Image Transforms for Robust Coding

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Image Transforms for Robust Coding

• Javier Pinilla-Dutoit
• Educational Technology Research Group

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Introduction

• The fundamental problem of communication is that
  of reproducing at one point either exactly or
  approximately a message selected at another
  point. (Claude Shannon, 1948)
• What information should be transmitted?
• How should it be transmitted?

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Communication System
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Source Coding

• Goals
• Reduction of redundancy
• Removal of irrelevancy (irreversible)
• Stages
• Transformation
• Quantization (lossy)
• Entropy coding

Image samples Transform coefficients Symbol
stream Bit stream
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Lossless and Lossy

• Lossless (21 to 81)
• Limited by the entropy of the message (Claude E.
  Shannon)
• Lossy (1001)

Entropy is a measure of information content the
more probable the message, the lower its
information content, the lower its entropy
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Transformation

• Example rotation of coordinate axis
• Improves statistical distribution of image
  elements
• Decomposition into components of differing
  variance

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Transformations for Image Coding

• Block transforms (blocks in the spatial domain)
• Fixed size
• Quadtrees
• Sub-band decompositions (blocks in the frequency
  domain)
• Uniform
• Logarithm
• Pyramids
• Wavelet
• Wavelet packets

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Quantization
Four-level digital
Eight-level digital
representation
representation
Two-bit resolution
Three-bit resolution
Can be used to exploit features of the human
visual system
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Quantization (bit allocation)

• Example "rounding to the nearest integer"
• Non-uniform (variable bit allocation)
• Based on the statistics of the source (Laplacian
  quantizer)
• Based on the human visual system
  (perceptually-tuned quantization)

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Entropy Coding

• Methods based on repeated characters run-length
  encoding
• The repeated character is replaced by the number
  of occurrences and by the character itself
• e.g. AAAABBBCCCCC (12 symbols) is coded as 4A3B5C
  (6 symbols) 21
• Methods based on probability of occurrence
  Huffman coding, arithmetic coding
• Symbols that occur more often are assigned
  shorter codes
• e.g. natural languages a, is, the (shorter
  words) are those with higher probability
• Dictionary-based methods Lempel-Zip coding
• Words and phrases within a text stream are likely
  to be repeated
• e.g. acronyms The Lempel-Zip encoding methods
  (LZ) are string-matching techniques. LZ were
  invented in 1977 and 1978

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Fourier Analysis
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Fourier Transform Bases
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Fourier Transform
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Short-time Fourier Analysis
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Short-time Fourier Transform Bases
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Wavelet Transform Bases
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Time vs. Frequency
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Application Local Analysis
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Wavelet Transform
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2-D Wavelet Transform
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Discrete Wavelet Transform and Filter Banks
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Haar Wavelet Compression
1
8 x
6 x
9 x
4 x
2 x
7 x
1 x
1 x
3 x
-1 x
-1 x
5 x
9
8
8
7
8
8
5
4
4
3
4
2
1
0
0
-1
Transformed
Original
Compressed
Reconstructed
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Cortex Transform(Watson, 1987)

• Radial and angular frequency bands
• Simulated response of neurons in the human visual
  cortex
• Invertible

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Modified Cortex Transform(Daly, 1993)
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Cortex Filters
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Cortex Layers
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Cortex Frequency Spectrum
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Assignment Haar Wavelet

• Consider the signal x(n)13, 11, -3, 7, 5, -13,
  15, 9.
• Apply the Haar decomposition to these
  coefficients. The Haar transform is implemented
  as a filter bank where the output of every
  lowpass filter is yL (n) ½ x (n) ½ x (n1)
  and the output of every highpass filter is yH (n)
  ½ x (n) ½ x (n1). After every filtered
  operation, the signal is downsampled by a factor
  of 2.
• Extensions
• Reconstruct the original signal from the
  transformed coefficients.
• Make a drawing of the analysis and synthesis
  filter banks.
• Plot the original, transformed and reconstructed
  signal and the Haar basis functions.

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Assignment Haar Wavelet

• Solution