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Department of Informatics, Aristotle University of Thessaloniki 1
Transform Based Watermarking
Solachidis Vassilios
Department of InformaticsAristotle University of Thessaloniki
Department of Informatics, Aristotle University of Thessaloniki 2
WatermarkingProof of ownership of digital data by embedding copyright statements
EmbedderDigital
dataKey
Watermarkeddigital
data
Detector
Digital data (possibly
watermarked)Key
•Watermarked•Not watermarked
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Basic idea
Spatial domain watermarking ♦not robust against compression and filtering♦should have lowpass characteristics
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–Robustness against attacks (filtering, compression)
Advantages of Transform Based Watermarking
♦ Watermark construction having specific frequency content
–Watermark perceptibility
♦Transform properties accelerates the detection (in geometrically distorted data)
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Watermarking in spatial / transform domain
TransformSignal Perceptual
analysis⊕
Watermark
Inverse Transform
Watermarked Signal
Signal Perceptual analysis⊕
WatermarkWatermarked Signal
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Watermark construction
1-D sequence → 2-D sequence
key Randomgenerator
1-D sequence of ♦ real numbers ~N(0,1) or
♦ ±1
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Watermark Embedding
– Modifications in the low frequencies cause visible changes in the spatial domain
– Compression and filtering affects the high frequencies of the transform and destroys the watermark
The watermark is added in the middle frequencies because
TransformSignal Perceptual
analysis⊕
Watermark
Inverse Transform
Watermarked Signal
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Low Low frequenciesfrequencies
Medium Medium frequenciesfrequencies
High High frequenciesfrequencies
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Watermark Detection
Correlation is used in most of the methods.
TransformSignal
Watermark
CorrelationDetectoroutput{0, not
watermarked1, watermarked
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Transform Domains
♦ Discrete cosine transform (DCT)♦ Discrete Fourier transform (DFT)♦ Fourier-Mellin transform♦ Discrete Wavelet transform (DWT)♦ Fourier descriptors
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DCT (discrete cosine transform)DCT (discrete cosine transform)
1 2
1 2
1 11 1 2 2
1 2 1 20 0 1 2
(2 1) (2 1)( , ) 4 ( , )cos cos
2 2
N N
n n
n k n kX k k x n n
N Nπ π− −
= =
+ += ∑ ∑
1 2 1 2( , ),x n n N N×
1 2
1 2
1 11 1 2 2
1 2 1 1 2 2 1 20 01 2 1 2
11 1
1 1
22 2
2 2
1 (2 1) (2 1)( , ) ( ) ( ) ( , )cos cos
2 2
1/2 0( )
1 1 1
1/2 0( )
1 1 1
N N
k k
n k n kx n n w k w k X k k
N N N N
kw k
k N
kw k
k N
π π− −
= =
+ +=
== ≤ ≤ −
== ≤ ≤ −
∑ ∑
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Watermark embedded in DCT (discrete cosine DCT (discrete cosine transform)transform) domain
Advantages
•Real output
•Resistance against JPEG compression
•Fast transform (especially when it is used in compressed images)
Disadvantages
•Not robust against geometric attacks
DCT (discrete cosine transform)DCT (discrete cosine transform)
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DCT can be performed at entire image
t, t`, original and watermarked signal
W watermark, a embedding power
•A pseudorandom sequence of real numbers is embedded in the frequency domain
•The coefficients of the N×N DCT are reordered in a vector using a zig-zag scan.
•Watermark is embedded according to:
t`= t + a | t |wPiva et al.
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}
8×8
•Select a block (pseudorandomly)
•Select a pair of midfrequency coefficients
•Modify the sign of their difference according to a bit value
•Select a block (Gaussiannetwork classifier decision)
•Using a DCT constraint or a circular DCT detection region modify the middle frequency coefficients
Kochet al.
DCT can be performed at each 8×8 block
Bors and Pitas
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Watermark embedded in DFT (discrete Fourier transform)DFT (discrete Fourier transform)domain
AdvantagesAdvantages
•Resistance against frequency attacks
•Properties that accelerates the detection of geometrically distorted image
DisadvantagesDisadvantages
•Complex output
•Calculating complexity (when size is not power of 2)
DFT (discrete Fourier transform)DFT (discrete Fourier transform)
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RotationRotation in spatial domain causes rotation of the Fourier domain by the same angle
CircularCircular shift in the spatial domain does not effect the magnitude of DFT
ScalingScaling in the spatial domain causes inverse scaling in the frequency domain
CroppingCropping in the spatial domain changes the frequency sampling step
Discrete Fourier transform properties
DFT (discrete Fourier transform)DFT (discrete Fourier transform)
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WatermarkWatermark: a ring that is separated in sectors and homocentric circles. The same value 1 or –1 is assigned in each watermark circular sector.
