Multiresolution analysis and wavelet bases
Outline :
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Multiresolution analysis
The scaling function and scaling equation
Orthogonal wavelets
Biorthogonal wavelets
Properties of wavelet bases
• A trous algorithm
• Pyramidal algorithm
The Continuous Wavelet Transform
• wavelet
• decomposition
The Continuous Wavelet Transform
• Example :
The mexican hat wavelet
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.
The Haar wavelet
•A basis for L2( R) :
Averaging and
differencing
The Haar wavelet
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. It’s a low pass filter.
• a basis in
is given by :
The Haar multiresolution
analysis
Example :
The Haar
multiresolution
analysis
Two 2-scale relations :
Defines the wavelet function.
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
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
=(
)
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 can’t be expressed analytically.
Fast algorithms (1)
• we start with
• we want to obtain
• we use the following relations between coefficients at different
scales:
• reconstruction is obtained with :
Fast algorithms using filter banks
2D Orthogonal wavelet transform
2D Orthogonal wavelet transform
Example :
Example :
Biorthogonal Wavelet Transform :
Biorthogonal Wavelet Transform :
The structure of the filter bank algorithm is the same.
Wavelet Packets
Scale 1
Scale 2
Scale 3
Scale 4
Scale 5
WT
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