1). (A.28)

Therefore, the "nal expressions for the Cherno!bounds are

PF)ek# (s)~sk5 # (s), ∀s'0 (A.29)

PD*q(1)(1!ek# (s)`(1~s)k5 # (s)), ∀s(1, (A.30)


where k5#(s)"g#ln q(0)/q(1). If the watermark car-

ries n#

codewords of the (n, k) block code andN

4pulses are reserved for synchronization pur-

poses, then the simpli"ed decoding regions havethe form E

l"RN4]D@

i1]2]D@

in#, i

j3M1,2,2kN,

∀j3M1,2, n#N, l3M1,2,2n#kN and the correspond-

ing moment generating function is

U0(s,E

l)"U

0(s,RN4)

n#<j/1

U0(s,D@

ij), (A.31)

where

U0(s,RN4 )"

q(s,RN4 )

q(0,RN4 )eN4g(s)"eN4g(s). (A.32)

Then, if we de"ne the function

k(s)OlnU0(s, E

l)#s ln

q(0,El)

q(1,El)

"n#k#(s)#N

4g(s), (A.33)

then the resulting Cherno! bounds are

PF)ek(s)~sk5 (s), ∀s'0, (A.34)

PD*q(1)n# (1!ek(s)`(1~s)k5 (s)), ∀s(1, (A.35)

where k5 (s)"g#n#ln q(0)/q(1). This completes the

proof of Theorem 1. h

Appendix B. Proof of Theorem 2

First, recall that the probability of detection isde"ned as

PD"

2k

+i/1

p(xi)2k

+j/1

PrMr3Dj,¸(r)'g D H

1, x

iN. (B.1)

If the block code is not perfect, the computation ofCherno! bounds based on the actual decoding re-gions D

imay be infeasible. As we did in Appen-

dix A, we will "nd bounds based on regions D@i

consisting of all the orthants Ol

satisfyingdH(y

l, y

i))xd

.*//2y, since any point r in D@

isuch

that ¸(r)'g also lies inside Diand, therefore,

PrMr3Dj,¸(r)'g DH

1, x

iN

*PrMr3D@j,¸(r)'g D H

1, x

iN,

∀j3M1,2,2kN. (B.2)

Now we can use the Cherno! bound to obtain theinequality

PrM¸(r)'g D H1, x

i, D@

jN*1!e~gsUi

1(s,D@

j),

∀s(0, (B.3)

where

Ui1(s,D@

j)OE[esL DH

1, x

i, D@

j]

"PD

@jAf (r DH

1, x

j)

f (r DH0) B

sf (r DH

1, x

i,D@

j) dr

"

1

P(D@jDH

1, x

i)

+Ol$D

@j

POl

expM!s/2p2DDr!abjDD2!1/2p2DDr!ab

iDD2#s/2p2

0DDrDD2N

(2p)n@2pn(s`1)p~ns0

dr.

(B.4)

After some manipulations, we can transform thefunction inside the integral into a Gaussian pdf,obtaining the expression

Ui1(s,D@

j)"

eng1(s)`h(s,dij )

P(D@jDH

1, x

i)

+Ol$D{j

POl

fij(r, s) dr, (B.5)

where

fij(r, s)&NA

ap20(sb

j#b

i)

(s#1)p20!sp2

,p2p2

0(s#1)p2

0!sp2

InB

and

g1(s)O

1

2lnA

p~2sp2(s`1)0

(s#1)p20!sp2B#

1

2

a2s(s#1)

(s#1)p20!sp2

,

(B.6)

h(s, dij)O!

2sa2dijp20/p2

(s#1)p20!sp2

. (B.7)

Comparing Eqs. (B.6) and (A.10) we can see thatg1(s)"g(s#1). Let us de"ne the function

A(s,dij)O +

Ol$D@jPOl

fij(r, s) dr. (B.8)

For a given pair i, j there are dij

components ofr with marginal pdf

f0(r)&NA

ap20(1!s)

(s#1)p20!sp2

xmi,

p2p20

(s#1)p20!sp2B


and n!dij

components whose marginal pdf is

f1(r)&NA

ap20(1#s)

(s#1)p20!sp2

xmi,

p2p20

(s#1)p20!sp2B.

