0.7 Generalizations of the basic multiresolution wavelet system  (Page 11/28)

Cohen-daubechies-feauveau family of biorthogonal spline wavelets

Splines have been widely used in approximation theory and numerical algorithms. Therefore, they may be desirable scaling functions,since they are symmetric, smooth, and have dyadic filter coefficients (see Section: Example Scaling Functions and Wavelets ). However, if we use them as scaling functions in orthogonal wavelet systems, the waveletshave to have infinite support [link] . On the other hand, it is very easy to use splines in biorthogonal wavelet systems. Choose $h$ to be a filter that can generate splines, then [link] and [link] are linear in the coefficients of $\tilde{h}$ . Thus we only have to solve a set of linear equations to get $\tilde{h}$ , and the resulting $\tilde{h}$ also have dyadic coefficients. In [link] , better methods are used to solve these equations indirectly.

The filter coefficients for some members of the Cohen-Daubechies-Feauveau family of biorthogonal spline wavelets are listed in [link] . Note that they are symmetric. It has been shown that as the length increases, the regularity of $\phi$ and $\tilde{\phi }$ of this family also increases [link] .

Cohen-daubechies-feauveau family of biorthogonal wavelets with less dissimilar filter length

The Cohen-Daubechies-Feauveau family of biorthogonal wavelets are perhaps the most widely used biorthogonal wavelets, since the scaling functionand wavelet are symmetric and have similar lengths. A member of the family is used in the FBI fingerprint compression standard [link] , [link] . The design method for this family is remarkably simple and elegant.

In the frequency domain, [link] can be written as

Recall from Chapter: Regularity, Moments, and Wavelet System Design that we have an explicit solution for ${\left|H\left(\text{ω}\right)\right|}^2=M\left(\text{ω}\right)$ such that

and the resulting compactly supported orthogonal wavelet has the maximum number of zero moments possible for its length. In the orthogonal case, we geta scaling filter by factoring $M\left(\text{ω}\right)$ as $H\left(\text{ω}\right)H^{*}\left(\text{ω}\right)$ . Here in the biorthogonal case, we can factor the same $M\left(\text{ω}\right)$ to get $H\left(\text{ω}\right)$ and $\tilde{H}\left(\text{ω}\right)$ .

Factorizations that lead to symmetric $h$ and $\tilde{h}$ with similar lengths have been found in [link] , and their coefficients are listed in [link] . Plots of the scaling and wavelet functions, which are members of the family used in the FBI fingerprint compressionstandard, are in [link] .

Tian-wells family of biorthogonal coiflets

The coiflet system is a family of compactly supported orthogonal wavelets with zero moments of both the scaling functions and wavelets described in Section: Coiflets and Related Wavelet Systems . Compared with Daubechies' wavelets with only zero wavelet moments, the coiflets are more symmetrical and may have betterapproximation properties when sampled data are used. However, finding the orthogonal coiflets involves solving a set of nonlinear equations. Noclosed form solutions have been found, and when the length increases, numerically solving these equations becomes less stable.