• ring→ middle frequencies
• sectors → resistant in slight rotation (±3 degrees)
full search only for degrees 6k, k=1,2,…,29
•Correlation for many
frequency steps can detect
the watermark in a cropped image
Solachidis and Pitas
DFT (discrete Fourier transform)DFT (discrete Fourier transform)
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Watermark embedded in FMT (FourierFMT (Fourier--MellinMellintransform)transform)
AdvantagesAdvantages
•Properties that accelerates the detection of geometrically distorted image
DisadvantagesDisadvantages
•Complex output
• Very big calculating complexity (2 fourier transforms – logpolar tranform)
•Not very accurate
Fourier Fourier MellinMellin transformtransform
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Cartesian coordinates
Log polarcoordinates
Fourier Fourier MellinMellin transformtransform
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•DFT
Amplitude resistant in
translation
•Cartesian → Log polar
(x,y) →(µ,?), x=eµcos(?), y=eµsin(?)
Scaling and rotation equals translation
Rotation by an angle ?’ (x,y) →(µ,?+?’)
Scaling by a factor ? (?x, ?y) →(µ+log(?),?)
•DFT
Amplitude resistant in translation,rotation, scaling
3 steps3 steps
Ruanaidh et al.
Fourier Fourier MellinMellin transformtransform
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Watermark embedded in waveletwavelet domain
•Spatial localization
•Frequency spreading
•Average values from each correlator from all the sub bands and levels
Tsekeridou and Pitas
DWT Discrete wavelet transformDWT Discrete wavelet transform
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Let LL be such a closed polygonal line that consists of NN vertices, each of them represented as a pair of coordinates ((xxii,,yyii).
We construct the complex signal:
1 1
2 2
n n
x iyx iy
z
x iy
+ + = +
M
Watermark embedded in the Fourier descriptorsFourier descriptors of a polygonal line
Solachidiset al.
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A watermark WW is added in the magnitude |Z||Z| of the Fourier coefficients of z
|Z? |=|Z| ⊕ pW , p power of the watermark
44TranslationTranslation affects only the DC term Z(0).Z(0). By not adding watermark to the DC term we obtain watermark immunity to translation.
44RotationRotation by an angle ?? results in phase shift of the Fourier descriptors. The magnitude of the FD remains invariant.
44ScalingScaling by a factor aa results in the scaling of the FD magnitude by the same factor. Normalized correlation overcomes this effect.
Fourier descriptorsFourier descriptors
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44Inversion of the traversal directionInversion of the traversal direction results in the same indexing reversal in the FD:
ZinvertionZinvertion(k)=Z(N(k)=Z(N--11--k)k)
4Solutions:
4Construct a symmetrical watermark
4Always embed the watermark in the same direction (e.g. clockwise). During detection determine the traversal direction and invert it, if needed.
44Change of the polygonal line starting pointChange of the polygonal line starting point affects only the phase of the FD.
44Reflection (mirroring)Reflection (mirroring) causes FD magnitude indexing reversal:
||ZreflectionZreflection(k)|=|Z(N(k)|=|Z(N--11--k)| k)|
4Solution: Construct a symmetrical watermark.
Fourier descriptorsFourier descriptors
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ReferencesReferences• A.Piva, M.Barni, E.Bartolini, and V.Cappellini “DCT-based watermarking recovering
without resorting to the uncorrupted original image”in Proc. IEEE Int.Conf.Image Processing (ICIP), vol 1, Santa Barbara, CA, 1997, p.520
• E.Koch, J.Rindfrey, and J.Zhao, “Copyright protection for multimedia data”, Digital media and electronic publishing, 1996
• A.Bors and I.Pitas, “Image watermarking using DCT domain constraints ” in Proc.Int.Conf.Image Processing (ICIP), Lausanne, Switzerland, Sept.1996
• V. Solachidis and I. Pitas, “Circularly symmetric watermark embedding in 2-D DFT domain”, IEEE Int. Conf. on Acoustics, Speech and Signal Processing (ICASSP'99), Phoenix, Arizona, USA, Vol.6, pages 3469-3472, 15-19 March 1999
• J.J.K.Ó Ruanaidh, F.M.Boland, and O.Sinnen, “Rotation, scale and translation invariant spread spectrum digital image watermarking”, Signal Processing (Special Issue on watermarking), vol.66, no.3, pp.303-318, May 1998
• S. Tsekeridou, I. Pitas, “Embedding Self-Similar Watermarks in the Wavelet Domain” , 2000 IEEE Int. Conf. on Acoustics, Systems and Signal Processing (ICASSP'00), vol. IV, pp. 1967-1970, Istanbul, Turkey, 5-9 June 2000
• V. Solachidis, N. Nikolaidis and I. Pitas, “Watermarking Polygonal Lines Using Fourier Descriptors”, IEEE Int. Conf. on Acoustics, Speech and Signal Processing (ICASSP'2000), Istanbul, Turkey, vol. IV, pp 1955-1958, 5-9 June 2000
• S.Katzenbeisser, F.Petitcolas, “Information hiding techniques for steganography and digital watermarking”, Artech house