Re: [WM]: watermark embedding: additive vs multiplicative

[Teddy Furon <teddy.furon@irisa.fr>]

Hi,
> Hi,
>
> As you know that, for spread spectrum-based watermarking, one can 
> embed the watermark in either an additive or multiplicative way (and 
> possibly other ways as well). In the case of BLIND watermark 
> extraction, some optimal detector should be employed.
> A number of authors think that multiplicative embedding is superior to 
> additive embedding in terms of wm robustness (or error rate).
> But my experients show a lack of support for this conclusion. My 
> questions are:
>
> 1) Have you ever compared the performance of these two embedding 
> methods? Do you really think multiplicative embedding is preferable?

Sure, in theory for Gaussian white host with no attack, the multiplicative scheme has an efficiency sqrt(2) higher than the additive
scheme.
- additive: e(NoNoise) = sqrt(Pw.L/Ph)
- multiplicative: e(NoNoise) = sqrt(2.Pw.L/Ph) with Pw (Ph) the power of the watermark signal (resp. host), L the length of the
vector, and e the efficiency (See classical book on detection for its definition, eg Vincent Poor, or Van Tree)

However, its efficiency decreases more rapidly with the attack distortion than for the additive case.
- additive: e = sqrt(Pw.L/Ph) * sqrt(Pn/Pn+Ph)
- multiplicative: e = sqrt(2.Pw.L/Ph)* (Pn/Pn+Ph)

With Pn power of the noise. So, it depends on Pn, whether additive is better than multiplicative.

>
> 2) For multiplicative embedding, if the signal features are modelled 
> by GGD, what is the optimum wm detector?
>
For the Gaussian case, the sufficient statistics of the Local Most Powerful test are based on the derivative of the likelihood:
- additive: t = sum_i y_i.w_i
- multiplicative: t=sum_i y_i^2.w_i

> Thanks for your opinions in advance.
>
I have written this from my remembrance, so be careful, there might be some errors.
> Robert

Teddy


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