Wavelet Toolbox

Battle-Lemarie Wavelets

See [Dau92] pages 146-148, 151.

These wavelets are not included in the toolbox, but we use the spline functions in the biorthogonal family.

There are two forms of the wavelet: one does not ensure the analysis to be orthogonal, while the other does. For N=1, the scaling functions are linear splines. For N=2, the scaling functions are quadratic B-spline with finite support. More generally, for an N-degree B-spline,

with if N is odd, if N is even.

This formula can be used to build the filters. The twin scale relation is

• For an even N, is symmetrical around, x = 1/2; is antisymmetrical around x = 1/2. For an odd N, is symmetrical around x = 0; is symmetrical around x = 1/2.
• The analysis becomes orthogonal if we transform the functions and somewhat. For N=1, for instance, let

• The supports of and are not finite, but the decrease of the functions and to 0 is exponential. The support of is compact. See [Dau92] p. 151.
• The functions have derivatives up to order N-1.

Meyer Wavelet: meyr

Mexican Hat Wavelet: mexh