Cryptography Reference
In-Depth Information
efficients representing the smallest frequencies will often be in
this set, but it isn't guaranteed.
Another solution is to order the coefficients according to their
visual significance. Figure 14.13 shows a zig-zag ordering used
to choose the coefficients with the lowest frequencies from a
two-dimensional transform. The JPEG and MPEG algorithms
use this approach to eliminate unnecessary coefficients. Some
authors suggest skipping the first
coefficients in this ordering
because they have such a big influence on the image. [PBBC97]
Choosing the next
l
coefficients produces candidates that are
important to the description of the image but not too impor-
tant.
k
3. Create a
, to be hidden.
These can be either simple bits or more information rich real
numbers. This information will not be recovered intact in all
cases, so it should be thought of more as an identification num-
ber, not a vector of crucial bits.
k
-element vector,
{b
1
,b
1
,b
2
,...b
k−1
}
4. Choose
, a coefficient that measures the strength of the em-
bedding process. This decision will probably be made via trial
and error. Larger values are more resistant to error, but they
also introduce more distortion.
α
5. Encode the bit vector in the data by modifying the coefficients
with one of these functions they suggest:
• y
i
=
y
i
+
αb
i
• y
i
=
y
i
(1 +
αb
i
)
• y
i
=
αb
i
y
i
e
6. ComputetheinverseDCTorFFTtoproducethefinalimageor
sound file.
The existence of the embedded data can be tested by reversing
the steps. This algorithm requires the presence of the original image,
a problem that severely restricts its usefulness in many situations.
The steps are:
1. ComputetheDCTorFFToftheimage.
2. Compute the DCT or FFT of the original image without embed-
ded data.
3. Compute the top
k
coefficients.
