# RST invariant watermarking by using radon transform

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```					   Rotation, scaling and
translation invariant digital
image watermarking

Yan Liu

1
Our proposed method
Belongs to the first category, RST invariant domain
based approaches.
Based on log-polar mapping and phase-only filtering
method.
Measure the RST parameters by computing the cross-
correlation between a template and the magnitude of an
image in the log-polar domain.
Can be used by any image watermarking algorithms to
be robust against RST transformations.

2
Discrete Fourier transform (DFT)
Assume:
i ( x, y) – original image           I (u, v) – Fourier Transform of i ( x, y)
i '( x, y)– RST version image       I '(u, v) – Fourier Transform of i '( x, y)
 ,  ,( x0 , y0 ) – RST parameters

Relationship:
i' ( x, y)  i( ( x cos   y sin  )  x0 ,  ( x sin   y cos  )  y0 )
2
I ' (u, v)          I ( 1 (u cos   v sin  ),  1 (u sin   v cos  ))

 The magnitude of DFT of an image is independent of the
translational parameters ( x0 , y0 ) .

3
Log-polar mapping (LPM)
Using log-polar coordinates:
u  e  cos 
v  e  sin 

The magnitude of the Fourier spectrum:
2
I ' (  , )          I (   ln  ,   )
 Image scaling results in a translational shift of ln  along
 Image rotation results in a cyclical shift of  along the
angle  axis.
 Image translation has no effects in LPM domain.

4
Example

y                            

x                            
Cartesian domain             LPM domain

5
Matching template
We cut a small block as a matching template from the
LPM domain or the spatial domain of the original image.

Original                                          Cut a
image
DFT      LPM              template

optional
Cut a          DFT: discrete Fourier transform
template         LPM: log-polar mapping

The key technique is to match the template in the log-
polar domain with the watermarked image having
undergone RST attacks.

6
Filter Design
The template g and image f , where g is smaller
than f , the two dimensional normalized cross-
correlation function is defined as:

r (u, v) 
  f ( x  u , y  v ) g ( x, y )
x       y

  ( f ( x  u, y  v))
x       y
2

filter
Its Fourier transform:
r  IFFT[ F ( ,  )  G* ( ,  )]
where
 j F (  , )
F ( ,  )  FFT[ f (  , )]  AF ( ,  )e
 j G (  , )
G( ,  )  FFT[ g (  , )]  AG ( ,  )e
7
1.   Classical matched filter                               5.   Binary phase-only filter
 j G (  , )                                j BPOF (  , )
G( ,  )  AG ( ,  )e                                GBPOF ( ,  )  e

2.   Amplitude-only filter                                       where
 00 ,       Gr  0
GA ( ,  )  AG ( ,  )                                 BPOF ( ,  )   0
180 ,       Gr  0
3.   Inverse filter                                              Gr stands for the real part of
 j ( , )
e G                                       the Fourier transform G ( ,  )
GI ( ,  ) 
AG ( ,  )
4.   Phase-only filter
 j G (  , )
G ( ,  )  e

8
The matching results of different filters
(without rotation or scaling)

(a) Classical matched filter          (b) Amplitude-only filter                   (c) Inverse filter

(d) Phase-only filter               (e) Binary phase-only filter
9
The matching results of different filters
(with rotation and scaling)

(g) Classical matched filter           (h) Amplitude-only filter                 (i) Inverse filter

(j) Phase-only filter                (k) Binary phase-only filter
10
Phase-only filtering method
Phase information is more important than the amplitude
information in preserving the visual intelligibility.
Correlation detection is only optimal in the case that the signal
can be modeled as additive white Gaussian noise.
Using only the phase information of the matching template
and the LPM spectrum of the watermarked image undergone
RST transformations.

r  IFFT [ F ( ,  )  G ( ,  )]
*

where                               j F (  , )
F ( ,  )  e
 j G (  , )
G ( ,  )  e

11
The matching results of our method

(a) Without rotation or scaling   (b) With rotation and scaling

12
Three filter comparison criteria
1.    Noise robustness: signal-to-noise ratio (SNR)

 
M       N
2
| C0 |                            i 1    j 1
C'ij 2
SNR  10 log10              with     MSE 
MSE                                 M N

2.    Peak sharpness: peak-to-correlation energy (PCE)
| C0 |2
PCE  10 log10                                 i 1  j 1
M       N
with    CPE                    Cij 2
CPE

