Watermarking Wigner Distribution A TimeFrequency Approach

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Watermarking Wigner DistributionA
Time-Frequency Approach

• Bijan G. Mobasseri
• ECE Department
• Villanova University
• Villanova, PA 19085

Funded by the US Air Force Office of Scientific
Research
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Outline

• Motivation
• Time-frequency distributions
• Wigner distribution
• Watermarking model
• Embedding and detection
• Capacity
• Future work

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Motivation

• 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

distinct keys
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Time-varying data hiding

• Watermarking of time-frequency distributions can
  follow a trajectory in time-frequency plane,
  potentially complicating steganalysis efforts
• Attacks with known T-F signatures can be
  circumvented
• An N-point signal has N2 TF distribution cells. A
  substantial fraction is available for watermarking

f
t
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Previous work

• The following is most likely the only work which
  follows a similar concept.
• 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

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Generating TFDWigner Distribution

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

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WD at work
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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

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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, find the
  corresponding signal x whose Wigner distribution
  is closest to Y in some sense

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Setting up the problem
?2HM ?1
?2
f2
?
?2? ?2
?1Mf1
f1
M
?
C(1)
?1Mf1
R(2)
?inadmissible
?admissible Mmapping function Htransformation
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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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Example time-frequency filtering
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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

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Error surfaces
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Error surfaces
Q30
Q50
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Which component to watermark?

• JPEG follows YUV(value-hue-saturation) color
  model.
• We have found that the TF signature of saturation
  band, when subjected to compression, is most
  robust

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MSE analysis
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Watermarking Geometry

• Tile the image
• Exhaustively
• Randomly (keyed)
• Embed one bit in WD of each block
• Use a unique key per block. Image is then tiled
  by a reference template

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Algorithm Summary
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Watermark strength vs. image PSNR
4x4 blocks, each carrying one bit
Q50
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Results
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Data hiding in saturation band16x16 blocks
Virtually identical performance across all
Q-factors
Q5
Q50
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Capacityare TF cells independent?

• Richard01 has shown that
• For all , the number of linearly independent
  components of discrete WD of x is upper bounded
  by for N even.
• For 8x8 blocks, there are 4096 components of
  which1056 are independent
• 8x8 DCT produces a maximum of 64 coefficients

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Payload numbers

• CapacityN2/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 Q25

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Conclusions

• A new transform domain for watermarking is
  introduced
• It features high capacity, low probability of
  intercept and low Q-factor operation
• Need work on blind detection
• More detailed comparison with DCT w/m
• Steganalysis benchmarking