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4 Conclusion
We have proved formulas for probabilities
P
m
and
P
fa
when an additive embed-
ding of WM, a linear filtering, an additive white noise attack and a correlation
detector are used. From these, there follows that whenever an attacker provides
some filtering of watermarked message without noticeable distortion of the CM
then either a degradation of the WM system appears if the WM sequence is white
noise, or no degradation appears if the WM sequence is selected as colored noise.
Thus, later approach in preferred when designing a WM system.
Our theoretical results are confirmed by a simulation of watermarked and
the resulting attacked images. They show a significant degradation of the WM
system after the filtering attack without significant distortion of CM. As for
the case of a colored noise WM sequence, the simulation results show some
improvement in comparison with white noise WM, but there is still a significant
degradation of the WM system in this case. This can be explained by another
type of frequency response that has been considered in theory. We have proved
that a more effective attack on the WM system is a combination of linear filtering
and additive colored noise attack. It remains as an open problem to prove the
formulas for
P
m
and
P
fa
in this case.
In order to simplify the situation we have presented a tile-based WM that
can be a real candidate for practical application of the WM system under the
condition of linear filtering attack. The statement concerning this model can be
extended to a general situation. Roughly speaking, one can claim that if CM can
be passed through low-pass filter with frequency response close to (26) without
noticeable distortions then there results a degradation of the best WM system
by a factor of
N/K
h
times. This statement is consistent with the well known fact
from communication theory [3] regarding
spread spectrum systems
. In this case,
the material presented in [1] is incomplete because the authors do not consider
the statistical properties of additive attack noise at all.
Besides the open problem mentioned above, there appear several open prob-
lems concerning the linear filtering attack. Among them there are the following:
the proof of the optimal receiver for the model of filtering and colored additive
noise attack, the extension of the theory to the case in which it is unknown for
the WM detector the frequency response of the attack filter, the extension of the
theory to the case of a blind WM detector, and the correction of theoretical re-
sults taking into account the property of the MSE criterion that underestimates
perceptual quality of the images. We will consider them in the near future.
References
1. Cox, I.J., Miller, M.L., Bloom, J.A.: Digital Watermarking. Morgan-Kaufmann
Publishers (2002)
2. Korjik, V., Morales-Luna, G., Marakov, D., Marakova, I.: A performance evaluation
of digital private watermarking under an additive noise condition. In: Proc. VII
Spanish Conf. on Cryptology and Information Security, U. Oviedo (2002) 461-470
3. Proakis, J.: Digital Communications, Fourth Edition. Mc Graw Hill (2001)
