Abstract—Digital image watermark can be detected in

transform domain using maximum likelihood (ML) detection,

whereby the decision threshold is obtained using the Neyman-

Pearson criterion. A probability distribution function (PDF) is

required to correctly model the statistical behavior of the

transform coefficients. In the literature, this detection method

has been considered by modeling magnitude of a set of discrete

Fourier transform (DFT) coefficients using a Weibull PDF. In

this paper, we propose extending the Weibull model to a

generalized gamma model. For the work here, we also propose

new estimators for parameters of the generalized gamma PDF.

I. INTRODUCTION

Maximum likelihood (ML) detection scheme based on

Bayes’ decision theory has been considered for image

watermarking in transform domain [1], [5]-[7]. The Neyman-

Pearson criterion is used to derive a decision threshold to

minimize the probability of missed detection subject to a

given probability of false alarm. To achieve optimum behavior

of the ML detector, a probability distribution function (PDF)

that models correctly the distribution of the transform

coefficients is required. In [1], magnitude of a set of discrete

Fourier transform (DFT) coefficients is modeled using a

Weibull PDF. Discrete wavelet transform (DWT)

coefficients are modeled using a Gaussian PDF in [5], a zero

mean generalized Gaussian PDF in [6], and a Laplacian PDF

in [7]. Here, we consider extending the Weibull model of [1]

to a generalized gamma model.

The paper is organized as follows. The embedding rule

used with ML detection is given in Section II. The decision

rule and threshold of ML detection are briefly explained in

Section III. A more detailed description can be found in [1]

and [6]. Section IV shows the modeling of the magnitude of

the DFT coefficients by a generalized gamma PDF. We

propose a new method to estimate parameters of the

generalized gamma PDF in Section V. Experimental results

are described in Section VI and our work is concluded in

Section VII.

The notation used is as follows. Non-bold letters are used to

represent scalar quantities, sets and functions. Bold letters are

used for vectors. All vectors are real-valued and expressed in

column form.

II. WATERMARK INSERTION

LetT

Nxxx 21x andT

Nyyy 21y be

N-vectors representing magnitude of a set of DFT coefficients

of an original image and the associated watermarked image,

respectively. A watermark T

Nwww **2

*1

*w ,

chosen from a given set, is embedded into x giving y. The

elements *, ii wx and iy are viewed as realizations of the

random variables ,iX *iW and iY , with underlying PDFs

)( iX xfi

, )( ** iW

wfi

and )( iY yfi

, respectively, for

Ni ,,2,1 .

In ML detection scheme, *w is usually embedded into x

using the multiplicative rule, defined as

),1( *iiii wxy ,,,2,1 Ni (1)

where i is a positive scalar representing the embedding

strength. The embedding strength is tuned to provide a trade-

off between robustness and imperceptibility of the watermark.

III. ML DETECTION

Elements of the watermark are assumed to be statistically

independent and uniformly distributed in the interval ]1,1[ .

DFT coefficients are also assumed to be statistically

independent. The embedding strength is assumed to be much

lower than 1. Under these assumptions, the ML detection

decision rule [1] states that the watermark *w is detected if

,)(ln1

ln)(

1*

N

i

iX

ii

iX yf

w

yfz

iiy (2)

where is the decision threshold.

The Neyman-Pearson criterion can be used to obtain in

Maximum Likelihood Detection in Image

Watermarking Using Generalized Gamma

Model

T. M. Ng H. K. Garg

Department of Electrical and Computer Engineering,

National University of Singapore,

4 Engineering Drive 3, Singapore 117576

16801­4244­0132­1/05/$20.00 ©2005 IEEE

such a way that the missed detection probability is minimized

subject to a fixed false alarm probability, say FAP . It is given

as [1]

)(2

)(1 2)2(erfc XX zzFAP , (3)

where

N

i

iX

ii

iX xf

w

xfzz

ii

1*

)(ln1

ln)()xy

y(x (4)

is the realization of the random variable )(Xz with mean

N

i

iX

ii

iXz xf

w

xfEzE

ii

1*)( )(ln

1ln)(XX

(5)

and variance

N

i

iX

ii

iXz xf

w

xfVzV

ii

1*

2)( )(ln

1ln)(XX .

(6)

Here, central limit theorem is used to assume that )(Xz has a

Gaussian PDF. As a rule of thumb, the value of N should be at

least 30 for the application of central limit theorem [2].

IV. GENERALIZED GAMMA MODEL

We consider modeling iX as a random variable having

generalized gamma PDF [8], defined as

i

p

i

i

i

pi

pi

iiX x

a

xx

a

pxf

iii

iii0,exp

)()(

1

, (7)

with positive shape ip , scale ia , and power i parameters,

and

0

1 0,)exp()( udtttu u is the gamma function.

Note that the exponential ( 1ii ), Weibull ( 1i ), and

gamma ( 1ip ) PDFs are included as special cases.

