Floating Costa Scheme with Fractal Structure for

Information Embedding

Ali KOMATY

Naval Academy Research Institute (IRENav),

CC600, 29240 Brest Armee cedex 9, France

Claude DELPHA and Aurelia FRAYSSE

Laboratoire des Signaux et Systemes (L2S)

CNRS - Supelec - Univ. Paris-Sud,

3, Rue Joliot Curie, 91192 Gif sur Yvette, France.

Abstract—Nowadays, multimedia data protection widely usesdata hiding technology like Digital Watermarking or steganogra-phy. Among the large offer of watermarking techniques severalones are now based on the side information scheme proposed byCosta. These techniques mainly use scalar quantization for em-bedding a given information. Such solutions have good propertiesin terms of robustness and capacity but has lack of statisticaltransparency. In this paper we propose a new watermarkingscheme based on a floating quantizer with a fractal structure.The aim of this study is to get rid of the unwanted fluctuationsin the probability density function (PDF) of the watermarkedsignal, caused by the use of a scalar quantizer in the ScalarCosta Scheme approach (SCS). We present here the structure ofa Fractal based Costa Scheme (FCS) and detail its performances.Afterwards, we compare the performances of our FCS proposedmethod with those obtained with the reference SCS. We showthat we are able to reduce the statistical distortions obtained withthis proposed scheme and keep good robustness properties.

I. INTRODUCTION

Digital watermarking is a solution to control unapproved

copying and redistribution of multimedia data. For many

applications watermarking techniques based on quantization

are often proposed for their good robustness against attacks

[1]- [2]- [3]. These methods are generally designed on a

communication scheme based on Costa’s work [4] where

the watermark is introduced in a coding scheme as a side

information at the encoder. One major advantage of this

method is that it is possible to prove that the channel capacity

is independent of the host signal power. Indeed this allows

to increase the channel capacity. To make such a scheme

fully realizable in practice, the use of a scalar quantization

approach has been introduced and efficient schemes were

proposed like the main Quantization Index Modulation

(QIM) [5], or the Scalar Costa Scheme (SCS) [6]. In these

cases, good robustness performances are proved for blind

detection and decoding of the embedded information [6].

With such methods, no statistical transparency is possible

except when using dithered quantization. It has been proved

that watermarking with the SCS (without dither) introduces

distortions on the signal probability density function (PDF)

of the signal [7]- [8]. These distortions are mainly due to the

regularity of the used quantizers. This is a real drawback: an

attacker will have more motivations to attack a watermarked

signal for removing the embedded information if he identifies

suspects distortions in its PDF. To avoid these problems,

several solutions have been proposed to reduce the distortions

and enhance the watermark’s undetectability. Some were

based on a treatment of the host signal with constraints [7],

using specific channel coding [9], embedding in a specific

domain [10], or embedding with a specific quantization

method (Treillis Coded Quantization: TCQ) [11]. In this last

case, using TCQ, the robustness of the scheme is severely

reduced. The aim of the present proposed scheme is to reduce

the distortion on the watermarked signal while keeping a good

robustness and correct capacity by using floating quantization

based on a fractal approach with Costa’s techniques.

In the following sections, we first recall Costa’s approach with

a scalar quantizer (practical realization). Secondly, we present

the improved floating fractal quantizer and then the fractal

Costa Scheme. The performances of these three schemes are

then presented when applied to images and compared before

conclusion.

II. QUANTIZATION-BASED WATERMARKING

In quantization-based approaches, the watermarking scheme

is considered as a communication procedure with side in-

formation at the encoder (Figure 1). In such scheme, let’s

consider a message m embedded in a host signal x of

power σ2x, to produce the watermarked data s (at the encoder

side). The difference w = s − x denotes the watermark. In

informed approaches, this watermarked signal is based on

the quantization error of the host signal encoded with the

message m. When this watermarked signal s is transmitted

to a receiver in a communication channel, an attacker can

do transformations to make the watermark unreadable by the

receiver. These attacks are modeled by an AWGN noise noted

v of power σ2v which is added to the watermarked signal s

to produce the received signal z. This signal is then used by

the decoder (at the receiver side) to extract the watermark and

estimate the embedded message m (Figure 1).

