Cryptography Reference
In-Depth Information
both parts. In the following, the matching for positive part is only described
(negative part can be treated in the same way).
Two quantizers, Q
0
(x)andQ
1
(x) are prepared: the former used to embed
zero and the latter to embed one,
Q
0
(x)=2j, t
j
<x<t
j+1
,j∈0, 1, 2,...,
(8.24)
Q
1
(x)=2j +1,t
j
<x<t
j+1
,j∈0, 1, 2,..., (8.25)
where x is a positive DCT coe
cient and t
0
= t
0
= 0. The decision threshold
values t
j
,j∈1, 2,...for Q
0
(x) are set so that they satisfy
h
0
1
2
N (t
j
<x<t
j+1
)=
for
j =0,
(8.26)
h
2j
for
j∈1, 2,...,
where N (t
j
<x<t
j+1
) depicts the number of coe
cients in the interval t
j
<
x<t
j+1
. Note that 1/2 in Eq. (8.26) means that half of relevant coe
cients
are used for embedding zero and its number is adjusted to h
0
or h
2j
of cover
image to preserve the histogram of cover image. The decision threshold values
t
j
,j∈1, 2,...for Q
1
(x) are similarly set as they satisfy
1
2
N (t
j
<x<t
j+1
)=h
2j+1
,j∈0, 1, 2,....
(8.27)
From Eqs. (8.26) and (8.27), it is found that the histogram preservation can be
realized if h
0
∞
j=1
h
2j
=
∞
j=0
h
2j+1
= N (0 <x<∞)/2 . The condition
+
∞
j=0
h
2j+1
, i.e., ♯even = ♯odd is infrequently approximately
satisfied in histograms for very low frequency components, but never hold
true in general. The relation ♯even > ♯odd generally holds true, and in high
frequency components, ♯even≫♯odd because h
0
is much larger than others.
Consider how to match after-embedding histogram with before-embedding
one under the relation of ♯even > ♯odd. We introduce a dead zone, 0 <x<
t
d
(t
d
< 0.5) in which DCT coe
cients are not used for embedding
4
. t
d
is
determined as it fulfills
∞
j=1
h
2j
=
h
0
+
N
d
= N (0 <x<t
d
)=♯even−♯odd.
(8.28)
Eq. (8.28) means that ♯odd is equal to ♯even with least N
d
coe
cients removed.
Then using t
d
and N
d
, the decision threshold values t
j
,t
j
,j∈1, 2,...for
Q
0
(x)andQ
1
(x) are set so that they satisfy Eqs. (8.29)(8.32),
4
In case that ♯even < ♯odd, we cannot introduce the dead zone, i.e., t
d
=0,
and cannot fully match two histograms. Partial matching is however possible
to match for smaller absolute value coe
cients (bins). In practice, matching for
larger absolute value bins is not needed since the number of samples in such a
bin is small and mismatch in such a bin is not distinguishable by steganalysis.
