Improved Spread Spectrum - A New Modulation Technique for Robust Watermarking

7/28/2019 Improved Spread Spectrum - A New Modulation Technique for Robust Watermarking

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Improved spread spectrum:a new modulation technique

for robust watermarkingIEEE Trans. On Signal Processing, April 2003

Prof. Ja-Ling Wu

Graduate Institute of Networking and Multimedia


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Spread-spectrum based watermarking,where b is the bit to be embedded


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2005-12-05 GINM, NTU 3

o u: the chip sequence (reference pattern), withzero mean and whose elements are equal to

+u or -u (1-bit message coding)

o inner product:

o norm: ||x||

o Embedding: s = x+ bu

o Distortion in the embedded signal:

D = ||s - x|| = ||bu|| = ||u|| = u2

o Channel noise: y= s + n

-

=

1

0

1 N

iiuix


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2005-12-05 GINM, NTU 4

Detection is performed by firstcomputing the (normalized) sufficient

statistic r: normalized correlation

u2

||u|| ||u||

and estimating the embedded bit by

b = sign( r )

r

b + + b + +x n~ ~

^


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2005-12-05 GINM, NTU 5

We usually assume simple statisticalmodels for the original signal x and the

attack noise n:

both to be samples from uncorrelated white

Gaussian random process

xi N(0,x2)

ni N(0,n2)


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2005-12-05 GINM, NTU 6

Then, it is easy to show that the sufficient

statistic ris also Gaussian, i.e.,

r N(mr,r2)

where

mr = E( r ) = b b {0,1}

x2 +n

2

Nu2

r2


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2005-12-05 GINM, NTU 7

Lets consider the casewhen b = 1.

Then, an error occurswhen r< 0, andtherefore, the errorprobabilityp is given by

forb = 1, mr

= E( r ) = b = 1

( )

( )

function.errorarycomplement

theiswhere

12

1

2

1

22

1

22

1

)1|0Pr(

2

2

2

2

22

2

+

=

+=

=

=


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2005-12-05 GINM, NTU 8

The same error probability is obtainedunder the assumption that

b = -1 but r> 0^

( mr/r )

If we want an errorprob. Better than 10-3,then we need

mr/

r> 3

Nu2

> 9(x2

+n2

)

-3


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2005-12-05 GINM, NTU 9

In general, to achieve an error probabilityp, we need

Nu2 > 2 (erfc-1(p))2(x

2+n2)

One can trade the length of the chip

sequence Nwith the energy of thesequence u

2 !!


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2005-12-05 GINM, NTU 10

Main idea:by using the encoder knowledge aboutthe signal x (or more precisely, theprojection ofx on the watermark), onecan enhance performance by modulatingthe energy ofthe inserted watermark tocompensate for the signal interference.

New Approach via Improved-SS


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2005-12-05 GINM, NTU 11

We vary the amplitude of the inserted chip

sequence by a function (x,b):

s = x+ (x,b)u

where, as before

x= / ||u|| : signal interference

SS is a special case of the ISS in whichthe function is made independent ofx.

~

~

~


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2005-12-05 GINM, NTU 12

Linear Approximation:

is a linear function of x

s = x+ (b-x)u

The parameters and control thedistortion level and the removal of thecarrier distortion on the detectionstatistics. Traditional SS is obtained bysetting = 1 and = 0.

~


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2005-12-05 GINM, NTU 13

With the same channel noise model as

before, the receiver sufficient statistic is

||u||

The closer we make to 1, the more theinfluence ofx is removed from r.

The detector is the same as in SS, i.e.,

the detected bit is sign(r).

r b + (1 -) x + n~ ~

~


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2005-12-05 GINM, NTU 14

The expected distortion of the new system isgiven by

To make the average distortion of the new systemto equal that of traditional SS, we force E[D]=u

2,and therefore

[ ]

[ ]2

2

222

22

1

~

u

u

x

u

N

xbE

xsEDE

ss

sla

sla

44 344 21

=

+=

-=

-=

2

222

u

xu

N

N

s

sls

a

-

=


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2005-12-05 GINM, NTU 15

To compute the error probability, all we need isthe mean and variance of the sufficient statistic r.

They are given by

Therefore, the error probabilityp is

2

2222 )1(

u

xn

r

r

N

bm

s

slss

a

-+=

=

{ }

-+-=

=

=


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2005-12-05 GINM, NTU 16

We can also writep as a function of the relativepower of the SS sequence Nu

2/x2 and

the SNR x2/n2

By proper selection of the parameter , theerror probability in the proposed method can be

made several orders of magnitude better thanusing traditional SS.

-+

--

=

-+

-

=

2

2

22

2

2

2

2

)1(1

1

2

1

2

1

)1(2

1

2

1

l

l

ls

s

ls

s

SNR

sspowererfc

N

erfcp

x

n

x

u


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2005-12-05 GINM, NTU 17

The three lines correspond to values to 5, 10,

and 20dB SNR (with higher values having smallererror probability).

Solid linesrepresent a 10-dB SIR and dash

lines represent a7-d SIR

SIR: Signal-to-interferenceratio


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2005-12-05 GINM, NTU 18

As can be inferred from the above figure, the errorprobability varies with , with the optimum valueusually close to 1.

The expression for the optimum value for can becomputed from the error probabilityp byand is given by

Note: for Nlarge enough, opt

1 as SNR

0=

l

p

-

++-

++=

2

22

2

2

2

2

2

2

2

2

4112

1

x

u

x

u

x

n

x

u

x

n

opt

NNN

s

s

s

s

s

s

s

s

s

sl

Improved Spread Spectrum - A New Modulation Technique for Robust Watermarking

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