A. SCALAR COSTA SCHEME

In Costa’s approach, a codebook with the quantization

information is shared between the encoder and the decoder

side [4]. As mentioned before, one major particularity of

Costa’s approach concerns the capacity (i.e. the total amount

of embedded information that can be extracted without errors

for a given distortion [12]) which is independent from the

19th International Conference on Telecommunications (ICT 2012)

978-1-4673-0747-5/12/$31.00 ©2012 IEEE

Message

( m )

Estimated

Message

( m )

Encoder

Host signal

( x )

Marked data

( s )

Attack

( v )

Received

signal

( z )

Watermark

( w )

Decoder

Embedding Extraction

Channel

Fig. 1. Watermarking scheme as a communication procedure

host signal power. To achieve this maximum capacity we must

use an infinite length codebook Lx, which makes this scheme

unrealizable in practice [4]. A suboptimal scheme proposed

by Eggers et al. is the Scalar Costa Scheme (SCS) [6], which

uses a suboptimal codebook ULx as a product of Lx one-

dimensional codebooks U1 :

ULx = U1 ◦ U1 ◦ . . . ◦ U1. (1)

Let us denote by ∆scs the quantization step, αscs the ro-

bustness optimization factor, d the elements of the message

m belonging to an alphabet of size D. The codebook U1 is

chosen to be representative of a scalar uniform quantizer with

step size αscs∆scs/D, which is formally denoted as:

U1 = {u = lαscs∆scs +d

Dαscs∆scs | d ∈ D, l ∈ Z} (2)

where l enumerates all quantizer types of a prototype scalar

quantizer with step size αscs∆scs and d introduces a shift of

the prototype quantizer. The dth sub-codebook of U1 can be

written as:

U1d = {u = lαscs∆scs +

d

Dαscs∆scs | l ∈ Z}. (3)

Each sub-codebook is then equivalent to the types of a scalar

uniform quantizer with step size αscs∆scs. We must also

introduce a quantization error qn formally denoted as:

qn = Q∆scs{xn −∆scs(

dn

D)} − {xn −∆scs(

dn

D)} (4)

where Q∆scs{.} means a scalar uniform quantization of step

size ∆scs and the index n (as in qn) denotes the nth element of

the quantization error vector q. So, the transmitted watermark

sequence is given by w = qαscs and the watermarked data is:

s = x + w = x + qαscs. (5)

Similarly, the decoder acts as a scalar uniform quantizer that

quantizes the received signal z (z = x + w + v) to the

closest codebook entry using the codebook U1 . The extraction

process is first done by extracting the signal y from the

received signal z. This extraction rule of the nth element is:

yn = Q∆scs{zn} − {zn} (6)

Then a hard decision decoding has to be made concerning the

value of |yn|. If |yn| is close to zero, we estimate dn = 0, and

if |yn| is close to ∆scs/2, we estimate dn = 1. This scheme

is robust against an AWGN attack, but it has the inconvenient

that the PDF of the watermarked data s is distorted compared

to the PDF of the host signal x. This is mainly due to the

scalar uniform quantizer (See Figure 8-a).

While using a secret key (dithered quantization), this problem

is fully solved. Nevertheless this secret information has to

be shared between the encoder and decoder, so it has to be

sent at first through the communication channel securely.

In this first transmission phase, a statistically transparent

operation has to be proposed to avoid alerting the attackers

on the transmission of hidden informations. A solution to this

problem is to introduce another type of quantization procedure

in Costa’s scheme or embedding in a specific domain [10],

[11], [9]. To reduce the regularity of the quantization step we

propose to use a floating quantization approach. The goal of

our work fully concerns this approach using a fractal structure.

