Transcript Fourier Transforms and Their Use in Data Compression

Fourier Transforms
and Their Use in
Data Compression
By Joseph Gehring
What is a Fourier Transform?
• From Simple Wikipedia:
–“A Fourier transform is a math
function that makes a sometimes less
useful function into another more
useful function.”
–“A Fourier transform really just shows
you what frequencies are in a signal.”
The Math
• The Fourier Transform is a generalization of
the Fourier Series
• Any periodic function can be represented as
an infinite sum of sines and cosines
Fourier Series
Fourier Transform
• Forward
• Inverse
• Symmetric Linear Transform
• a=0, b=-2*pi for signal processing
Fourier Transform
• Every function f(x) has a forward and inverse
Fourier Transform such that
• Given:
– Integral of f(x) exists
– Discontinuous at a finite number of points
– Function has a bounded variation
Discrete Fourier Transform
• For given input data:
– Reveals periodic elements
– Shows the relative strength of those periodic
elements
• Input sequence of real numbers results in
Fourier Transform output of complex numbers
• Efficiently computed using Fast Fourier
Transform
Some Clarification
• Fourier Series uses an infinite sum of sines
and cosines
• Fourier Transform uses an integral over an
infinite range to develop an approximation
• Discrete Fourier Transform uses a finite sum of
sines and cosines over a given range, based on
sampling rates and sample length
– In music, the sample rate is usually set to 44,100
samples/second based on CD quality
Approximating a Square Wave
Fast Fourier Transform
• Efficient algorithm reducing the number of
computations required to determine the
discrete Fourier Transform of a function from
O(n^2) to O(n*log2(n))
• Has been used in mp3 and JPG compression
• Ultimately, even the FFT could not compete
with the Discrete Cosine Transform, which is
the cosine portion of the Fourier Transform,
and uses only real values
Compression
• The compression ratio offered by use of the
Fourier Transform is dependent on the quality
required by the application
• The higher quality the result needs to be, the
lower the compression ratio will be
• To create a more accurate output, more
coefficients are needed and the data cannot
be compressed as significantly
MP3
• Input file is sampled, usually at 44.1 kHz, and the
file is split into chunks of 576 samples each
(~.013 seconds)
• FFT or DCT is performed to convert time domain
to frequency domain
• Frequencies outside range of human hearing are
removed
• Coefficient data is stored in conjunction with a
32-bit header containing sound quality (frame)
• Multiple frames are combined to make a single
mp3 file
•
http://www.indiana.edu/~acoustic/s522/fourapdkp.html
JPG
The original image is broken up into
64 pixel blocks, each 8x8 pixels.
The DCT is taken of each 8x8 group
using a set of 64 basis functions.
Each numerical value in the group is
replaced with a new, smaller
number representing a coefficient
for a basis function.
Because these numbers are smaller,
the number of bits required to
represent them can be reduced. So,
each value in the group is truncated
to a lower number of bits.
By storing this lower number of bits
instead, the total amount of
information is compressed.
JPG (cont’d)
• In this image, we see how many coefficients
are required to achieve an approximation of
the original image. Using 15 coefficients for
the Fourier Transform instead of 64 original
values, a good
approximation
can be made
of the initial
image
JPG (Cont’d)
• The last step of JPG compression involves the
use of Huffman Encoding, which is a form of
variable bit length encoding that uses fewer
bits to represent values that occur more
frequently than those that occur more rarely.
• The 64 encoded values are then converted to
a linear sequence of values rather than an
array
In Conclusion
• All these methods have undergone periodic
updates depending on the complexity of input
data and the computing power available to
perform the tasks.
• As storage space becomes cheaper,
compression ratios can become less strict to
create closer approximations to original
information
This cat has some serious periodic components.
Works Cited
“Fourier Transform.” Simple Wikipedia. Web. 04 April 2011.
http://simple.wikipedia.org/wiki/Fourier_transform.
Various Articles. Wolfram MathWorld: The Web’s Most Extensive Mathematics
Resource. Web. 03 April 2011. http://mathworld.wolfram.com.
“MP3.” Wikipedia, The Free Encyclopedia. Web. 04 April 2011.
http://en.wikipedia.org/wiki/MP3.
Smith, Steven W. “JPEG (Transform Compression).” The Scientist and Engineer’s Guide
to Digital Signal Processing. Web. 04 April 2011.
http://www.dspguide.com/ch27/6.htm.
Yoo, Yerin. “Tutorial on Fourier Theory.” Department of Computer Science. University
of Otago. Web. 05 April 2011.
http://www.cs.otago.ac.nz/cosc453/student_tutorials/fourier_analysis.pdf
Handley, Mark. “3: Fourier Transforms.” Department of Computer Science. Columbia
University. Web. 05 April 2011.
http://www.cs.columbia.edu/~hgs/teaching/ais/slides/03-fourier.pdf
Munroe, Randall. “Fourier.” xkcd. Web. 05 April 2011. http://xkcd.com/26.