Image transforms of Image compression

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Image transforms of Image compression

• Presenter Cheng-Jin Kuo ???
• Advisor Jian-Jiun Ding, Ph. D.
• Professor ?????
• Digital Image Signal Processing Lab
• Graduate Institute of Communication Engineering
• National Taiwan University, Taipei, Taiwan, ROC

2
Outline

• Introduction
• Image compression scheme
• Image Transform
• Orthogonal Transform
• DCT transform
• Subband Transform
• Haar Wavelet transform

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Introduction

• Image types
• bi-level image
• grayscale image
• color image e.g. RGB, YCbCr
• continuous-tone image
• -natural scene
• -image noise
• -clouds, mountains, surface of lakes

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Introduction

• discrete-tone image(graphical image or synthetic
  image)
• -artificial image
• -sharp and well-defined edges
• -high contrasted from the background
• cartoon-like image
• -uniform color

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Introduction

• The principle of Image compression
• removing the redundancy
• -the neighboring pixels are highly correlated
• -the correlation is called spatial redundancy

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Image compression scheme

• 
  
  Arithmetic coding,
• 
  
  Huffman coding,
• 1.Orthogonal
  transform(Walsh-Hamadard transform,
  RLE, .
• 
  DCT, )
  
• 2.Subband
  transform(wavelet transform, )
• 
  
  quantization error
• 
  
  
• 
  
  

image
transform
quantizer
encoder
Compressed image file
decoder
Inverse transform
image
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Image transform

• Two properties and main goals
• -to reduce image redundancy
• -to isolate the various freq. of the image
• (identify the important parts of the image)

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Image transform

• Two main types
• -orthogonal transform
• e.g. Walsh-Hdamard transform, DCT
• -subband transform
• e.g. Wavelet transform

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Orthogonal transform

• Orthogonal matrix W
• ? CW.D
• Reducing redundancy
• Isolating frequencies

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Orthogonal transform

• One choice of W
• (Walsh-Hadamard
  transform)
• CW.D
• W should be Invertible (for inverse transform)
• Other properties?

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Orthogonal transform

• Reducing redundancy (Energy weighted)
• example d5 6 7 8
• after multiply by W/2? c13 -2 0 -1
• energy of d energy of c 174
• energy ratio of the first index
• d25/174 14
• c169/174 97

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Orthogonal transform

• Reducing redundancy (Energy weighted)
• d4 6 5 2 c8.5 1.5 -2.5 0.5 E81
• In general, we ignore several smallest
  elements in d, and get c8.5 0 -2.5 0
• quantize it and get the inverse
• c3 5.5 5.5 3
• E81.75
• Property 1 should be large while others,
  small.

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Orthogonal transform

• Isolating frequencies (freq. weighted)
• example
• d1 0 0 1?c2 0 2 0 W
• d0.51 1 1 10.51 -1 -1 1
• d0 0.33 -0.33 -1?c0 2.66 0 1.33
• d0.661 1 -1 -10.331 -1 1 -1

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Orthogonal transform

• Isolating frequencies (freq. weighted)
• Property 2 should correspond to zero freq.
  while other coefficients correspond to higher and
  higher freq.
  
• W , W
• (Walsh-Hadamard transform)

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Orthogonal transform

• So how do we choose W?
• Invertible matrix
• Coefficients in the first row are all positive
• Each row represents the different freq.
• Orthogonal matrix

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Orthogonal transform

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Discrete Cosine Transform

• W matrix of DCT
• W

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Discrete Cosine Transform

• 1D DCT ,
  for f07
• 
  , f0
• 
  1 , fgt0
• Inverse DCT(IDCT)

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Discrete Cosine Transform

• 2D DCT
• Inverse DCT(IDCT)

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Discrete Cosine Transform

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Subband Transform

• Separate the high freq. and the low freq. by
  subband decomposition

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Subband Transform

• Filter each row and downsample the filter output
  to obtain two N x M/2 images.
• Filter each column and downsample the filter
  output to obtain four N/2 x M/2 images

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Haar wavelet transform

• Haar wavelet transform
• Average resolution
• Difference detail
• Example for one dimension

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Haar wavelet transform

• Example data(5 7 6 5 3 4 6 9)
• -average(57)/2, (65)/2, (34)/2, (69)/2
• -detail coefficients
• (5-7)/2, (6-5)/2, (3-4)/2, (6-9)/2
• n (6 5.5 3.5 7.5 -1 0.5 -0.5 -1.5)
• n (23/4 22/4 0.25 -2 -1 0.5 -0.5
  -1.5)
• n (45/8 1/8 0.25 -2 -1 0.5 -0.5 -1.5)

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Haar wavelet transform
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Subband Transform
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Subband Transform

• The standard image wavelet transform
• The Pyramid image wavelet transform

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Subband Transform

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Subband Transform
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Reference

• David Salomon, Coding for Data and Computer
  Communication, Springer, 2005.
• A. Uhl, A. Pommer, Image and Video Encryption,
  Springer, 2005
• David Salomon, Data Compression - The Complete
  Reference 3rd Edition, Springer, 2004.
• Khalid Sayood, Introduction to Data Compression
  2nd Edition, Morgan Kaufmann, 2000.
• J.Goswami, A.Chan, Fundamentals of Wavelets
  Theory, Algorithms, and Application, Wiley
  Interscience, 1999
• C.S. Burrus, R. A. Gopinath, H. Guo, Introduction
  to Wavelets and Wavelet Transforms A Primer,
  Prentice-Hall, 1998