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In this section we describe our data embedding system in detail. It is well known
that most natural images are low frequency signals [8], i.e., most of
the signal energy and significant components are contained in the low frequency
coefficients. Current image transforms such the DCT or DFT
compact the image energy into a relatively small number of coefficients.
Therefore, if one tries to embed the information directly into an image using
these transforms, there may not be enough coefficients to embed all the information
we are interested in hiding. One way to increase the number of coefficients that
can be utilized to embed information is to introduce some significant mid and high
frequency components into the image. That way, the host image spectrum will have
significant frequency components that are distributed over the entire spectrum.
One way to do this is to decorrelate the image samples using some key .
The key will scramble the image pixels in such a way the resulting image
would look like white noise to the viewer. Let be some natural image
in which some information is to be embedded and let be some scrambling
operator, operating on in a lossless and one-to-one fashion.
The output of this operator is
The key generates a vector of random integers of the same size as the
cover image, i.e., , where is the size of the image. The
cover image is also converted into a one dimensional vector .
The key scrambles , which is then converted into a two
dimensional signal . Figure 1-a shows an example of an
input image to the scrambler and Figure 1-b shows the output image
.
Note that the first order statistics and pixel values of
are the same as for . The next step is the embedding process. To do that we take the DCT of
, although other image transforms can be used as well. The
two-dimensional DCT of is given by [8]
Assume the image pixels are i.i.d uniformly distributed between {0-255} i.e.,
where , represents the spatial coefficient at position
. Each DCT coefficient is expressed as a weighted sum of i.i.d random
variables. By the central limit theorem [9], it is easy to show
that , follows a Gaussian distribution with zero
mean, i.e.,
The result is confirmed by computing the first and second order statistics and
by computing the kurtosis of the distribution. The kurtosis is a measure of the
degree of departure of a probability distribution from the Gaussian distribution,
and it is given by the following formula
where is the number of samples. If the value of the kurtosis is 3
then the pdf is Gaussian distributed. The average value of the kurtosis for
, which indicates that the probability density
function is indeed Gaussian distributed.
Since the channel in which the embedding will take place is Gaussian, we can model
our embedding system as transmitting digital information through a Gaussian
channel. Our goal is to maximize the number of bits that can be embedded in a secure
way, i.e., without being detectable. To do that we adjust the amplitude of the DCT
coefficients in small steps such that the resulting image in the spatial domain
will look perceptually identical to the original cover image. Our embedding scheme
does not assume access to the original image. Therefore, we embed the coefficients
in such a way the we can determine unambiguously whether a binary "0" or "1" is
embedded. To do that we use basic quantization techniques to determine the value
of the embedded bit. For example, when a coefficient is divided by some threshold
, and the resulting number is even then that will indicate a binary "1" ,
and if the result is odd that indicates a binary "0". When the embedding
process is completed we take the inverse DCT transform and then descramble the image
using the same key , to get the stego image .
When data is embedded in the transform domain, the space of pixel values changes
from integers to real numbers i.e., . Since the
pixel values of 8-bit gray
level digital images fall in the range , we need to quantize the
resulting image to restore its dynamic range. For small changes in transform
coefficients the real pixel values
of will fall in the range [0-255]. The simplest type of
quantization is one that rounds the pixel values to the nearest integer, i.e.,
where represents the noise due to rounding operation. The noise
distribution can be approximated by a uniform distribution
between [-0.5,0.5] in the spatial domain and it follows a Gaussian distribution
in the transform domain. Figure 2 shows
the noise distribution in the spatial domain and DCT domain.
Figure:
Noise distribution.
|
Note that the ripples are
due to the fact that we are dealing with discrete samples. The noise distribution
in the transform domain follows approximately zero mean Gaussian distribution.
This noise is inherent in the embedding process and the hidden information
must have enough power to survive this noise. Suppose the input message consists of binary data which is mapped to a polar format [10] with
amplitude and occur equally likely. The power in the hidden information
is
where is the power in each bit and is the length of the hidden message.
The message is embedded inside the signal , in an additive way, i.e.,
Where is a quantized value of
set in such a way that one can determine the value of the
embedded bit. The amplitude of is small compared to the amplitude of
.
Therefore the distribution of will also be Gaussian distributed with
zero mean. Assuming and are independent then the variance of
is
The decoding process consists of dividing the space of the coefficients into two
non-overlapping regions, one indicating binary "1"; and the other for a binary
"0" . The DCT coefficients are adjusted in such a way that after adding the data,
the result of dividing by some threshold , is either an even or an odd
number as shown below
where represents the received coefficient.
This technique suppresses the noise due to the host image completely. Therefore,
the cover image acts as a carrier to the information. Hence, the only
noise that needs to be considered is that introduced by the rounding operation.
This noise is zero mean Gaussian with a variance . The input data
take on equally likely values. Therefore, the distribution of the hidden message
consist of two impulses as shown in Figure 3-a.
The distribution of the carrier signal is Gaussian. Therefore, when we add two
independent signals the resulting signal will have a probability density function
(pdf) which is the convolution of the two [9] pdf's . Since the
amplitude of is very small compared with the amplitude of the DCT
coefficients of the host signal , the
resulting distribution will be approximately Gaussian as shown in Figure 3-b.
Therefore, we model our data hiding system as transmitting information carried
by a signal with a
Gaussian pdf through an additive white Gaussian noise channel. It is known that
for Gaussian channels maximum capacity is achieved if the input is also
Gaussian [11]. Therefore, we believe that this system can maintain high
data embedding rate capacity, with input power in the hidden signal which
depends on the perceptual quality of the host signal.
Next: System Security
Up: SECURE HIGH DATA EMBEDDING
Previous: Introduction
Muhammad Zubair Ikram
7/14/2000