Let us denote the "rst group by A0

and the secondone by A

1. Consider now all the orthants O

lsuch

that dH( y

l, y

i)"d where 0)d)xd

.*//2y. There

are (nd) possible bit con"gurations in which y

land

yi

can di!er. If we assume that k di!ering bits liewithin A

0and d!k within A

1, the number of

possible con"gurations is (dijk

)(n~dijd~k

). Each of thesecon"gurations has d

ij!k di!ering bits in A

0, k

equal bits in A0, d!k di!ering bits in A

1and

n!dij!d#k equal bits in A

1. Adding all the

contributions for all the possible values of k andd we have

A(s,dij)"

xd.*/ @2y+d/0

.*/(d,dij )+k/0

(dijk

)(n~dijd~k

)p0(s)dij~k

](1!p0(s))kp

1(s)d~k(1!p

1(s))n~dij~d`k,

(B.9)

where

p1(s)OQA

a(1#s)p0/p

J(1#s)p20!sp2B, (B.10)

p0(s)OQA

a(1!s)p0/p

J(1#s)p20!sp2B. (B.11)

Note that P(D@jDH

1, x

i)"A(0, d

ij), so

Ui1(s,D@

j)"

A(s, dij)

A(0, dij)eng(s`1)`h(s,dij ). (B.12)

Therefore, PD

can be bounded as

PD*

2k

+i/1

p(xi)2k

+j/1

A(0, dij)A1!min

s:0

e~gsUi1(s,D@

j)B.

(B.13)

We can observe that all the codewords xjwith the

same Hamming distance to xi, have the same mo-

ment generating function Ui1(s,D@

j). In fact, the

function Ui1(s, D@

j) has d

ijas the only parameter that

determines its shape. Hence, the minimum in Eq.(B.13) is achieved at the same s for those momentgenerating functions with the same d

ij. Further-

more, if the code is linear, the set of possible valuesof d

ijis the same for every x

iand is determined

by the Hamming weight distribution of thecode. Let n(w) be the number of codewordswith Hamming weight w. Let us de"neU

1(s,w)O(A(s,w)/A(0,w))eng(s`1)`h(s,w). Then,

PD*

n+

w/0

n(w)A(0,w)A1!mins:0

e~gsU1(s, w)B, (B.14)

where we have used the fact that +2k

i/1p(x

i)"1.

Therefore, it is necessary to compute as manyvalues of s as di!erent values of w exist in the weightdistribution. In a practical situation, this techniqueis computationally complex. We can instead ma-nipulate Eq. (B.14) to get

PD*

n+

w/0

n(w)A(0,w)A1!mins:0

e~gsU1(s, w)B (B.15)

*

n+

w/0

n(w)A(0,w)!mins:0

e~gsn+

w/0

n(w)A(0,w)

!U1(s,w). (B.16)

This new bound requires the computation of onlyone minimum, resulting in a considerable reductionin computational complexity. Let us de"ne thefunction

kc1

(s)Olnn+

w/0

n(w)A(0,w)U1(s,w) (B.17)

*ng(s#1)#lnn+

w/0

n(w)A(s,w)eh(s,w). (B.18)

Then, the minimum in Eq. (B.16) is achieved at thevalue of s which satis"es k5

1(s)"g and, hence,

PD*

n+

w/0

n(w)A(0,w)!ek1 (s)~sk5 1 (s), (B.19)

where k51(s)"g. Now suppose that the watermark

carries n#

codewords belonging to the (n, k) blockcode plus N

4bits reserved for synchronization pur-

poses. Following arguments similar to those usedin Appendix A, we can de"ne the function

k1(s)On

#kc1

(s)#N4g(s#1). (B.20)


With this de"nition, the Cherno! bound in thiscase is

PD*A

n+

w/0

n(w)A(0,w)Bn#!ek1 (s)~sk5 1(s), (B.21)

where k51(s)"g, and this completes the proof of

Theorem 2. h

References

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[3] I.J. Cox, J. Kilian, T. Leighton, T. Shamoon, Secure spreadspectrum watermarking for multimedia, Technical Report95-10, NEC Research Institute, Princeton, NJ, USA, 1995.

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[8] F.J. MacWilliams, N.J.A. Sloane, The Theory of ErrorCorrecting Codes, Vol. 16, North-Holland MathematicalLibrary, Amsterdam, The Netherlands, 1977.

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[12] M.D. Swanson, B. Zhu, A.H. Tew"k, Transparent robustimage watermarking, Proceedings IEEE InternationalConference on Image Processing, Vol. III, Lausanne, Swit-zerland, September 1996, pp. 211}214.

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Improving the performance of spatial watermarking of images using channel coding

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