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

The Continuous Wavelet Transform

• decomposition

• wavelet

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

• is the scaling function. It’s a low pass filter.

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

The Haar multiresolution analysis :

with :

• And a sequence of orthogonal complements, details’ subspaces :

such that

• 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

Most wavelets we use can’t be expressed analytically.

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

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

h h h h h

WT