Substituting (7) in (2), we obtain the decision rule as

N

ip

iipi

piip

iii

i

i

wa

wyz

1*

*

)1(

1)1()(y . (8)

In view of (3), the mean and variance of

N

ip

iipi

piip

iii

i

i

wa

wXz

1*

*

)1(

1)1()(X (9)

are required to obtain the decision threshold . By

considering the rth moment of iX [8],

,)(

)( 1

i

iiri

ri

rpaXE (10)

it is straightforward to show that

N

ip

iii

piii

zi

i

w

w

1*

*

()1)((

]1)1)[(1(X) (11)

and

N

ip

iii

piiiii

zi

i

w

w

12*2

2*22

)()1)((

]1)1)][(1()()2([X .

(12)

Taking 1i , we obtain X)(z and 2)(Xz under the Weibull

model of [1]. The present of the power parameter i in (11)

and (12) provides more degree of freedom in the adjustment

of decision threshold to improve watermark detection.

V. PARAMETERS ESTIMATION

In applying the decision rule and threshold, the parameters

,, iia and ip need to be pre-determined. For blind detection

[1] of the watermark, these parameters can be estimated from

magnitude of the DFT coefficients of the watermarked image

instead of the original image. In other words, the original

image is not required in the detection process.

Estimating all three parameters of the generalized gamma

PDF is rather complex [8]. Here, we consider fixing either i

or ip , and then estimate the other two parameters. The

estimators are derived from the mean and variance of iX

given as

)(

)(][

1

i

iiii

paXE (13)

and

)(

)(

)(

)2(][

2

122

12

i

iii

i

iiii

pa

paXV , (14)

respectively.

First, if i is fixed, say 0i , by use of (13) and (14) we

define

1681

][][

][

)()2(

)()(

2

2

01

10

2

ii

i

io

ii

XEXV

XE

p

pp . (15)

As in [1], magnitude of the DFT spectrum is divided into

different regions for watermark embedding. Let R be the

region containing ix and having RN coefficients. All

coefficients in R are assumed to be identically distributed, i.e.,

they have identical PDF. The unbiased estimators of ][ iXE

and ][ iXV are

RyRi y

N

1ˆ and 22 )ˆ(

1

1ˆ

i

RyRi y

N,

respectively, where y is the corresponding magnitude of the

DFT coefficient of the watermarked image (possibly distorted)

in R. With these, we propose estimating ip as

22

21

ˆˆ

ˆˆ

ii

iip , (16)

and then ia as

)(

)(ˆˆ

10

0

i

iip

a . (17)

Similarly, if ip is fixed, say 0ppi , then we define

another function

][][

][

)()2(

)()(

2

2

10

10

2

ii

i

ii

ii

XEXV

XE

p

p (18)

and propose estimating i and ia as

22

21

ˆˆ

ˆˆ

ii

ii (19)

and

)(

)(ˆˆ

10p

a

i

iii , (20)

respectively.

One way to solve (16) and (19) is to approximate the

inverse functions 1 and 1 using any of the well-known

function interpolation methods [3]. Note that for the

interpolation process to be practical, the range of the functions

and must be finite. In Appendix A, we show that both

functions are indeed of finite range.

VI. EXPERIMENTAL RESULTS

Using 512512 test images as shown in Fig. 1, three

special cases of the generalized gamma model are explored,

(i) 1i (ii) 1.1i (iii) 1ip . In (i), we have the Weibull

model. This is similar to the model of [1] except that in [1] ia

and ip are estimated using maximum likelihood method

which requires the use of iterative procedure like the Newton-

Raphson method. In this regard, the proposed estimators are

considered simpler. We refer to (ii) as GG model

with 1.1i . In (iii), we have the gamma model. All these

models are selected because the plot of their PDFs closely

resembles the shape of the histograms derived from subsets of

Nxxx ,,, 21 .

In [1], the DFT magnitude spectrum is divided into 16

regions. Here, for simplicity of testing, we consider only two

regions, shown in Fig. 2 as two squares at the upper half of the

DFT matrix. Each region has 2500 identically distributed

coefficients. The watermark is embedded into these

coefficients and then duplicated to the corresponding

coefficients in the two regions at the lower half of the DFT

matrix. This is done to preserve the symmetry property of the

DFT magnitude spectrum. A constant embedding strength

3.0 is used for all the coefficients.

With 610FAP , robustness of the watermark is tested

under different standard image processing operations. Table I

shows the results for watermarked images compressed by

JPEG with 70% quality factor. In Table II, each watermarked

image is cropped to retain only 400400 pixels at the center.

The missing portion is replaced by zero pixels so that the size

of each image remains at 512512 . In Table III, they are

corrupted by Gaussian noise of zero mean and variance equals

to 0.5. These results are based on 10,000 trials. In each trial,

we embed into each image a watermark *w chosen from a set

W of 100 randomly generated watermarks. A detection is said

to be successful only if the decision threshold is exceeded for

the watermark *w but not for any other watermarks in W. In

general, our simulation results reveal that watermark in JPEG

compressed images can be detected more effectively under

GG model with 1.1i . However, the Weibull model

performs better when the images are cropped. All models

perform equally well under image scaling.

