Orthonormal Wavelet Basis 2.2 2002/12/04 2003/02/11 Ivan Selesnick selesi@taco.poly.edu Charlet Reedstrom charlet@rice.edu Ivan Selesnick selesi@taco.poly.edu orthogonal wavelet basis An orthonormal wavelet basis is an orthonormal basis of the form B 2 j 2 ψ 2 j t k j k The function ψ t is called the wavelet. The problem is how to find a function ψ t so that the set B is an orthonormal set. Haar Wavelet The Haar basis (described by Haar in 1910) is an orthonormal basis with wavelet ψ t ψ t 1 0 t 12 -1 12 t 1 0 For the Haar wavelet, it is easy to verify that the set B is an orthonormal set ().

Notation: ψ j , k t 2 j 2 ψ 2 j t k where j is an index of scale and k is an index of location. If B is an orthonormal set then we have the wavelet series. Wavelet series x t j k d j k ψ j , k t d j k t x t ψ j , k t