B. IMPROVED FLOATING QUANTIZATION SCHEME

WITH FRACTAL STRUCTURE

This type of quantizers, noted as Fractal Floating Quantizer

(FFQ), was first proposed by P. Bas [13]. It was implemented

to improve the robustness against non-linear distortions. We

propose here an improved version of the FFQ, the so called

IFFQ, which increases the robustness against an AWGN

attack. It should be noted also that this scheme is a substitutive

blind quantization scheme. In this case the quantization step

size, noted ∆f , is not fixed, but it depends on the values of

the host signal x. Specifically, it depends on the values of

neighboring values of a given sample. For example, taking

the ith sample xi, we select its two neighbors xi−1 and xi+1,

and the selected samples are sorted in ascending order; then

we find ∆f using the following equation:

∆f =r

2(N1(r)+1)if N(rmid) > 0 (7)

= ∆ if N(rmid) = 0

where r is the distance between the maximal sample value

xmax and the minimum one xmin and rmid = xmid − xmin

(xmid and xmin are the median and the minimal sample

values respectively among three values). ∆ is an arbitrarily

fixed quantization step. Note that the functions N1 and N are

given by the following equations using the positive integer part

function ε+():

N(rmid) =ε+(log2(rmid

α∆)) (8)

N1(r) =ε+(log2(r

α∆)) (9)

N1(r) is used to calculate ∆f and N(rmid) corresponds to

the number of the selected quantizer. The motivation behind

the definition of N(rmid) is to reduce the effect of the AWGN

at the decoder by taking into consideration the value of the

median sample xmid. Thus, we choose the parameters of the

quantizer by using all the samples of the signal and not only

the extreme ones as in [13]. It should be noted that in order

to have a bounded quantization step size ∆f , we define the

possible variation of ∆f by the relation: α1∆ ≤ ∆f ≤ α2∆,

with α1 ≤12 and α2 ≤

12 . We chose scale factors α1 = 1

4

Fig. 2. Building of the set of quantizers. Quantizers that are in the gray arearepresent a fractal structure. The choice of the quantizer is determined by thevalue of rmid.

Fig. 3. Repartition of Q in function of rmid.

and α2 = 12 to avoid quantization cells overlapping. Figure 2

depicts the set of quantizers which mainly presents a fractal

structure and Figure 3 represents the repartition of the set of

quantizers Q in function of the range value rmid. The encoding

process is then as follows:

1) Select the triplet {xi, xj , xk}.2) Arrange the triplet as follows : {xmin, xmid, xmax}, and

compute rmid = xmid − xmin, r = xmax − xmin, ∆f ,

N1(r) and N(rmid) using equations (7), (8) and (9)

defining the quantizer.

3) If Qi 6= Q0, quantize rmid (xmid) using Qi and the

value of the bit b.

4) If Qi = Q0, if b = 0 set rmid and r to 0, and if b = 1set rmid and r to ∆.

5) Select another triplet which has not been selected and

insert another bit.

And the decoding process is summarized as follows:

1) Select the triplet {xi, xj , xk}.2) compute rmid, r, ∆f , N1(r) and N(rmid).3) If N(rmid) = 0, two cases arise:

{

if rmid ≈ 0 estimate 0

else estimate 1

4) If N(rmid) 6= 0, find the nearest neighbor to xmid and

estimate the corresponding bit.

This quantization scheme is noted Improved Floating Quanti-

zation Scheme with Fractal structure (IFFQ) hereafter.

III. FLOATING COSTA SCHEME WITH FRACTAL

STRUCTURE

We replace the scalar quantizer by the floating quantizer

described in the previous section (the IFFQ). The codebook

ULx is chosen as a product of Lx one-dimensional non

identical codebooks U i, with i ∈ {1, . . . , Lx}, such as each

U i is chosen to be representative of a floating quantizer with

step size αscs∆fi/D, which is formally denoted as:

U i = {u = lαscs∆fi+

d

Dαscs∆fi

| d ∈ D, l ∈ Z} (10)

The encoding algorithm is as follows:

1) Select three neighboring samples {xi, xj , xk}.2) Arrange the triplet as follows : {xmin, xmid, xmax}, and

compute rmid = xmid − xmin, r = xmax − xmin, ∆f ,

N1(r) and N(rmid) using equations (7), (8) and (9)

defining the quantizer.

3) Quantize xmid using the selected quantizer, to the near-

est quantization cell.

