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Therefore we will base our theory on the simpler mean square error (MSE) cri-
terion but we will use the human visual testing on our simulation investigations.
We will consider only a zero-bit private WM system with informed detector .
In such a system, no WM message provides any further information than its
own presence or absence, the WM signal is kept secret and known only by the
authorized detector and the CM is known by the authorized detector (both WM
and CM can be regarded as a secret key of legal users).
The goal of any dishonest user ( attacker ) is either to remove the WM signal
or to disable it without noticeable distortion of the CM. In order to reach this
goal, an attacker can process the watermarked message performing different
actions: noise addition, filtering, recompression, geometric transformations (e.g.
rotation, translation, cropping or scaling), and so on. In the current paper we
consider only a linear filtering combined with an addition of random noise . This
model provides an extension of our results developed in [2] for the case of an
additive attack noise, without any filtering. It is worth to point-out that the
linear filtering attack already reported in the literature is very poor.
Some experimental results concerning a linear filtering attack are shown in [1].
Nevertheless they are incomplete and some of the authors conclusions contradict
our results, proven in our analysis and confirmed experimentally.
The performance of any WM system can be estimated using two probabilities:
P m . The probability of WM- missing : the WM detector fails to detect the WM
in spite of its presence.
P fa . The probability of WM- false alarm : the WM detector falsely claims the
WM presence.
The number of WM elements, N , embedded in the CM, which determines the
desired probabilities P m and P fa , is also very important (if the CM is an image
then N is upper bounded by the number of image pixels).
There are several different methods to embed a WM into the CM, but we will
consider only the so-called communication-based model with additive embedding ,
in which the watermarked message, namely the stegomessage (SM), is:
S ( n )= C ( n )+ W ( n ) ,
n =0 ,...,N āˆ’
1
(1)
where SM is S =( S ( n )) Nāˆ’ 1
n =0 ,CMis C =( C ( n )) Nāˆ’ 1
and WM is W =( W ( n )) Nāˆ’ 1
n =0 ,
the index n corresponds to discrete time instances and N is the WM length.
Our analysis can be applied to any kind of CM. However, without loss of
generality, we will keep in mind images as CM. Each entry C ( n ) corresponds to
the luminance of the n -th pixel in the image and n is the vector coordinates in
the 2D-message.
The paper is organized as follows: In Section 2 we prove the formulas of the
probabilities P m and P fa , in their general forms and in some particular cases
of different WM and additive noise sequences. Section 3 contains the results of
simulations for different watermarking of images and different attacks against
WM systems. In Section 4 we summarize the main results and we formulate
some open problems.
n =0
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