3.    Horner efficiency:

H 
 | f ( x , y ) g * ( x , y )|2 dxdy

 | f ( x , y)|2 dxdy

13
Performance results
SNR (dB)            PCE (dB)            Horner efficiency (%)
Filters
No RS       RS      No RS        RS         No RS             RS
New        44.04     37.86     44.09      31.34         100             100
POF        44.04     37.86     34.09      25.39         100             100
BPOF        40.09     36.32     30.20      23.46         100             100
Classical    29.32     32.47     16.17      16.11        58.09           51.66
AOF        28.39     31.07     15.22      15.39        5809            51.66
Inverse     32.54     22.45     29.23      20.02     2.65 x 10-5      1.04 x 10-5

New: new Phase-only filtering method     Classical: Classical filter
POF: Phase-only filter                   AOF: Amplitude-only filter
BPOF: Binary Phase-only filter           Inverse: Inverse filter

14
Rectification scheme
1)   Calculation of rotation and scaling parameters:
Compute the cross-correlation between a template and the
watermarked image in the LPM domain.
Suppose the coordinates of the peak is ( 1 ,1 ) . We know the
original position of the template is ( 0 ,0 ) .
So, the translations in the LPM domain are:
    1  0

    1  0
Therefore, the rotation and scaling parameters in the spatial
domain are:
      360 0   
 ' 
           N'
ln( rmax )  

 ' e
M'

15
Rectification scheme
2)   Calculation of translation parameters:
Compute the cross-correlation between a template and the
watermarked image in the spatial domain.
Suppose the coordinates of the peak is ( x1 , y1 ) . We know the
original position of the template is ( x0 , y0 ) .
So, the translation parameters in the spatial domain are:
  x  x1  x0

  y  y1  y0
The detected translation parameters are quite accurate when
shifting the image by the integer pixels. If shifting the pixels by
non-integer pixels, the imprecision of the translation factors are at
most 0.5.

16
Detection accuracy of rotation
             '        |   '|    To increase the computation
precision, we set the sampling
10      2     1.05470    0.05470
number along the  axis to be
20      6     2.10940    0.10940     1024, along  axis to be 512.
100     29    10.19530    0.19530
200     57    20.03910    0.03910        : real rotation angles.
450     129   45.35160    0.35160         : detected translational
700     200   70.31250    0.31250      shift in the LPM domain.
900     256      900         00          ' : theoretical rotation
1000     285   100.19530   0.19530      angles.
1200     341   119.88280   0.11720       |   '| : the imprecision
of the rotation angles
1400     398   139.92190   0.07810
between the real one and
1800     512     1800         00        the theoretical one. It is
180.50   513   180.35160   0.14840      almost always below 0.50 .

17
Detection accuracy of scaling

          '      |   '|    : real scaling ratios.
0.6   -45   0.5958   0.0042        : detected translational
shift in the LPM domain.
0.7   -31   0.7000      0
0.8   -20   0.7944   0.0056
 ' : theoretical scaling
ratios.
0.9   -9    0.9016   0.0016
|   '| : the imprecision
1     0       1         0        of the scaling ratios
1.1   8     1.0964   0.0036      between the real one and
1.2   16    1.2022   0.0022      the theoretical one. It is
1.3   23    1.3033   0.0030
always below 0.01.

18
Watermark embedding process
Watermarked                  Original       Template 2   1
image                      image
1                      1   Watermark
3      2                                    data 1

IDFT                     DFT

2                      2   Watermark
3                                           data 2

ILPM                     LPM

3                      3   Watermark
data 3
1 : Appl. 1, Spatial domain
Template 1
2 : Appl. 2. Fourier domain
1     2    3
3 : Appl. 3. LPM domain                                           19
RST parameters detection
Watermarked     1
image

Rectified
Watermarked
DFT
image
1
LPM                         Template 2
1 2 3
Template 1
Correlation 2
Correlation 1
Original    1 2 3                               1     Original
location of                                           location of
template 1                                            template 2
R.S.         Translational
1 : Application 1.     parameters       parameters

2 : Application 2.
3 : Application 3.                                             20
Watermark detection process
Watermarked
image
R.S.              1 2
parameters
Rectified
Translational     1         watermarked
parameters                    image
1                  2

Watermark                                            DFT
Watermark
data 1                                                          data 2
Similarity 1                 Similarity 2

Threshold 1                                                      Threshold 2

1 : Application 1.
Watermark ?
2 : Application 2.
21
Watermark detection process
Watermarked   3    Extract
image          watermark
sequence
Watermark
data 3
DFT

Similarity 3
LPM
R.S.                                              Threshold 3
parameters
Rectify
watermark        Watermark ?
sequence

3 : Application 3.
22
Scaling without rotation

1                                        2

3

1 : Application 1.
2 : Application 2.
3 : Application 3.                                      23
Rotation with cropping

1                                      2

3

1 : Application 1.
2 : Application 2.
3 : Application 3.                                    24
Scaling and Rotation