4) Select another triplet which has not been selected and

insert another bit.

In the following, we work without optimization of the capacity.

Therefore, the scale factor αscs is set to 1. After the quantiza-

tion, we add the value d∆f

2 to xmid. Then we proceed in the

same manner as in the SCS. The procedure is summarized by

equations (3), (4) and (5).

At the decoding stage, we receive a signal z (z = s + v) and

our goal is to extract the hidden bit. To do this, we quantify

the signal z using three quantization levels as described in the

following:

1) Select the triplet {xi, xj , xk}.2) Compute rmid, r, ∆f , N1(r) and N(rmid).3) If N(rmid) = 0, use the quantizer Q0 for decoding.

4) If N(rmid) 6= 0, use three quantizers [QN−1QNQN+1]to quantize rmid, it is quantized to the nearest cell of

the corresponding quantizer.

5) compute |rmid − Q∆f(rmid)| and estimate d = 1 if it

is close to∆f

2 , and d = 0 if it is close to zero.

IV. EXPERIMENTAL RESULTS

To evaluate the performances of the proposed algorithm,

we have studied the evolution of the Bit Error Rate against

an AWGN attack (Robustness), the ability to decode properly

the bits of the message without errors for a given distortion

(Capacity) and the statistical and perceptual invisibility (Trans-

parency) of the embedded information on the watermarked

signal. The performances were evaluated and averaged on a

database of 140 grey scale real images of size 512×512 using

the Monte Carlo method.

A. ROBUSTNESS

To evaluate the effectiveness of our algorithm, we first plot

the BER curve shown in Figure 4 for several values of ∆.

It is observed that the more ∆ increases, the better the

performance will be in terms of BER. Nevertheless, we

obtain increasing perceptual distortion on the considered wa-

termarked image if ∆ increases. For several values of ∆,

we have plotted the PSNR for different watermarked images

with the FCS proposed scheme without attacks (Figure 5).

The results obtained prove that to be perceptually transparent,

-10 -8 -6 -4 -2 0 2 4 6 8 1010

-4

10-3

10-2

10-1

100

WNR [dB]

BE

R

∆=1

∆=2

∆=4

Fig. 4. Bit Error Rate (BER) of blind watermarking facing an AWGN attackfor binary FCS with different value of ∆.

the watermark procedure should be done with low values

of ∆. In the following, the FCS watermarking procedure

is implemented with a value of ∆ ensuring a satisfying

perceptual transparency (High PSNR).

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1626.9

26.95

27

27.05

27.1

27.15

27.2

27.25

27.3

∆∆∆∆

PS

NR

Fig. 5. PSNR value of blind watermarking for binary FCS in function ofthe ∆ value.

For a given value of ∆ ensuring a satisfying perceptual

transparency of the message embedded in the image, we have

compared the robustness obtained with the FFQ [13], the

proposed FCS scheme, the SCS and the contributed IFFQ in

a substitutive watermarking scheme (Figure 6).

To compute these results, a message with the same number of

bits was used for the mentioned schemes (IFFQ, FFQ, FCS

and SCS). Thus, a BER-comparison can be done under the

same watermarking conditions. So for the SCS, a message

of one-third-length of the total number of pixels was used in

order to achieve a fair comparison in terms of robustness. Of

course, the SCS performances obtained here are better than in

the case where the message and the signal have same lengths

(an embedding rate of 1 bit per sample).

The results on Figure 6 highlight that the FCS and the

FFQ have very close robustness performances. These results

are less good than for SCS and IFFQ, but the obtained BER

-10 -5 0 5 1010

-4

10-3

10-2

10-1

100

WNR [dB]

BE

R

binary IFFQ

binary FCS

binary FFQ

binary SCS

Fig. 6. Bit Error Rate (BER) of blind watermarking facing an AWGN attackfor binary SCS, binary IFFQ, and binary FCS.

levels are satisfying. For WNR values higher than 5dB, the

BER is smaller than 10%. As mentioned in section III, the

robustness optimization factor is fixed in the case of FCS and

not optimized. An improvement of our FCS proposal could be

obtained by optimizing this parameter as for SCS.

