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A New Detector for Spread-Spectrum Based
Image Watermarking using Underdetermined ICA
Hafiz Malik, Ashfaq Khokhar, Rashid Ansari
Multimedia Systems Lab
University of Illinois At Chicago
{ hmalik, ashfaq, ansari }@ece.uic.edu
Multimedia Systems Lab - UIC 1
Organization
Introduction
Motivation
Blind Detection for SS-based Watermarking
using Underdetermined ICA (UICA)
Simulation Results
Conclusion
Multimedia Systems Lab - UIC 2
Information Hiding Model
HOST MEDIA
S
INPUT
MESSAGE EXTRACTED MESSAGE
ATTACK
EMBEDDING EXTRACTION
CHANNEL
b x x
% ˆ
b
K
General Data Hiding Model
EMBEDDING KEY
Multimedia Systems Lab - UIC 3
Blind Additive Embedding
Embedder does not exploit any information on the
host signal or attack-channel distortion level
x = s + α ⋅w
Ex: Spread Spectrum (SS) based embedding
Multimedia Systems Lab - UIC 4
Detection Methods
Blind Detection
Host signal is not used during information detection
process
Informed Detection
The host signal and/or watermark is used for
information extraction and/or detection
Multimedia Systems Lab - UIC 5
SS-based Watermarking System
Watermark Embedder Attack Channel Watermark Detector
INPUT MESSAGE MESSAGE H1 , H 0
∑ ∑
MESSAGE DETECTOR
X
b
ENCODER w αw b x x
%
α
PERCEPTUAL MESSAGE
MASK ESTIMATION s v DECODER
ˆ
b
K
DATA ADVERSARY
HOST MEDIA
EMBEDDING KEY ATTACK
Multimedia Systems Lab - UIC 6
SS-based Watermarking System
Salient Features
Simplicity
Robustness
Limitations
Host signal acts as an interference at the blind
detector
Low embedding capacity
Multimedia Systems Lab - UIC 7
SS-based Watermarking System
Salient Features
Simplicity
Robustness
Limitations
Host signal acts as an interference at the blind
detector
Low embedding capacity
Multimedia Systems Lab - UIC 8
Motivation
Reduce the host-signal interference at the blind detector
Improve detection/decoding performance of the blind
detector in the presence of attack channel distortion
Multimedia Systems Lab - UIC 9
Blind Detection Using UICA
The proposed detector exploits,
1. Independence between the host signal and the
watermark, and
2. Non-Gaussian distribution of the host signal
in order to estimate the watermark from the
watermarked media using ICA framework
Multimedia Systems Lab - UIC 10
Independent Component Analysis (ICA)
ICA is a statistical framework for estimating underlying
hidden factors or components of multivariate statistical
data
x = As +v
A ~ n x m mixing matrix
s ~ m -dimensional vector of latent random variables know as
independent components
x ~ n -dimensional observation
v ~ n -dimensional random noise vector
Goal: Estimate both A and s using x only
Multimedia Systems Lab - UIC 11
Independent Component Analysis (ICA)
Assumptions:
The si are mutually independent
The si are non-Gaussian
The mixing matrix A is constant i.e. static mixing
Multimedia Systems Lab - UIC 12
Blind Source Separation (BSS) using ICA
Estimate both A and s using observation x only
Standard ICA
number of observation ≥ number of sources,
Estimated demixing matrix W is used to estimated
sources
Underdetermined ICA
more sources than observations
Source extraction from underdetermined mixtures is a
non-trivial problem
Multimedia Systems Lab - UIC 13
The simple “Cocktail Party” Problem
Mixing matrix A
s1 a11 x1
a12 a21 Observations
Sources
x2
a22
s2
x = As + v
Objective:
separate speakers s from the microphone recordings x
Multimedia Systems Lab - UIC 14
BSS for Linear Mixture using ICA
Linearly Mixing Process
⎡ A11 L A1m ⎤ ⎡ s1 (t ) ⎤ ⎡ x1 (t ) ⎤
⎢ M O ⎥⋅⎢ M ⎥ = ⎢ M ⎥
M ⎥ ⎢
⎢ ⎥ ⎢ ⎥
⎢ An1 L
⎣ Anm ⎥ ⎢ sm (t ) ⎥ ⎢ xn (t ) ⎥
⎦ ⎣ ⎦ ⎣ ⎦
Mixing Matrix Source Observed
Separation Process
Separated Demixing Matrix
Cost Function
⎡ y1 ( t ) ⎤ ⎡ W11 L W1 n ⎤ ⎡ x1 ( t ) ⎤
Independent?
