Improved Spread Spectrum: A New Modulation Technique for Robust Watermarking IEEE Trans. On Signal Processing, April 2003 Multimedia Security

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

D = ||s - x|| = ||bu|| = ||u|| = σu 2 u: the chip sequence (reference pattern), with zero mean and whose elements are equal to +σu or -σu (1-bit message coding) inner product: <x,u> ＝ norm: <x,x> ＝ ||x|| Embedding: s = x + bu Distortion in the embedded signal: D = ||s - x|| = ||bu|| = ||u|| = σu 2 Channel noise: y = s + n

and estimating the embedded bit by Detection is performed by first computing the (normalized) sufficient statistic r : normalized correlation <y,u> <bu+x+n,u> <u,u> σu 2 <x,u> <n,u> ||u|| ||u|| and estimating the embedded bit by b = sign( r ) r ＝ ＝ ~ ~ ＝ b + + ＝ b + x + n ^

both to be samples from uncorrelated white We usually assume simple statistical models for the original signal x and the attack noise n: both to be samples from uncorrelated white Gaussian random process xi ～ N(0,σx2) ni ～ N(0,σn2)

Then, it is easy to show that the sufficient statistic r is also Gaussian, i.e., r ～ N(mr,σr2) where mr = E( r ) = b b {0,1} σx2 +σn2 Nσu2 σr2 ＝

Let’s consider the case when b = 1. Then, an error occurs when r < 0, and therefore, the error probability p is given by for b = 1, mr = E( r ) = b = 1

The same error probability is obtained under the assumption that b = -1 but r > 0 ^ If we want an error prob. Better than 10-3, then we need mr/σr > 3  Nσu2 > 9(σx2+σn2) -3 ( mr/σr )

In general, to achieve an error probability p, we need Nσu2 > 2 (erfc-1(p))2(σx2+σn2)  One can trade the length of the chip sequence N with the energy of the sequence σu2 !!

New Approach via Improved-SS Main idea: by using the encoder knowledge about the signal x (or more precisely, the projection of x on the watermark), one can enhance performance by modulating the energy of the inserted watermark to compensate for the signal interference.

x = <x,u> / ||u|| : signal interference We vary the amplitude of the inserted chip sequence by a function μ(x,b): s = x + μ(x,b)u where, as before x = <x,u> / ||u|| : signal interference  SS is a special case of the ISS in which the function μ is made independent of x. ~ ~ ~

Linear Approximation: μ is a linear function of x s = x + (αb-λx)u The parameters α and λ control the distortion level and the removal of the carrier distortion on the detection statistics. Traditional SS is obtained by setting α= 1 and λ= 0. ~

With the same channel noise model as before, the receiver sufficient statistic is <y,u> ||u|| The closer we make λ to 1, the more the influence of x is removed from r. The detector is the same as in SS, i.e., the detected bit is sign(r). αb + (1 - λ) x + n ~ ~ r ＝ ＝ ~

The expected distortion of the new system is given by To make the average distortion of the new system to equal that of traditional SS, we force E[D]=σu2, and therefore

To compute the error probability, all we need is the mean and variance of the sufficient statistic r. They are given by Therefore, the error probability p is

We can also write p as a function of the relative power of the SS sequence Nσu2/σx2 and the SNR σx2/σn2 By proper selection of the parameter λ, the error probability in the proposed method can be made several orders of magnitude better than using traditional SS.

Solid lines represent a 10-dB SIR and dash lines represent a 7-d SIR SIR: Signal-to-interference ratio The three lines correspond to values to 5, 10, and 20dB SNR (with higher values having smaller error probability).

As can be inferred from the above figure, the error probability varies with λ, with the optimum value usually close to 1. The expression for the optimum value for λ can be computed from the error probability p by and is given by Note: for N large enough, λopt → 1 as SNR →