AFOSR PROGRAM REVIEWJUNE 5-7, 2003PRINCETON, NJ

DATA HIDING IN TIME-FREQUENCY DISTRIBUTION OF

IMAGESBijan Mobasseri

ECE DepartmentVillanova UniversityVillanova, PA 19085

2

Outline

• Data hiding definition and modalities

• Motivation for using TF distributions

• Wigner distribution

• Watermarking model

• Embedding and detection

• Capacity

• Future work

3

Data hiding requirements

• Data hiding must meet at least the following three conditions:– Transparency; no visible impact on the cover

signal– Robustness; Survive “friendly fire”: filtering,

compression, cropping but break under attacks – Security; hidden data should not be easily

removed or replaced

4

Data hiding modalities

• Watermarking– Message itself is not secret: owner identification, copyright

protection, fingerprinting– Transparency, robustness and security still apply.– Embedding capacity not a major issue– In authentication applications, watermark must be content-

dependent, secure but somewhat brittle

• Steganography– Used as a covert channel, the message is secret and its

very presence within the host data must not be detectable

5

Information hiding as a game

• Information hiding has been stated as a game between two cooperative players (embedder and decoder) and an opponent (attacker)

• The first party tries to maximize a payoff function and the opponent tries to minimize it (Moulin, O’Sullivan)

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Data hiding paradigm

€

E h N ,m,k N( )

€

T y x( )

€

D yN ,k N( )

embedder attacker decoder

€

xN

€

yN

€

ˆ m

€

m : message

€

hN : host data

€

k N : side information

P. Moulan, J, O’Sullivan, IEEE Trans IT, March 2003

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New domain

• Digital watermarking has heretofore been applied in either spectral or temporal/spatial domains but not in both simultaneously.

• The ability to watermark joint time-frequency cells provides additional control, capacity and security

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DCT TFD t

f

distinct keys

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Time-varying data hiding

t

f• Watermark can be designed

to follow a trajectory in time-frequency plane

• Attackers have a harder time targeting watermarked bins or have to flood the whole TF plane

• Attacks with known T-F signatures can be circumvented

• An N-point signal has N2 TF distribution cells a substantial fraction of which is available for watermarking

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Previous work

S. Stankovic, I. Djurovic, I. Pitas, “Watermarking in the space/spatial-frequency domain using two-dimensional Radon-Wigner distribution, “IEEE Transaction on Image Processing, vol. 10, no. 4, pp.650-658, April 2001.

• They add a sinusoidal pattern to the image in a way that is only detectable in time-frequency domain. It is presented as a watermarking algorithm

• Our approach hides data in the transform domain instead

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Generating TFD:Wigner Distribution

• WD of function x is Fourier transform of its local autocorrelation function. The discrete-time WD of a 1-D signal is given below

Wx

nT , f( ) = 2 T x n + m( )

m

∑ x

*

n − m( ) (exp − j 4 π fmT ), f ≤

1

4 T

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WD

at

work 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

TIME(sec)

LINEAR FM CHIRP WITH GAUSSIAN AMPLITUDE

Figure 1 – Linear FM chirp with Gaussianenvelope. Figure 2 – Wigner distribution of Figure 1.

Distribution of frequencies along time is clearlyvisible.

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SCAN LINE

PIXEL

Figure 3 – Scan line of the190th column of clown.

Figure 4 – Wigner distribution of Figure 3. Thehotspot centered around the 20th pixel indicates highconcentration of DC content. This fact is not readilyobvious from the column plot alone

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Watermarking model

• We parallel DCT watermarking by additively modifying selected T-F cells of WD.

• This simple model will not work unless certain precautions are taken into account

Y t , f( ) = X t , f( ) + w t , f( ) ; t , f ∈ Ω{ }

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The Inverse Wigner

• Not every two dimensional function is an allowed time-frequency representation

• It is possible that no signal may be found that has the given TFD

• This is a synthesis problem and can be stated as follows

Given a target (watermarked) WD Y, find the corresponding signal x whose Wigner distribution is closest to Y in some sense

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A time-frequency filtering problem

C(1)

R(2)

f1

f2

1=Mf1

’2

M*

*:inadmissible

:admissibleM:mapping functionH:transformation

2=HM 1

’2 2

1=Mf1

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Solutions

• There are a number of solutions to this problem.

For DTWD:

V. Kumar et al, “Discrete Wigner synthesis,” Signal Processing, vol. 11, pp. 277-304, 1986.

For DWD:

S. Nelatury, B. Mobasseri,” Synthesis of discrete-time discrete-frequency Wigner Distribution “ IEEE Signal Processing Letters, in press.

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Exam

ple

: ti

me-f

requ

ency

filt

eri

ng

Figure 1- Filtering in time-frequency plane. Anexample of low pass filtering with finite temporalsupport.

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-10

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T-F FILTERED SIGNAL

Figure 2- Inverse approximation to Figure 5. Thisplot should be compared with the original in Figure4.

Figure 3- W igner distribution of Figure 6. This isnot equal to the target distribution in Figure 5 but isa close approximation.