⎢ M ⎥=⎢ M O M ⎥⋅⎢ M ⎥
⎢ ⎥ ⎢ ⎥ ⎢ ⎥
⎢ y m ( t ) ⎥ ⎢W m 1
⎣ ⎦ ⎣ L W mn ⎥ ⎢ xn (t ) ⎥
⎦ ⎣ ⎦
Multimedia Systems Lab - UIC 15
Optimize
Blind Detection Model for SS-Based
Watermarking
Mixing Model for SS-based embedding for b =1
x = s + α ⋅w
One observation, x, and two underlying
independent sources, w and s or m > n
BSS for underdetermined linear mixtures can
be used to estimate watermark from such
mixtures
Multimedia Systems Lab - UIC 16
Blind Detection using UICA
Can we estimate watermark from Iw using BSS based on
UICA?
No ! It is not possible
Why ?
Single channel separation is not possible, even when
mixing matrix is know, if sources obey heavy tailed
densities, i.e. Laplacian distribution, Gaussian
distribution etc.
However, existing BSS schemes based on UICA can
separate 3 or more sources from 2 observation, given
the underlying sources obey heavy tailed distributions
Multimedia Systems Lab - UIC 17
Blind Detection using UICA
Such BSS schemes for underdetermined mixtures can be
used to estimate watermark from the watermarked
image but with following assumptions
Assumptions
s and w are mutually independent
s and w obey non-Gaussian distributions
Repeated embedding into r non-overlapping blocks,
where r ≥ 2
Multimedia Systems Lab - UIC 18
Blind Watermark Estimation for SS-based
Watermarking Using UICA
The watermarked signal is a linear mixture of r +
1 independent sources, i.e., s1, s2 … sr and w
r independent observations are generated from
the received watermarked image Iw
BSS for underdetermined mixtures can be used to
estimate watermark from the observation x
Multimedia Systems Lab - UIC 19
Existing Schemes for BSS based on UICA
Multi-linear analysis and higher order statistics [P.
Comon, 2002, Lathauwer et al, 1999]
Sparse decomposition and over-complete basis [Pajunen
1997, Bofill et al 2001, Li et al 2004]
Statistical ICA based on the mean-field theory [Pedro et
al 2002]
Multimedia Systems Lab - UIC 20
Existing Schemes for BSS based on UICA
Multi-linear analysis and higher order statistics [P.
Comon, 2002, Lathauwer et al, 1999]
Sparse decomposition and over-complete basis [Pajunen
1997, Bofill et al 2001, Li et al 2004]
Statistical ICA based on the mean-field theory [Pedro et
al 2002]
Multimedia Systems Lab - UIC 21
SS-based Watermarking Model Using
Proposed Detector
Watermark Embedder Attack Channel Watermark Detector
INPUT MESSAGE MESSAGE H1 , H 0
∑ ∑
MESSAGE DETECTOR
X
b
ENCODER wb αw b x x
%
α
MESSAGE
PERCEPTUAL
MASK ESTIMATION s v DECODER
ˆ
b
K
DATA ADVERSARY
HOST MEDIA
EMBEDDING KEY ATTACK
ˆ
s
1 H1 , H 0
MESSAGE
INPUT MESSAGE DETECTOR
BSS
MESSAGE
ENCODER
X ∑ ∑ USING
HYPOTHESIS
TESTING ˆ
w
b
wb αw b x x
% UICA
α
MESSAGE
PERCEPTUAL ˆ
s
MASK ESTIMATION s v r +1 DECODER
ˆ
b
K
DATA ADVERSARY
HOST MEDIA
EMBEDDING KEY ATTACK
Multimedia Systems Lab - UIC 22
Performance Measure of a BSS Scheme
Estimated watermark w can be expressed as
ˆ
w = η 1 α w b + s interf
ˆ
where 0 ≤ η1 ≤ 1 , and s i n t e r f is interference due to the host signal. Let
s in terf = η 2 s, 0 ≤ η 2 ≤ 1
therefore,
w = η 1α w b + η 2 s
ˆ
The relative distortion due to interference in the estimated watermark is defined as,
D interf = (η 1 η 2 ) 2
and WIR = 10 ⋅ log( Dinterf )
In general, Dinterf > 1 Multimedia Systems Lab - UIC 23
Theoretical Results: Decoding
