Cohen-Daubechies-Feauveau wavelet

Cohen–Daubechies–Feauveau wavelet are the historically first family of biorthogonal wavelets, which was made popular by Ingrid Daubechies. These are not the same as the orthogonal Daubechies wavelets, and also not very similar in shape and properties. However, their construction idea is the same.

The JPEG 2000 compression standard uses the biorthogonal CDF 5/3 wavelet (also called the LeGall 5/3 wavelet) for lossless compression and a CDF 9/7 wavelet for lossy compression.

For every positive integer A there exists a unique polynomial ${\displaystyle Q_{A}(X)}$ of degree A − 1 satisfying the identity

This is the same polynomial as used in the construction of the Daubechies wavelets. But, instead of a spectral factorization, here we try to factor

where the factors are polynomials with real coefficients and constant coefficient 1. Then

and

form a biorthogonal pair of scaling sequences. d is some integer used to center the symmetric sequences at zero or to make the corresponding discrete filters causal.

Depending on the roots of ${\displaystyle Q_{A}(X)}$, there may be up to ${\displaystyle 2^{A-1}}$ different factorizations. A simple factorization is ${\displaystyle q_{\text{prim}}(X)=1}$ and ${\displaystyle q_{\text{dual}}(X)=Q_{A}(X)}$, then the primary scaling function is the B-spline of order A − 1. For A = 1 one obtains the orthogonal Haar wavelet.

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