Cohen–Daubechies–Feauveau wavelet

[1][2] These are not the same as the orthogonal Daubechies wavelets, and also not very similar in shape and properties.

The JPEG 2000 compression standard uses the biorthogonal Le Gall–Tabatabai (LGT) 5/3 wavelet (developed by D. Le Gall and Ali J. Tabatabai)[3][4][5] for lossless compression and a CDF 9/7 wavelet for lossy compression.

For every positive integer A there exists a unique polynomial

of degree A − 1 satisfying the identity This is the same polynomial as used in the construction of the Daubechies wavelets.

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.

, then the primary scaling function is the B-spline of order A − 1.

, this polynomial has exactly one real root, thus it is the product of a linear factor

The coefficient c, which is the inverse of the root, has an approximate value of −1.4603482098.

For the coefficients of the centered scaling and wavelet sequences one gets numerical values in an implementation–friendly form (1/2 adual) (bdual) (aprim) (1/2 bprim) There are two concurring numbering schemes for wavelets of the CDF family: The first numbering was used in Daubechies' book Ten lectures on wavelets.

The number of vanishing moments does not tell about the chosen factorization.

A filter bank with filter sizes 7 and 9 can have 6 and 2 vanishing moments when using the trivial factorization, or 4 and 4 vanishing moments as it is the case for the JPEG 2000 wavelet.

The same wavelet may therefore be referred to as "CDF 9/7" (based on the filter sizes) or "biorthogonal 4, 4" (based on the vanishing moments).

Similarly, the same wavelet may therefore be referred to as "CDF 5/3" (based on the filter sizes) or "biorthogonal 2, 2" (based on the vanishing moments).

For the trivially factorized filterbanks a lifting decomposition can be explicitly given.

be the number of smoothness factors in the B-spline lowpass filter, which shall be even.

Then define recursively The lifting filters are Conclusively, the interim results of the lifting are which leads to The filters

Then define recursively The lifting filters are Conclusively, the interim results of the lifting are which leads to where we neglect the translation and the constant factor.

[7] A standard for compressing fingerprints in this way was developed by Tom Hopper (FBI), Jonathan Bradley (Los Alamos National Laboratory) and Chris Brislawn (Los Alamos National Laboratory).

[7] By using wavelets, a compression ratio of around 20 to 1 can be achieved, meaning a 10 MB image could be reduced to as little as 500 kB while still passing recognition tests.