Introduction to Wavelet Transform

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Introduction to Wavelet Transform

• ??? 2004.03.26

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comment

• Giving some helpful Pre-knowledge for reading
  other paper about wavelets in the future.
• This presentation will focus on
• Brief introductions for basic wavelets theory
• The implement of Discrete Wavelet transform

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Outline

• Basic Wavelets Theory
• Discrete Wavelet transform
• Wavelets in Mpeg-4 and Jpeg-2000
• Discussion

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Outline

• Basic Wavelets Theory
• Discrete Wavelet transform
• Wavelets in Mpeg-4 and Jpeg-2000
• Discussion

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Basic Wavelets Theory

• Wavelet transform(WT), like Fourier transform
  (FT), is a powerful mathematical tool for many
  problem in science and engineering.
• FT the basis function are sines and
  cosines.
• WT a hierarchical set of wavelet functions.

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Basic Wavelets Theory

• Wavelet
• little wave or localized wave
• Oscillatory
• Little decay quick to zero
• No DC component

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Basic Wavelets Theory

• Mother wavelet, is a continuous function
  which has two properties.
• The function integrates to zero
• Its square integrable or, equivalently, has
  finite energy

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Basic Wavelets Theory

• Wavelets are dilations and translations of the
  mother wavelet.
• By parameters(a,b), the is dilated by a
  factor a and translated by factor b

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Basic Wavelets Theory

• For many applications a restricted set is
• used and (j,kintegers)
• wavelet at scale j with shift k

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Basic Wavelets Theory

• The simplest wavelet is the Haar mother wavelet
  which defined on 0,1)

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Basic Wavelets Theory

• Wavelets at different j and k
• j0 j1

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Basic Wavelets Theory

• Any admissible functions can be expressed
  as
• The coefficients

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Basic Wavelets Theory

• However, the Haar wavelets of scales
• j 0 are not enough to represent all
  functions (like constant function).
• A father wavelets is used to remedied the
  deficiency

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Basic Wavelets Theory

• In the Haar case, the father wavelet is defined
  to be
• The father wavelets is sometimes named the
  scaling function.
• Constant functions can be represented easily by a
  multiple of the father wavelets.

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Basic Wavelets Theory

• A function can be represented more accurately
  below
• In connect with filter banks, it is thehigh-pass
  filterthat leads to the low-pass
  filter lead to scaling function

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Outline

• Basic Wavelets Theory
• Discrete Wavelet transform
• Wavelets in Mpeg-4 and Jpeg-2000
• Discussion

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Discrete Wavelet transform

• In discrete wavelet transform(DWT), the DWT turns
  a data sequence into a set of discrete wavelets
  coefficients.
• The DWT consists of three main components
• Low-pass filter
• High-pass filter
• Sampling operator

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Discrete Wavelet transform

• dsf

Analysis
Synthesis
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Discrete Wavelet transform

• Why need downsample the signal?
• h0 low-pass filter h1 high-pass filter
• Yy0y1h0x h1x (without downsampling)
• The volume of data Y is the double of data X
  (without downsampling).

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Discrete Wavelet transform

• Downsample is represented by the symbol ,and
  this operation is not invertible.

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Discrete Wavelet transform

• In order to perfect reconstruction(PR) the
  original signal, the filters must have some
  properties
• (1)
• (2)

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Discrete Wavelet transform

• Example
• Input signal
• Use(5,3)filter bank

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Discrete Wavelet transform

• Example
• After filter
• h0
• h1
• Downsample and combine two data together

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Discrete Wavelet transform

• 2D-DWT

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• ll

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Discrete Wavelet transform

• The synthesis stage

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Outline

• Basic Wavelets Theory
• Discrete Wavelet transform
• Wavelets in Mpeg-4 and Jpeg-2000
• Discussion

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Wavelets in Mpeg-4 and Jpeg-2000

• Mpeg-4
• Support the coding of still textures in the
  visual texture coding mode of MPEG-4.
• Jpeg-2000
• Compress different types of still images with
  different characteristics.

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Wavelets in Mpeg-4 and Jpeg-2000

• Mpeg-4(Visual texture coding)
• Efficient compression
• Arbitrarily shape coding
• Spatial and quality scalability
• Error robustness
• tiling

• Jpeg-2000(Types of still image)
• Types of still image
• Lossless and lossy compression
• Spatial and quality scalability
• Error resilient coding
• Region of interest coding
• Sequential build up and tiling
• Random code-stream access and processing

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Wavelets in Mpeg-4 and Jpeg-2000
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Discussion

• The main advantages or properties of Wavelets
• Easy to implement because of recursive process
• Achieve the multiresolusion analysis(MRA) concept
  both in time and frequency intuitively
• Lower bit-rate and higher performance for image
  compression
• PR concept lead to lossless compression