Cryptography Reference
In-Depth Information
Using these localized functions can help resolve problems that
occur when signals change over time or location. The frequencies in
audio files containing music or voice change with time and one of the
popular wavelet techniques is to analyze small portions of the file. A
wavelet transform of an audio file might first use wavelets defined
between
0
and
2
seconds, then use wavelets defined between
2
and
4
seconds, etc. If the result finds some frequencies in the first window
but not in the second, then some researchers say that the wavelet
transform has “localized” the signal.
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Theories of Everything
lists topics “not by the
same author” including
Nothing But Cosines
.
The simplest wavelet transforms are just the DCT and FFT com-
puted on small windows of the data. Splitting the data into smaller
windows is just a natural extension of these algorithms.
More sophisticated windows use multiple functions defined at
multiple sizes in a process called
multi-resolution analysis.
The easi-
est way to illustrate the process is with an example. Imagine a sound
file that is 16 seconds long. In the first pass, the wavelet transform
might be computed on the entire block. In the second pass, the
wavelet transform would be computed on two blocks between
0
and
7
seconds and between
8
and
15
seconds. In the third pass, the trans-
form would be applied to the blocks
0
to
3
,
4
to
7
,
8
to
11
,and
12
to
15
. This is three stage, multi-resolution analysis. Clearly, it is easier
to simply divide each window or block by two after each stage, but
there is no reason why extremely complicated schemes with multi-
ple windows overlapping at multiple sizes can't be dreamed up.
Multiple resolution analysis can be quite useful for compression,
a topic that is closely related to steganography. Some good wavelet-
based compression functions use this basic recursive approach:
1. Use a wavelet transform to model the data on a window.
2. Find the largest and most significant coefficients.
3. Construct the inverse wavelet transform for these large coeffi-
cients.
4. Subtract this version from the original. What is left over is the
smaller details that couldn't be predicted well by the wavelet
transform. Sometimes this is significant and sometimes it isn't.
5. If the differences are small enough to be perceptually insignif-
icant, then stop. Otherwise, split the window into a number of
smaller windows and recursively apply this same procedure to
the leftover noise.
This recursive, multi-resolution analysis does a good job of com-
pressing many images and sound files. The wider range of choices