VII. CONCLUSION

An ML watermark detection scheme based on using a

generalized gamma PDF to model magnitude of a set of DFT

coefficients is formulated. By use of the Neyman-Pearson

criterion, a decision threshold is explicitly derived. We have

also proposed a new method to estimate parameters of the

generalized gamma PDF. These estimators are also useful in

area like reliability analysis where the generalized gamma

PDF is widely used.

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Further research will be focused on giving a more

comprehensive performance evaluation of the generalized

gamma model that includes robustness under other standard

image processing operations. In particular, it is desirable to

look for an optimum value of i that can yield good detection

over a wide range of images.

APPENDIX A

The Weierstrass formula [4] for gamma function is given as

1

1

/1)(

k

kuu ek

uueu , (21)

where }ln)/13/12/11{(lim mmm

is a positive

constant called the Euler-Mascheroni constant. Using (21), we

can write the function )( ip in (15) as

1

1

2

0

)2(

0

2

1

1

0

)1(

0

00

00

21

)2(

11

)1(

)(

k

kp

p

i

ip

p

i

i

k

kp

p

i

ip

p

i

i

i

i

i

i

i

i

i

i

i

ekp

pe

p

p

ekp

pe

p

p

p

1

1

/00

00 1

1

k

ke

ke

1

20

002

0

00

)1(

))(2(

)1(

)2(

k ii

iii

i

ii

pkp

kpkpp

p

pp.

(22)

The first derivatives of 2000 )1/()2( iii ppp and

2000 )1/())(2( iiiii pkpkpkpp can be shown

to be positive. This implies that is a strictly increasing

function. Furthermore, it can be verified from (15) and (22)

that 1)(lim ip

pi

and 0)(lim0

ip

pi

, respectively. Hence,

we conclude that the range of is finite. Using the same

approach, we can also show that the range of is also finite.

REFERENCES

[1] M. Barni, F. Bartolini, A. De Rosa and A. Piva, “A new decoder for the

optimum recovery of nonadditive watermarks,” IEEE Trans. Image

Processing, vol. 10, no. 5, pp. 755-766, 2001.

[2] J. L. Devore, Probability and Statistics for Engineering and the

Sciences, Brooks/Cole, USA, 2004.

[3] D. Kincaid and W. Cheney, Numerical Analysis, 2nd ed., Brooks/Cole

Pub. Co., 1996.

[4] S. G. Krantz, Handbook of Complex Variables, Birkhauser, Boston,

1999.

[5] S.-G. Kwon, S.-H. Lee, K.-K. Kwon, K.-R. Kwon and K.-I. Lee,

“Watermark detection algorithm using statistical decision theory,” Proc.

IEEE Int. Conf. on Multimedia and Expo, Vol. 1, pp. 561-564, 2002.

[6] T. M. Ng and H. K. Garg, “Wavelet domain watermarking using

maximum-likelihood detection,” Proc. SPIE Conf. on Security,

Steganography, and Watermarking of Multimedia Contents VI, vol.

5306, San Jose, pp. 816-826, 2004.

[7] T. M. Ng and H. K. Garg, “Maximum-likelihood detection in DWT

domain image watermarking using Laplacian modeling”, IEEE Signal

Processing Letters, pp. 285-288, 2005.

[8] E. W. Stacy and G. A. Mihram, “Parameter estimation for a generalized

gamma distribution,” Technometrics, vol. 7, no. 3, pp. 349-358, 1965.

Baboon Barbara Bridge F16

Lena Fishing Boat Peppers Pentagon

Fig 1. Test Images

Watermarked region

Symmetric coefficients

Fig 2. Watermark region in DFT (magnitude) matrix

TABLE I. PERCENTAGE OF SUCCESSFUL DETECTIONS UNDER JPEG

COMPRESSION

Image Weibull GG ( 1.1i ) Gamma

Baboon 100.00 100.00 100.00

Barbara 97.88 99.07 94.09

Bridge 100.00 100.00 100.00

F16 98.15 98.87 97.33

Lena 93.61 99.53 92.04

Fishingboat 94.59 99.02 90.17

Peppers 94.58 98.82 91.64

Pentagon 100.00 100.00 100.00

TABLE II. PERCENTAGE OF SUCCESSFUL DETECTIONS UNDER IMAGE CROPPING

Image Weibull GG ( 1.1i ) Gamma

Baboon 100.00 100.00 100.00

Barbara 99.81 99.07 99.19

Bridge 100.00 100.00 100.00

F16 100.00 100.00 100.00

Lena 100.00 100.00 100.00

Fishingboat 99.93 99.12 99.89

1683

Peppers 100.00 100.00 100.00

Pentagon 90.32 89.57 89.49

TABLE III. PERCENTAGE OF SUCCESSFUL DETECTIONS UNDER IMAGE SCALING

Image Weibull GG ( 1.1i ) Gamma

Baboon 100.00 100.00 100.00

Barbara 100.00 100.00 100.00

Bridge 100.00 100.00 100.00

F16 100.00 100.00 100.00

Lena 100.00 100.00 100.00

Fishingboat 100.00 100.00 100.00

Peppers 100.00 100.00 100.00

Pentagon 100.00 100.00 100.00

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