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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)
6
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
7
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
50 100 150 200 250 300
20
40
60
80
100
120
140
160
180
200
10
20
30
40
50
60
DCT TFD t
f
distinct keys
8
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
9
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
10
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
11
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.
0 20 40 60 80 100 120 140
0
10
20
30
40
50
60
70
80
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
12
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{ }
13
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
14
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
15
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.
16
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.
0 20 40 60 80 100 120 140
-20
-10
0
10
20
30
40
50
60
70
80
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
17
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( )
⎡ ⎣ ⎢
⎤ ⎦ ⎥
18
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
19
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
21
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( )( )
24
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
26
LUMINANCE
Figure 1-The luminance component image.
10 20 30 40 50 60 70 80 90
5
10
15
20
25
30
35
40
45
50
55
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.
10 20 30 40 50 60 70 80 90
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
MSE FOR HUE
JPEG Q-factor
MSE TextEnd
Figure 4- MSE for the hue component.
SATURATION
Figure 5- The saturation component.
10 20 30 40 50 60 70 80 90
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
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
27
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
28
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 >
29
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
30
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
31
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
0
5
10
15
20
25
30
35
0
0.5
1
1.5
x 10-3
0
5
10
15
20
25
30
35
0
0.005
0.01
0.015
0.024x4 blocks, each carrying one bit
Q=50
32
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
0
5
10
15
20
25
30
35
-0.03
-0.02
-0.01
0
0.01
0.02
0.03
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
0
5
10
15
20
25
30
35
-0.01
-0.005
0
0.005
0.01
0.015
0.02
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
33
Data hiding in saturation band:16x16 blocks
Q=5
Q=50
Virtually identical performance acrossall Q-factors
34
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
35
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
36
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