The construction of perfect reconstruction filter banks is simpler than the construction of conjugate mirror filters because the quadrature condition is replaced by a Bezout identity:
In particular, spectral factorization is no longer required.
A theorem by Cohen, Daubechies and Fauveau gives sufficient conditions for building biorthogonal wavelets.
One quite interesting example is given by biorthogonal spline wavelets. It is iteresting because it has symmetric scaling functions, and because there existe a closed form formula for the filters.
The h filter is taken to be
with e=0 if p is even and e =1 if p is odd. The scaling function is a B-spline of degree p-1 (this can verified by using the recursion which relates B-splines of different degrees). It is a symmetric function with respect to 0 if p is odd, and symmetric with respect to 1/2 if p is odd. The corresponding wavelet is respectively symmetric or antisymmetric. The dual wavelet has p vanishing moments.
The only constraint on the number of vanishing moments of the primal wavelet is that it should have the same parity as p. Hence the symmetries are the same as in the previous case. For q=(p+p2)/2, the biorthogonal filter h2 of minimum length is given by
Here is an example for p=3 and p2=7
The same filters are used to implement
the
dyadic wavelet transform