1                                    2

3

1 : Application 1.
2 : Application 2.
3 : Application 3.                                  25
JPEG compression

1                                2

3

1 : Application 1.
2 : Application 2.
3 : Application 3.                              26
Noise pollution
Gaussian noise pollution
Application I      Application II      Application III
Variance
Cor. 1   Cor. 2     Cor. 1   Cor. 2   Cor. 1       Cor. 2
0.001     5.7378   0.1255     0.9747   0.0242   0.9844       0.1491
0.01     5.6323   0.1763     0.6286   0.0428   0.8804       0.1533

Salt pepper noise pollution
Noise      Application I      Application II      Application III
density    Cor. 1   Cor. 2     Cor. 1   Cor. 2   Cor. 1       Cor. 2
0.01     5.2223   0.1179     0.8431   0.0122   0.9744         0
0.1      5.1579   0.2503     0.3634   0.0463   0.9692       0.0638

Cor. 1 : watermarked image. Cor. 2 : unwatermarked image.
27
Performance on different images
3.5                                                                                                       0.6

3                                                                                                        0.5

2.5                                                                                                       0.4
Correlation

2

Correlation
0.3
watermarked
unwatermarked
1.5                                                                                                       0.2
watermarked
unwatermarked
1                                                                                                        0.1

0.5                                                                                                            0

0                                                                                                        -0.1
0   10   20   30   40     50     60                  70    80    90   100                                    0        10        20        30     40     50     60   70   80    90   100
Images                                                                                                                        Images

1                                     1
2
0.9

0.8

0.7

0.6
3
Correlation

Watermarked
0.5
Unwatermarked

0.4

1 : Application 1.                                                   0.3

0.2
2 : Application 2.
0.1

3 : Application 3.                                                    0
0    10    20    30    40     50               60           70        80        90        100                                        28
Images
(a) Rotated by 45 degrees counter-               (c) Scaled by 0.7               (e) Rotated by 45 degrees counter-
clockwise without scaling                                                      clockwise after being scaled by
0.7062

(b) Correlation with (a). The peak is    (d) Correlation with (c). The peak is   (f) Correlation with (e). The peak is
at (185,371).                            at (250,401).                           at (186,401).
29
* The original template position is (250,370)
More results of our method on different
template size (without rotation or scaling)

(a) 64x64, 8bpp   (b) 64x64, 4bpp   (c) 64x64, 2bpp

(d) 64x64, 1bpp   (e) 32x32, 8bpp   (f) 16x16, 8bpp
30
More results of our method on different
template size (with rotation or scaling)

(g) 64x64, 8bpp   (h) 64x64, 4bpp   (i) 64x64, 2bpp

(j) 64x64, 1bpp   (k) 32x32, 8bpp   (l) 16x16, 8bpp
31
Detection of image flipping
Four possible flipped version of an image:
        f FH ( x, y)  f ( M  1  x, y)

        f FV ( x, y)  f ( x, N  1  y)

              f   FD1
( x, y)  f ( y, x)
f

FD 2
( x, y)  f ( N  1  y, M  1  x)

The relationship in the Fourier transform:
        F FH (u, v )  F (  u, v )e j 2u ( M 1)

         F FV (u, v )  F (u, v )e j 2v ( N 1)

              F FD1 (u, v )  F (v , u)
 F FD 2 (u, v)  F (  v , u)e j 2v ( N 1) e j 2u ( M 1)


32
Detection of image flipping
 u  e  cos
Rewrite the above equations by:                      
 v  e  sin 

The relationship in the Fourier transform:
| F FH (u, v )|  | F (  u, v )|
 | F (  e  cos , e  sin  )|
 | F (e  cos(180 0   ), e  sin(180 0   ))|
| F FV (u, v )|  | F (u, v )|
 | F (e  cos , e  sin  )|
 | F (e  cos(   ), e  sin(   ))|
| F FD1 (u, v )|  | F (v , u)|
 | F (e  sin  , e  cos )|
 | F (e  cos(90 0   ), e  sin(90 0   ))|
| F FD2 (u, v )|  | F (  v , u)|
 | F (  e  sin  , e  cos )|
 | F (e  cos(270 0   ), e  sin(270 0   ))|
33
Detection of image flipping
The relationship in the LPM domain:
 LFH ( , )  L( ,180 0   )

 L (  ,  )  L (  ,  )
FV

 FD1
  L ( , )  L( ,90 0   )
 LFD2 ( , )  L( ,270 0   )


We represent the above equations in pixels:

 LFH ( , )  L( , N '   1)
 FV
 L ( , )  L( , N '   1)
                     1
 LFD1 ( , )  L( , N '   1)
                     4
 FD2                 3
 L ( , )  L( , N '   1)
                     4

34

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