Study of the distribution of DCT coefficients in image coding

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Study of the distribution of DCT coefficients in
image coding

• Johnna Anderson
• Shang Xue
• Sehee Kim

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Outline

• Introduction to problem
• Example
• Project goals
• Materials and Methods
• Results and Conclusions

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Introduction
Each cell represents a pixel 8x8 block of
grayscale values
Image size 1024 x 1024 pixels
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DCT

• Discrete Cosine Transformation
• Reduces the amount of information that needs to
  be stored to produce image.
• Partition image into 8x8 blocks of pixels
• Apply DCT to each of these 8x8 matrices
• Apply quantization to every entry in matrix
  (rounds every number to an integer value).
• Results in matrix with a large number of zeros.

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DCT example

• 8x8 image block is turned into a block of DCT
  coefficients which consists of mostly 0s.

8x8 image block
8x8 DCT coefficients
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DCT Distribution Theory

• DCT coefficients are believed to have a
  distribution
• Centered at 0
• Heavy Tailed
• Why is knowing the distribution helpful?
• Allows for more efficient coding of values and
  quantization methods

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DCT Distribution Theory

• First position should be Normal (0,s2).
• All other positions should be Laplacian (Double
  Exponential) with mean 0 and variance 2b2.
• Each position has its own distribution.

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Project Goals

• Apply DCT to eight images.
• Use visualization techniques to examine the
  distributions of coefficients.
• Histograms
• QQplots
• Examine distribution for each position across all
  eight images.

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Images
1
2
3
4
10
Images
5
6
7
8
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Discrete Cosine Transformation

• The variance of the DCT coefficients for each
  position is proportional to the variance of the
  pixels in each 8x8 block.

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Materials and Methods

• Quantization is not applied to the DCT values.
• To make the distributions comparable across all
  eight images we standardized the DCT
  coefficients for each image.
• Divide each DCT value by its block pixel standard
  deviation.
• QQplots plot the percentiles of a Laplacian
  distribution against the percentiles of the
  images standardized DCT values.
• Theoretical distribution has mean 0 and a
  variance equal to the variance of the DCT values
  pooled across all eight images.

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Results and Conclusions

• 43 out of 64 positions are well approximated by
  their theoretical distributions.
• Roughly 67.
• Images 6 and 8 tend to have highest peaks.

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Position 1
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Position 1
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Position 2
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Position 2
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Position 15
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Position 15
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Position 17
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Position 17
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Position 29
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Position 29
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Position 57
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Position 57
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Position 64
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Position 64
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Questions?