biorthogonal wavelet transformation

multiresolution Biorthogonal wavelet transform (BWT). Wavelet coefficients at different decomposion

A biorthogonal wavelet is a wavelet where the associated wavelet transform is invertible but not necessarily orthogonal. Designing biorthogonal wavelets allows more degrees of freedom than orthogonal wavelets. One additional degree of freedom is the possibility to construct symmetric wavelet functions.
In the biorthogonal case, there are two scaling functions \phi,\tilde\phi, which may generate different multiresolution analyses, and accordingly two different wavelet functions \psi,\tilde\psi. So the numbers M and N of coefficients in the scaling sequences a,\tilde a may differ. The scaling sequences must satisfy the following biorthogonality condition