The IFFQ offers quite different robustness performances: we

can discuss the behavior obtained before and after WNR =−3dB. For low WNR values (WNR < −3dB), the IFFQ’s

BER is better than FCS and FFQ ones, but the SCS takes

a slight advantage on it. In the case of middle and high

WNR values (WNR > −3dB), the BER levels are better

(smaller values) than for the three other schemes. For example

for WNR > 5dB, the BER is smaller than 0.1%. These

better performances in terms of BER are mainly due to the

improvement introduced in section II-B (see Equation (8)) and

the substitutive approach of this watermarking scheme.

B. CAPACITY

As reminded by Cox in [12] the channel capacity can be

defined as the maximum of amount information that we are

able to embed and decode properly for a given distorsion

level. Considering this definition, we have also computed the

capacity of the SCS, IFFQ and FCS for some given WNRvalues (Figure 7).

We can note that the maximum value of the capacity is

slightly lower for the FCS and IFFQ than for the SCS : the

SCS’s rate can attain 1 bit per sample while in our proposed

schemes (IFFQ, FCS), the maximum rate is 1/3 bit per sample.

In fact the three schemes are evaluated using the maximum

achievable rate: for SCS this rate is three times higher than

for the two proposed schemes. As explained in the previous

section III, the embedding procedure induces insertion of the

watermark using a triplet of pixels, this implies the use of 1/3of the total number of available samples.

In the middle range of WNR (-10dB to 0dB), we obtain the

same capacity for the three schemes. In the highest level of

noise (WNR < −10dB), IFFQ has the lowest capacity. In the

same range FCS and SCS have quite similar capacity values.

-20 -15 -10 -5 0 5 10 1510

-4

10-3

10-2

10-1

100

WNR [dB]

C [

bit

s/e

lem

en

t]

SCS

IFFQ

Fig. 7. Capacity of blind watermarking facing an AWGN attack for binarySCS, binary IFFQ, and binary FCS.

0 50 100 150 200 2500

0.002

0.004

0.006

0.008

0.01

0.012

0.014PDFs of the original and watermarked signals

PDF of the original image

PDF of the watermarked image

(a)

0 50 100 150 200 2500

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009

0.01PDF of the original and watermarked signals

PDF of the original image

PDF de watermarked image

(b)

0 50 100 150 200 2500

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009

0.01

PDF of the original image

PDF of the watermarked image

(c)

Fig. 8. PDF of the original and watermarked images for (a) SCS, (b) FCSand (c) IFFQ

Note that in the opposite of SCS, the capacity and robustness

optimization factor for FCS was not optimally calibrated. With

such an optimized factor, we should be able to enhance the

system’s performances.

C. TRANSPARENCY

To evaluate the transparency, the same watermarking

conditions as for the capacity evaluation were used for

each schemes: the maximum possible amount of bits were

embedded. The statistical and perceptual transparency were

evaluated in different embedding conditions of ∆.

1) Statistical transparency: We have evaluated the statis-

tical imperceptibility of the embedded information on the

watermarked signal’s PDF (Figure 8).

We can easily notice the improvement in the watermarked

signal’s PDF in the case of the FCS algorithm compared to

the SCS algorithm. This visually-observed improvement is

0 5 10 15 20 25 3010

-4

10-3

10-2

10-1

100

101

102

DWR [dB]

DK

L

binary SCS

binary IFFQ (∆∆∆∆=4)

binary IFFQ (∆∆∆∆=1)

binary FCS (∆∆∆∆=4)

binary FCS (∆∆∆∆=1)

Fig. 9. Comparative study of the KLD for SCS, IFFQ and FCS, influenceof the quantization step ∆

also verified statistically by computing the Kullback-Leibler

Divergence (KLD) for SCS, FCS and IFFQ (Figure 9)

between the original and watermarked signal’s PDF for a

range of DWR (Document-to-Watermark-Ratio) values and

for different ∆ conditions values. We notice that both FCS

and IFFQ had better performances than SCS for different

DWR values. Moreover, in function of the quantization

step ∆, we show that for the fractal approaches (FCS and

IFFQ), the results are better for the smaller values of ∆: the

KLD is smaller. These improvements are due to the floating

quantization which reduces the regularity of the quantization

steps. Such algorithms will then be useful if robustness and

statistical invisibility watermarking performance are necessary

in order to reduce the suspicion of the presence of embedded

information in a media.

