Discrete Cosine Transform DCT

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Discrete Cosine Transform (DCT)

• Richard Kelley

Many people would sooner die than think. In fact
they do. - Bertrand Russell
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Review

• Convert from RGB to YCbCr
• Pixels are grouped into 8x8 pixels called data
  units
• Discrete Cosine Transform(DCT) applied to each
  data unit to create an 8x8 map of frequency
  components that represent the average pixel value
  and successive higher frequency changes within a
  group

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Overview

• What are image transforms?
• What are they used for?
• Wave transforms (cos, sin)
• DCT

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What is an image transform?

• Computers store images as an NxN matrix of values
  that represent pixels
• For example
• 256 gray-scale image each pixel is stored as a
  value between 0 255
• 0 black pixel
• 255 white pixel
• Value between are shades of gray

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What are they used for?

• We can apply mathematical functions to the
  matrices to rotate, skew, compressin other words
  TRANSFORM an image
• Remember quad trees?
• Lets look at an example

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Image Transform Example
M This is obviously not the real matrix for
this image, just pretend for the sake of the
example
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Image Transform Example
Suppose (x) transform function
In this case its an invert function
(M)
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More practical example
M
M
Apply some arbitrary transform on M
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More practical example
M
M
Notice how the higher values (low frequency) are
now positioned toward the top left and the lower
values (high frequency) are positioned toward the
bottom right
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Wave Transforms

• DCT and Fourier transforms convert images from
  time-domain to frequency-domain to decorrelate
  pixels
• Time-domain
• x-axis time, y-axis amplitude
• Frequency-domain
• x-axis frequency, y-axis amplitude

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Wave Transforms
Amplitude
Frequency
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DCT One Dimensional
where
n size p pixel G coefficients
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DCT 2D
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DCT

• Remember that JPEG breaks an image into 8x8 units
• So for DCT n 8
• Each pixel is scanned and the transform is
  applied
• Just like our example in the beginning We get a
  matrix with new values

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DCT Frequency Distro
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DCT Frequency Distro
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DCT Why does it do this?

• DCT takes advantage of redundancies in the data
  by grouping pixels with similar frequencies
  together
• Higher frequencies lower number
• Lower frequencies higher number
• If lossy compression is acceptable, then each
  data unit can then be divided by quantization
  coefficient (QC)

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DCT (cont)
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Summary

• What are image transforms?
• What are they used for?
• Wave transforms (cos, sin)
• DCT

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Sources

• Salomon D. A Guide to Data Compression Methods,
  2002
• Salomon D. Data Compression The Complete
  Reference, 2000
• Lehar S. An Intuitive Explanation of Fourier
  Theory. http//cns-alumni.bu.edu/slehar/fourier/
  fourier.html February 2005
• Marshall D. Discrete Cosine Transform,
  http//www.cs.cf.ac.uk/Dave/Multimedia/node231.htm
  lDCTbasis February 2005
• Cabeen K Gent P. Image Compression and the
  Discrete Cosine Transform. http//online.redwoods
  .cc.ca.us/instruct/darnold/laproj/Fall98/PKen/dct.
  pdf February 2005

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Questions?