Multiresolution analysis and wavelet bases

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Multiresolution analysis and wavelet bases

• Outline
• Multiresolution analysis
• The scaling function and scaling equation
• Orthogonal wavelets
• Biorthogonal wavelets
• Properties of wavelet bases
• A trous algorithm
• Pyramidal algorithm

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The Continuous Wavelet Transform

• wavelet

• decomposition

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The Continuous Wavelet Transform

• Example

The mexican hat wavelet
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The Continuous Wavelet Transform

• reconstruction

• admissible wavelet

• simpler condition zero mean wavelet

Practically speaking, the reconstruction formula
is of no use. Need for discrete wavelet
transforms wich preserve exact reconstruction.
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The Haar wavelet

• A basis for L2( R)

Averaging and differencing
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The Haar wavelet
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The Haar multiresolution analysis

• A sequence of embedded approximation subsets of
  L2( R)

with

• And a sequence of orthogonal complements,
  details subspaces

such that

• is the scaling function. Its a low pass
  filter.

• a basis in is given by

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The Haar multiresolution analysis
Example
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The Haar multiresolution analysis
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Two 2-scale relations
Defines the wavelet function.
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Orthogonal wavelet bases (1)

• Find an orthogonal basis of
• Two-scale equations
• orthogonality requires

if k 0, otherwise 0
N number of vanishing moments of the wavelet
function
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Orthogonal wavelet bases (2)

• Other way around , find a set of coefficients
  that satisfy the above equations.
• Since the solution is not unique, other favorable
  properties can be asked for compact support,
  regularity, number of vanishing moments of the
  wavelet function.

• then solve the two-scale equations.

• Example Daubechies seeks wavelets with minimum
  size compact support for any specified number of
  vanishing moments.

The Daubechies D2 scaling and wavelet functions
(
)
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Orthogonal wavelet bases (2)

• Other way around , find a set of coefficients
  that satisfy the above equations.
• Since the solution is not unique, other favorable
  properties can be asked for compact support,
  regularity, number of vanishing moments of the
  wavelet function.

• then solve the two-scale equations.

• Example Daubechies seeks wavelets with minimum
  size compact support for any specified number of
  vanishing moments.

The Daubechies D2 scaling and wavelet functions
Most wavelets we use cant be expressed
analytically.
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Fast algorithms (1)

• we start with

• we want to obtain

• we use the following relations between
  coefficients at different scales

• reconstruction is obtained with

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Fast algorithms using filter banks
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2D Orthogonal wavelet transform
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2D Orthogonal wavelet transform
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Example
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Example
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Biorthogonal Wavelet Transform
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Biorthogonal Wavelet Transform
The structure of the filter bank algorithm is the
same.
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Wavelet Packets
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