Cryptography Reference
In-Depth Information
big changes in the coefficients produced by the transform. Some
researchers combat this with a polar transform that produces the
same coefficients in all orientations of the document. This solution,
though, often breaks if the document is cropped thus changing the
center. Every model has strengths and weaknesses.
16.3.1 Choosing the Coefficients
Another challenge is choosing the coefficients to change. Some sug-
gest changing only the largest and most salient. In one of the first
papers to propose a watermark scheme like this, Ingemar Cox, Joe
Kilian, Tom Leighton, and Talal Shamoon suggested choosing the
largest coefficients from a discrete cosine transform of the image.
The size guaranteed that these coefficients contributed more to the
final image than the small ones. Concentrating the message in this
part of the image made it more likely that the message would survive
compression or change.[CKLS96]
Others suggest concentrating in a particular range for perceptual
reasons. Choosing the right range of discrete cosine coefficients can
introduce some resistance to cropping. The function
cos(2
πx
)
,forin-
stance, repeats every unit while
cos(2π
1000x)
repeats every 1000 units. Awa-
termark that uses smaller, shorter waves is more likely to resist crop-
ping than one that relies on larger ones. These shorter waves also
introduce smaller, more localized distortions during the creation of
the watermark.
16.4 An Averaging Watermark
Cropping is one of the problems confronting image watermark cre-
ators. Artists frequently borrow photographs and crop them as
needed. A watermark designed to corral these artists must withstand
cropping.
An easy solution is to repeat the watermark a number of times
throughout the image. The first solution in this chapter accom-
plishes this to some extent by using decomposition techniques like
the discrete cosine transform. Many of the same coefficents usually
emerge even after cropping.
This example is a more ordinary approach to repeating the wa-
termark. It is not elegant or mathematically sophisticated, but the
simplicity has some advantages.
A watermark consists of an
block of small integers. For the
sake of simplicity, let's assume the block is
4
m × n
×
4
and constructed of
