Cohen-Daubechies-Feauveau-Wavelet

Cohen- Daubechies wavelets Feauveau ( CDF wavelets) are the historically first family of biorthogonal wavelets. They were designed by Albert Cohen, Ingrid Daubechies and Jean -Christophe Feauveau and presented in 1990. CDF wavelets are to be distinguished from the orthogonal Daubechies wavelets that have different shapes and characteristics. Both Wavelettypen go back to the same design idea, CDF wavelets renounce in favor of symmetry orthogonality of the wavelets (with Daubechies wavelets is it vice versa).

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

Properties

• The Primgenerator is a B -spline, when the simple factorization (see below ) is selected
• Dual generator has the maximum number of factors, smoothness, which is possible for the length
• All generators and wavelets of this family are symmetric.

Construction

For any positive integer, it is a unique polynomial of the degree satisfying the identity.

Is the same polynomial that is used in the construction of the Daubechies wavelets. Instead of a spectral factorization, however, an attempt is

Where the factors are polynomials with real coefficients and the constant 1.

In this case forms

And

A pair of biorthogonal scaling consequences. is an integer, which is used for centering the symmetrical sequence to zero, or to make the corresponding discrete causal filter.

Depending on the roots, there are up to various factorizations. A simple factorization is and. In this case, the primary scaling is the B-spline function of order. For obtaining the orthogonal Haar wavelet.

Coefficient table

For obtaining the LeGall-5/3-Wavelet:

For obtaining the 9/7-CDF-Wavelet. Is obtained. This polynomial has exactly one real root and is thus the product of the linear factor and a quadratic factor. The coefficient, which is the inverse of the root, has a value of about -1.4603482098.

For the coefficients of the centered scaling and wavelet sequences numerical values ​​obtained in implementation- friendly form:

(1/2 adual )

( bDual )

( Aprim )

(1/2 bprim )

Number Name

There are two parallel numbering schemes for the wavelets of the CDF family.

• The number of factors of the smoothness of the low-pass filter, or ( equivalently ), the number of vanishing moments of the high-pass filter, for example, 2.2
• The lengths of the low-pass filter, or ( equivalently ), the lengths of the high -pass filter, for example, 5.3

The first scheme was used in Daubechies ' book " Ten lectures on wavelets ". None of the names is unique. The number of vanishing moments says nothing about the chosen factorization. A filter bank, the filter lengths 7 and 9 be, having 6 and two vanishing moments when using a trivial factorization or vanishing moments 4 and 4, as in the case of the JPEG 2000 wavelet. The same wavelet can therefore " 4/4 bi-orthogonal " hot as " CDF 9/7 " (based on the filter lengths) or (based on vanishing moments ).

Lifting decomposition

For the trivial factored filter banks a lifting decomposition can be explicitly given.