Multiresolution analysis and wavelet bases Outline Multiresolution analysis

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 • 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 L 2( R) : Averaging and differencing

The Haar wavelet

The Haar multiresolution analysis : • A sequence of embedded approximation subsets of L 2( 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 D 2 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 D 2 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 :