Cryptography Reference
In-Depth Information
The authors use the discrimination function to capture the smoothness of the
groups. For example, the variation of the group of pixels (x
1
,x
2
,,x
n
)can
be chosen as the discrimination function f ():
n−1
f (x
1
,x
2
,,x
n
)=
x
i+1
−x
i
.
(13.2)
i=1
The purpose of the discrimination function is to capture the smoothness or
regularity of the group of pixels G. An invertible operation F with the
amplitude A can be applied to the groups. It can map a gray level value to
another gray level value. It is reversible since after applying it to a gray level
value twice produces the original gray level value. That is, F has the property
that F
2
=Identity or F (F (x)) = x, for all x∈P , where P =0, 1,, 255, for
an 8-bit gray-scale image. This invertible operation is called flipping F .The
difference between the flipped values and the original values is A.
A suitably chosen discrimination function f () and the flipping operation
F are utilized to define three types of pixel groups: Regular R, Singular S,
and Unusable U .
Regular groups:
G∈R⇔f (F (G)) >f(G);
Singular groups:
G∈S⇔f (F (G)) <f(G);
Unusable groups:
G∈U⇔f (F (G)) = f (G).
From the definitions of the R, S,andU groups, it is apparent that if G is
regular, F (G) is singular, if G is singular, F (G) is regular, and if G is unusable,
F (G) is unusable. Thus, the R and S groups are flipped into each other using
the flipping operation F . The unusable groups U do not change their status.
In a symbolic form, F (R)=S, F (S)=R and F (U )=U .
In the expression F (G), the flipping function F may be applied to all or to
selected components of the vector G (x
1
,x
2
,,x
n
). The noisier the group of
pixels G (x
1
,x
2
,,x
n
) is, the larger the value of the discrimination function
becomes. The purpose of the flipping F is to perturb the pixel values in an
invertible way by a small amount thus simulating the act of Invertible Noise
Adding. In typical pictures, adding small amount of noise, or flipping by a
small amount, will lead to an increase in the discrimination function rather
than to a decrease. Although this bias may be small, it will enable us to embed
a large amount of information in an invertible manner.
As explained above, F is a permutation that consists entirely of two-
cycles. For example, the permutation F
LSB
is defined as 0↔1, 2↔
3, , 254↔255 corresponds to flipping or negating the LSB in each gray
level. The permutation corresponds to an invertible noise with a larger ampli-
tude than two. The amplitude A of the flipping permutation F is defined as
0↔2, 1↔3, , 253↔255. The average change of under the application
of F is:
1
P
A =
x−F (x).
(13.3)
x∈P
