Any discussion of wavelets begins with Haar wavelet, the first and simplest. The Haar wavelet is discontinuous, and resembles a step function. It represents the same wavelet as Daubechies db1.
Ingrid Daubechies, one of the brightest stars in the world of wavelet research, invented what are called compactly supported orthonormal wavelets — thus making discrete wavelet analysis practicable. The db1 wavelet, as mentioned above, is the same as Haar wavelet. Here are the wavelet functions psi of the next nine members of the family:. This family of wavelets exhibits the property of linear phase, which is needed for signal and image reconstruction.
By using two wavelets, one for decomposition on the left side and the other for reconstruction on the right side instead of the same single one, interesting properties are derived. See Biorthogonal Wavelet Pairs: biorNr. Nd in the Wavelet Toolbox User's Guide for more detail. Built by I. Daubechies at the request of R.
The wavelet function has 2 N moments equal to 0 and the scaling function has 2 N -1 moments equal to 0. The two functions have a support of length 6 N The symlets are nearly symmetrical wavelets proposed by Daubechies as modifications to the db family. Miner, N.
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