10-05-2012, 02:01 PM
SECURE STEGANOGRAPHY: STATISTICAL RESTORATION OF THE SECOND ORDER DEPENDENCIES FOR IMPROVED SECURITY
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INTRODUCTION
We consider the problem of secure communication or steganography,
in which a secret message is hidden into an innocuous looking
host or cover to get a composite or stego image such that the very
presence of communication is not revealed. To detect the presence
of embedded data, steganalysis techniques exploit the changes in
the host statistics due to hiding. In this paper, we present practical
techniques for image steganography that provide improved security
in comparison to many prior schemes by closely matching
the second-order distribution rather than just the marginal statistics.
The techniques are based on a general steganography framework
called statistical restoration, initially proposed in our prior publications
([1, 2, 3]) and applied to marginal statistics .
PROBLEM FORMULATION
In the statistical restoration framework, the set of host symbols X is
divided into two disjoint sets: H for hiding and C for compensation.
Data is embedded using the hiding function f1 into the hiding set H
to get ^H , as shown in (1) later. We divide the host symbol set X into
2-D bins and find their respective bin-counts (number of terms per
bin). We use BX(i; j) to denote the bin-count of the (i; j)th bin of
X. Since the normalized bin-count gives the PMF, compensating for
the bin-counts is equivalent to restoring the PMF.
STATISTICAL RESTORATION AND THE EARTH MOVER’S DISTANCE
The EMD [9] between two PMF’s is defined as the minimum “work”
done in converting one PMF to the other. Here, work refers to the redistribution
of weights among the various bins in the discrete distribution.
The solution to the EMD problem returns the optimal transportation
flows among the bins. For our statistical restoration problem,
we have to convert a 2-D histogram BC to B^C , according to
(5), the normalized histogram being the PMF. Thus, EMD provides
the optimum way of redistributing weights in BC to obtain B^C .
Let S and T denote two 2-D signatures, each having M clusters.
The weight of each cluster is the fraction of points it contains. Let
the center for the kth (k = (i; j)) cluster of S be fsi; sjg while the
`th (` = (m; n)) cluster center of T is denoted by ftm; tng. The
square Euclidean distance between the kth cluster center of S and
the `th cluster center of T is called dk`.