Cryptography Reference
In-Depth Information
1
2
N (t
d
<x<t
1
)=h
0
−N
d
,
(8.29)
1
2
N (t
j
<x<t
j+1
)=h
2j
,
j∈1, 2,...,
(8.30)
1
2
N (t
d
<x<t
1
)=h
1
,
(8.31)
1
2
N (t
j
<x<t
j+1
)=h
2j+1
,
j∈1, 2,...,
(8.32)
respectively. Eqs. (8.29) to (8.32) indicate that ♯odd and ♯even−N
d
are equal
to N (t
d
<x<∞)/2 and then histogram matching becomes possible. Note
that in HM-JPEG steganography, the XOR operation should be applied to
embedding message as applied in QIM-JPEG steganography.
The above-mentioned HM-JPEG steganography is a histogram matching
method using two quantizers, where the representatives of each quantizer are
given in advance and the intervals for each representative are set so as to
preserve the histogram of cover image. Though the usage of two quantizers
is common to HM-JPEG and QIM-JPEG, the histogram matching method is
different from QIM in that the chosen representative of a relevant quantizer
is not necessarily closest to a given input. Therefore stego images by HM-
JPEG might be more distorted than those by QIM-JPEG.
8.6.4 Experimental Results
Embedding performance of the proposed methods were evaluated comparing
with F5 using three standard images: Lena, Barbara and Mandrill.
These images are 512512 pixels in size, 8 bpp gray images, and were com-
pressed with quality factor 80. The least 21 frequency components, including
the DC component, in the zigzag scan of 64 components were used for embed-
ding experiments. The histogram change can be measured by Kullback-Leibler
divergence.
Experimental results are shown in Table 8.2. Fig. 8.15 shows four his-
tograms of (1, 1) frequency component for Lena: histogram of cover im-
age (before embedding) and after-embedding histograms by QIM-JPEG, HM-
JPEG and F5. The KL divergence value shown in Table 8.2 is the mean for
the least 21 frequency components except the DC component. The KL diver-
gence value by F5 is much larger than those by QIM-JPEG and HM-JPEG.
Smaller KL divergence values represent better histogram preservation. Fur-
thermore, QIM-JPEG and HM-JPEG achieved higher embedding rates with
less degradation of image quality (higher PSNR vales) than F5. Among the
two proposed methods, QIM-JPEG produced a bit higher PSNR stego images
than HM-JPEG, probably because in QIM-JPEG the closest representative
of a relevant quantizer to a given DCT coe
cient is chosen as a quantized
coe
cient.
