Cryptography Reference
In-Depth Information
Distribution of
correlation value
c
k
of
watermarked region
(
b
k
=1)
Distribution of
correlation value
c
k
of
watermarked region
(
b
k
=0)
Distribution of
watermarked region
after video processing
0
Fig. 7.13.
Transition of distribution by video processing.
Estimating Distribution from Correlation Values
To estimate mean µ
(f,r)
and variance σ
2(f,r)
from the, K correlation values
c
(f,r
k
s, we consider the case where all embedded bits are known, i.e., b
k
=1,∀k,
and the general case where all embedded bits are unknown:
(1) All embedded bits are 1: Fig. 7.14(a) shows the histogram of K cor-
relation values c
(f,r
k
s. The horizontal axis of the histogram represents
the value of c
(f,r
k
, and the vertical axis represents the frequency of that
value. The histogram describes the normal distribution, N (µ
(f,r)
,σ
2(f,r)
),
because K correlation values c
(f,r
k
s follow on N (µ
(f,r)
,σ
2(f,r)
) and are
independent on each k: the values of µ
(f,r)
and σ
2(f,r)
could therefore be
estimated using the following formulas if the number of embedded bits,
K, is large enough:
1
K
c
(f,r)
k
µ
(f,r)
=
,
(7.30)
k
1
K
c
(f,r)2
k
σ
2(f,r)
−µ
(f,r)2
.
=
(7.31)
k
(2) The values of all embedded bits are unknown: As shown in Fig. 7.14(b),
the K correlation values independently follow a distribution described by
either N (µ
(f,r)
,σ
2(f,r)
)orN (−µ
(f,r)
,σ
2(f,r)
) based on the embedded bit.
Thus, the histogram describes a mixture normal distribution comprising
two normal distributions.
In the following we take up the expectation-maximization (EM) algorithm
from inferential statistics supposing the case (2) and use it to estimate the
BER for a region.
