Cryptography Reference
In-Depth Information
for images with large very smooth regions, the payload capacity limit of the
G-LSB method does not exceed 1 bpp. The DE method could easily embed
more than 1 bpp. The payload capacity limit of the RS method is lower than
both the G-LSB and the DE method. By embedding a payload of the same
bit length, the embedded Lena image by the DE method is about 2 to 3 dB
higher than both the G-LSB and the RS method.
13.3.4 The Reversible Method Based on the Difference Expansion
Method of Vectors
Alatter [12] presented a high-capacity, data-hiding algorithm. The proposed
algorithm is based on a generalized, reversible, integer transform, which cal-
culates the average and pair-wise differences between the elements of a vector
extracted from the pixels of the image. Several conditions are derived and
used in selecting the appropriate difference values. Watermark bits are em-
bedded into either the LSBs (least significant bits) of selected differences or
alternatively the LSBs are one bit left-shifted versions of selected differences.
Derived conditions can identify which difference is selected after embedding
to ensure new vector computed from average and embedded difference has
grayscale values. To ensure reversibility, the locations of shifted differences
and the original LSBs must be embedded before embedding the payload. The
proposed algorithm can embed N−1 bits in every vector with a size of N1.
The proposed algorithm is based on a Generalized, Reversible, Integer
Transform (GRIT). We will now introduce the theorem for GRIT.
= a, d
1
,d
2
,,d
N−1
]
T
,if
v
Theorem 3
For
Du
= ⌊Du⌋
, then
u
=
D
−1
⌊v⌋
.
v
and
u
form a GRIT pair, where
D
is an
NN
full-rank matrix
with an inverse
D
−1
,
u
is an
N1
integer column vector,
a
is the weighted
average value of the elements of
u
,
d
1
,d
2
,,d
N−1
. These are the indepen-
dent pair-wise differences between the elements of
u
,
⌈⌉
and
⌊⌋
respectively.
It indicates round up or down to the nearest integer.
The proof is given in Alatter [12].
Alatter generalized the algorithm based on GRIT to vectors of length of
more than 3. Alatter uses an example in which N = 4. One possible value of
D is given by
⎡
⎣
a
0
/
c
a
1
/
c
a
2
/
c
a
3
/
c
⎤
⎦
−11 0 0
0−11 0
00−11
D =
,
(13.18)
N−1
where c =
a
i
, 0≤i≤3, and
i=0