Table 1- Filtering in time-frequency domain. Themean of the signal is lowered due to the removal ofDC. This reduction, however, is limited to a finitetemporal window

Effect of T-F filtering on mean

Before filtering After filtering

<15,25>pixels

<26,end>pixels

<15,25>pixels

<26,end>pixels

Mean72.45

Mean23.49

Mean45.42

Mean23.62

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2D Wigner

• Formally, the 2D Wigner Distribution is a 4-D function,

• In this work, we avoid this by applying a 1D Wigner to each block of image

€

WD n1,n2,k1,k2( ) = I n1 + m1,n2 + m2( )m21 =−

N

2

N

2−1

∑m1 =−

N

2

N

2−1

∑ I * n1 − m1,n2 − m2( )exp − j4π

Nm1k1 + m2k2( )

⎡ ⎣ ⎢

⎤ ⎦ ⎥

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1D Wigner distribution of 2D block

• Let define an NxN image block

• Define an “equivalent” linear array then do a 1D Wigner on it €

X,xij , i, j( )∈ N{ }

€

rX

€

º º º º

º º º º

º º º º

º º º º

€

º º º º

º º º º

º º º º

º º º º

€

º º º º

º º º º

º º º º

º º º º

Column-wise zigzag random

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Picking the order

• Since WD reflects local autocorrelation of the signal, different pixel arrangements produce distinctly different TFDs

• However, we are only interested in the integrity of signal synthesis. In this sense, it makes no difference how is found from X

€

rX

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Evidence

SNR>300 dB

original reconstructed

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Compression effect on time-frequency signature

• If robustness to compression is desired, only compression-resistant TF cells must be watermarked. We evaluate a simple error measure and apply it across JPEG Q-factor

€

e=WD x( )−WD JPEG x( )( )

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Error surfaces

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Error surfaces

Q=30Q=50

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Which component to watermark?

• JPEG follows YUV(luma-hue-saturation) color model.

• We have found that the TF signature of saturation band is most robust to compression

LUMINANCEHUE SATURATION

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Band energy

LUM HUE SAT

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LUMINANCE

Figure 1-The luminance component image.

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MSE FOR LUMINANCE

JPEG Q-factor

Figure 2-The mean square error arising from

the comparison of the TFDs of compressed and

uncompressed image for the luminance

component.

HUE

Figure 3-The hue component.

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MSE FOR HUE

JPEG Q-factor

MSE TextEnd

Figure 4- MSE for the hue component.

SATURATION

Figure 5- The saturation component.

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0.1

MSE FOR SATURATION

JPEG Q-factor

MSE TextEnd

Figure 6- The time-frequency signature of

the saturation is least affected by compression.

MSE a

naly

sis

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Watermarking Geometry

• Tile the image:– Exhaustively– Randomly (keyed)

• Embed one bit, spread spectrum-wise, in the WD of each block

• Use a unique key per block. Image is then tiled by a reference template

ORIGINAL

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Algorithm Summary

€

Embedding

x Wigner ⏐ → ⏐ ⏐ X;

Y t, f( ) = X t, f( ) +αpk t, f( ); k ∈1,2{ }

pk : spreading_ sequence

Y t, f( ) Wigner −1

⏐ → ⏐ ⏐ ⏐ xwm

xwmJPEG ⏐ → ⏐ ⏐ xwm _ comp

Extraction

xwm _ compWigner ⏐ → ⏐ ⏐ Xwm _ comp

d = Xwm _ comp t, f( ) − Y t, f( ); t, f ∈Ω{ }

ρ =< d ,pk >

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Watermark detection

• Watermark is detected based on the following hypothesis testing– Ho: – H1:

• Rejecting the null hypothesis, when it is true, amounts to the probability of false alarm(picture is incorrectly decided to carry watermark)

€

ρ ≠0

€

ρ =0

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Test statistic

• For candidate TF cells, evaluate the following test statistic

• The null hypothesis will be rejected at significance level if

€

z = 0.5 n − 3( )0.5

ln 1+ ρk

1− ρk

( )

€

z > thr

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Watermark strength vs. image PSNR

WSR(dB) SNR(dB) ρ-13 58 0.6

-15 60 0.52

-18 63 0.42

-21 66 0.32

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x 10-3

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0.005

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0.015

0.024x4 blocks, each carrying one bit

Q=50

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Resu

lts

ORIGINAL

Figure 1- B locks shown are time-frequencywatermarked, each with a unique key.

WATERMARKED-LUMINA CHANNEL

Figure 2- Watermarked- PSNR=58 dB

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Detector response for Q=5

Figure 3- Detection response for Q=5.

QuickTime™ and aPhoto - JPEG decompressorare needed to see this picture.

Fi gu re 4 - F i g u r e 1 7 co mp re sse d w i t h Q = 5

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Detector response for Q=50

Figure 5- Detector response sharpens for Q=50

QuickTime™ and aPhoto - JPEG decompressorare needed to see this picture.

Fi gu re 6 - F i g u r e 1 7 co mpr e ss ed w i th Q= 5 0

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Data hiding in saturation band:16x16 blocks

Q=5

Q=50

Virtually identical performance acrossall Q-factors

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Capacity:are TF cells independent?

• [Richard’01] has shown that:

For all , the number of linearly independent components of discrete WD of x is upper bounded by

(N even)

• For 8x8 blocks, there are 4096 components of which1056 are independent

• 8x8 DCT produces a maximum of 64 coefficients

€

x∈RN

€

N2 +2N( ) 4

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Payload numbers

• Capacity=N2/block_size

• Larger block size provides bigger PG and watermark survival at lower Q

• In lena(2562), we can embed 4096 bits using 4x4 blocks at WSR= -13dB

• Reliable detection is possible down to Q=25

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Conclusions and future work

• A new transform domain for sata hiding is introduced

• It features high capacity, low probability of intercept and low JPEG Q-factor operation

• Need work on blind detection

• Robustness to geometric transformations

• Capacity and Steganalysis benchmarking

THE END