Decoding Bit Error Probability Performance Improvement due to Host-Signal-Interference
Cancellation at the Detector (1 bps embedding case)
Multimedia Systems Lab - UIC 24
Theoretical Results: Decoding
Decoding Bit Error Probability Performance of the Proposed ICA-based Detector for
different values of WIR and payload 0.2 bps (Left) , 0.1 bps (Right)
Multimedia Systems Lab - UIC 25
Theoretical Results: Detection
ROC Performance of the Proposed Detector and the detector operating without
canceling the host signal, for different values of WIR, WSR = 13 dB, and 0.2 bps
embedding
Multimedia Systems Lab - UIC 26
Theoretical Results:
Watermarking-Rate
Consider estimated watermark in the presence of Gaussian noise
w [ i ] = η 1 [ i ] α [ i ]w [ i ] b + η 2 [ i ]s[ i ] + η 3 [ i ]n[ i ]
ˆ
Variance of the estimated watermark can be expressed as
σ w[i ] = η 12 [ i ]σ w[i ] + η 22 [ i ]σ s[i ] + η 32 [ i ]σ n[i ]
2
ˆ
2 2 2
The maximum watermarking-rate
1 ⎛ σ 2
⎞
= ⎜1 +
w [i ]
⎟
Blind Correlation based detector R C or
2
lo g 2
⎜
⎝ σ 2
n [i ] +σ 2
s[i ]
⎟
⎠
1 ⎛ σ 2
⎞
Informed Detector R In fo r m e d = lo g ⎜1 +
w [i ]
⎟
2
2 ⎜ σ 2 ⎟
⎝ n [i ] ⎠
1 ⎛ η 12 [ i ] σ w [ i ]
2
⎞
Blind ICA based Detector R IC A = lo g 2 ⎜1 + 2 ⎟
2 ⎜ η 2 [ i ] σ s2[ i ] + η 32 [ i ] σ n2[ i ] ⎟
⎝ ⎠
σ w [i ]
2
σ w[
η 12 [ i ] σ w [i ] Multimedia Systems Labi ]- UIC
2 2
< 2 < 2 ⇒ R Inform ed > R ICA > R Cor 27
σ n[i ] η 2 [ i ] σ s [i ] + η 3 [ i ] σ n[i ] σ n[i ] + σ s[i ]
2 2 2 2 2
Simulation Results
Watermark obeys Laplacian distribution with decay rate
= 0.1
Same watermark is embedded into 4 segments of the
host image i.e. r = 4
Watermark embedding/detection in DCT domain
Watermark is embedded into upper triangle matrix
coefficients excluding DC coefficient
Statistical ICA scheme based on Mean-Field Theory for
underdetermined mixtures is used to estimate
watermark from the watermarked image [Pedro et al,
2002]
Multimedia Systems Lab - UIC 28
Original image Original image Original image
Watermarked image Watermarked image Watermarked image
PSNR = 41 (dB) PSNR = 35 (dB) PSNR = 41 (dB)
Multimedia Systems Lab - UIC 29
Robustness Performance:
Additive White Gaussian Noise Attack
Multimedia Systems Lab - UIC 30
Robustness Performance:
JPEG Compression Attack
Multimedia Systems Lab - UIC 31
Conclusion
A blind detector for SS-based watermarking based on
UICA is presented with following features and
constraints,
Salient Features
Host-signal interference reduction capability
Improved detection/decoding performance
Very low error probability is possible
Applicable to all embedding domains and all media types
Limitations
Lower embedding capacity i.e. by a factor of r
Higher computational cost
Multimedia Systems Lab - UIC 32
Future Directions
Performance evaluation against watermark removal
attack from SS-based watermarking using proposed
detector
For more realistic comparison of the proposed scheme,
true estimate of η and η from the estimated
1 2
watermark is needed
Develop BSS scheme for underdetermined mixtures
based on estimating A and si separately
Multimedia Systems Lab - UIC 33
Questions ?
Thank you for Attention
More simulation results are available at
http://multimedia.ece.uic.edu/~hafiz/WM_UICA.htm
Multimedia Systems Lab - UIC 34
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