2) Perceptual transparency (visual imperceptibility): To

see the perceptual influence of our watermarking procedure

on images, we have perceptually compared the watermarked

images between the SCS, the IFFQ, and the FCS with the

original image. We show in Figure 10 an example of the results

obtained with the Lena image. On this figure, we can notice

very slight visual distortion on the edges of the watermarked

image using the fractal quantizer (Figure 10-b, Figure 10-d).

This is due to the fact that, in the edges regions, the correlation

assumption between neighboring samples does not hold. If

∆ increases, the distortions will increase (see Figure 5). One

solution in this case consists in using a perceptual mask which

will allow us to determine the maximum possible changes that

can be applied to a pixel before being visually noticeable.

D. GENERAL PERFORMANCES

In fine, the general performances obtained here prove the

efficiency of a floating quantization approach in watermarking:

good capacity (3×10−1 bits by samples with FCS and IFFQ)

for WNR=0dB, correct BER (3×10−2 with IFFQ and 3×10−1

with FCS) for this WNR and best statistical transparency

performances (KLD ≈ 10−4) for a sufficiently high DWR

(a) (b)

(c) (d)

Fig. 10. (a) original image, (b) watermarked image by the IFFQ, (c)watermarked image by SCS, and (d) watermarked image by FCS with ∆ = 1

(for all watermarked images DWR=35 dB)

(30dB) ensuring good imperceptibility (PSNR ≈ 27.22dB).

Compared to other approaches, for example using channel

coding like spread transform [9] or insertion in a specific

domain with the scalar quantization approaches [10], the

presented results are even more interesting. It is proved

that using specific channel coding techniques enhance

the watermarking performances [9]. Thus with our new

quantization proposed approach with the Costa’s scheme,

adding specific channel coding or insertion in another domain

will certainly enhance the performances obtained.

V. CONCLUSION

In this paper, we have proposed a new quantization-based

watermarking scheme building up on the well known Costa’s

approach. The proposed solution is based on a floating fractal

quantization structure. We have shown how to manage with

such a type of structure and also how to use it for watermark-

ing (alone and in a Costa-based approach). The substitutive

floating fractal structure was noted IFFQ and the additive

informed Costa watermarking scheme was noted Fractal Costa

Scheme (FCS). We have studied the performances of the

proposed schemes and compared them to the Scalar Costa

Scheme (SCS) while respecting perceptual transparency.

The main difference induced by the proposed floating fractal

quantization methods concerns the statistical transparency of

the embedded information: no distortion is observed on the

PDF of the watermarked signals and very low Kullback-

Leibler Divergence is obtained. In terms of robustness, the

FCS scheme has a satisfying behavior but this could be

improved by tuning the optimization embedding factor in

the same way than in the SCS case. Compared to the other

schemes, the IFFQ is more robust, thus seems more efficient

mainly in real life applications WNR range [−3dB; 3dB].In terms of capacity, the maximum value achieved by the

FCS and IFFQ is slightly lower than that of the SCS due

to the embedding procedure using one sample among three

(the encoding and decoding procedure requires both neighbors

of a given sample for embedding and extraction). However,

this trade-off study between robustness, capacity and statistical

transparency showed that the proposed FCS and IFFQ are

very efficient schemes compared to others considered. Thus,

confronted to such watermarking schemes, an attacker will

have very little information on the possible existence of a

watermark in a protected signal. Of course the robustness

performances for FCS can be improved for example by using

channel coding techniques.

This work will be continued using higher order statistics with

a theoretical evaluation of the performances, optimizing the

watermarking embedding power, taking into account other

types of attacks and facing synchronization problems.

ACKNOWLEDGMENT

The authors1 would like to thank MEDIEVALS project from

ANR for funding.

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1A. Komaty contributed to this work while he was at the L2S