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nel. Section 4 discuss the results of simulation-based study of the influence of attrib-
utes specifying the SVD-based embedding procedure on both the capacity of the in-
visibly embedded data and robustness of its transmission through steganographic
communication channel distorted by
JPEG
compression of different quality. The
Conclusion summarizes the paper results and future work.
2
SVD of Digital Images and Information Hiding Technique
Assume that a digital image in the bitmap format is specified by a
m
×
n
matrix,
A
=
{
a
,
}
. If an image is represented in
RGB
format then it is specified by three
such matrices,
A
,
A
and
A
.
An arbitrary matrix
A
of size
m
×
n
can be represented by its SVD [8] in the form
i
j
m
,
n
(1)
∑
A
=
X
Y
i
i
=
=
r
T
T
λ
X
Y
=
i
i
i
1
X
,
X
,...,
X
where
X
,
Y
are orthogonal
m
×
m
and
n
×
n
matrices respectively,
and
1
2
m
Y
,
2
Y
,...,
Y
are their columns,
is a diagonal matrix with non-negative elements,
1
r
≤
min{
m
,
n
}
and
is the rank of the matrix
A
. Diagonal terms
λ
,
λ
,...,
λ
of
1
2
r
matrix
are called
singular values
(SV
)
of the matrix
A
and
r
is
the total number of
non-zero singular values. Columns of the matrices
X
,
Y
are known as
left
and
right
singular vectors of matrix
A
respectively.
There are several ways to calculate SVs
λ
,
λ
,...,
λ
. The most simple way is to
1
2
r
λ =
µ
µ
calculate them as
i=1,2,…,r,
where
is the
i-th
eigenvalue of the matrix
i
i
i
T
A
T
X
i
,
i
=
,
AA
A
, or
.
The left singular vector
is equal to the eigen-
T
µ
AA
vector of the matrix
corresponding to
. Similarly, the right singular vec-
i
Y
,
i=1, 2,.., r
, is equal to the eigenvector of the matrix
T
A
A
tor
that corresponds
µ
to its eigenvalue
. If an image is given in
RGB
format then it can be represented
by three such SVDs in the form (1). Thus, SVD of an image is decomposition of each
its matrix,
i
A
and
A
A
,
λ
X
Y
T
λ
X
Y
T
λ
X
Y
T
, into layers
,
,...,
. As a
B
1
1
1
2
2
2
s
r
r
λ >
λ
rule, singular values are enumerated in descending mode: if
then
i<j ,
and
i
j
λ
is the largest one.
SVD of an image was originally considered in [2]. In this work the author pro-
posed to use it for a
lossless
compression of images. Later a number of authors also
used SVD of digital image transform combined with other transforms in order to
increase the ratio of an image lossless compression. For example, to code images, in
[31] SVD transform is combined with Vector Quantization approach, and in [30] a
combination of SVD and Karhunen-Loeve transform is used to develop a hybrid
compression. In [9] a format for SVD-based
lossy compression
of digital images was
1
