A Theory for Multiresolution Signal Decomposition: The Wavelet Representation

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A Theory for Multiresolution Signal
Decomposition The Wavelet Representation

• Stephane G. Mallat,
• IEEE Transactions on Pattern Analysis and Machine
  Intelligence, Vol. 11, No. 7, July, 1989, pp.
  674-693

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Outline

• Introduction
• Multiresolution transform
• Multiresolution approximation
• Implementation of multiresolution transform
• The wavelet representation
• The detail signal
• Implementation of orthogonal wavelet
  representation
• Conclusion

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Introduction

• Multiresolution approximation is wanted
• It is natural to analyze first the image at a
  coarse resolution then gradually increase the
  resolution

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Introduction

• Low pass filter can do this job
• Ex subsampling
• Methodology
• Laplacian pyramid
• Wavelet pyramid
• and so on (maybe)

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Outline

• Introduction
• Multiresolution transform
• Multiresolution approximation
• Implementation of multiresolution transform
• The wavelet representation
• The detail signal
• Implementation of orthogonal wavelet
  representation
• Conclusion

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Multiresolution transform

• Refresh some definitions
• Vector space
• Basis
• Inner product
• Notation lta,bgt
• Orthogonal
• Orthonormal

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Multiresolution transform

• Definitions

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Multiresolution transform -approximation

• There exist scaling function ?(x)
• Is an orthonormal basis of

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Multiresolution transform -approximation

• Examples of ?(x)

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Multiresolution transform -approximation
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Multiresolution transform -approximation

• can be interpreted as a low-pass filtering
  of f(x) followed by a uniform sampling at the
  rate
• Original discrete signal is defined as
• This filter is special because of its orthogonal
  family

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Multiresolution transform -approximation
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Multiresolution transform -Implementation
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Multiresolution transform -Implementation

• This operation is called a pyramid transform

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Multiresolution transform -Implementation

• ?(x) ?? h(n) ?? approximation method
• Prefer ?(x) which is continuous differentiable
  and asymptotic decay
• Good localization properties in both freq. and
  spatial domain
• It is possible to choose H(?) to obtain a ?(x)

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Outline

• Introduction
• Multiresolution transform
• Multiresolution approximation
• Implementation of multiresolution transform
• The wavelet representation
• The detail signal
• Implementation of orthogonal wavelet
  representation
• Conclusion

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The wavelet representation

• Definitions
• Detail signal difference between

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The wavelet representation
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The wavelet representation
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The wavelet representation

• Called orthogonal wavelet representation

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The wavelet representation

• Examples of ?(x)

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The wavelet representation

• ? is a band pass filter, which decompose the
  signals into a set of independent frequency
  channels
• Because overlap of frequency channels, ? can only
  provide intuitive approach to the model.

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The wavelet representation - implementation
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The wavelet representation - implementation
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The wavelet representation - implementation
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The wavelet representation
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Reconstruction
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Conclusion

• Theory of orthogonal wavelet representation has
  been showed
• Didnt mention the extension 2D transform
• Application
• Texture discrimination
• fractal analysis