Conversion between orthogonal and biorthogonal wavelets

We consider the problem to define "appropriate" mappings between orthogonal wavelets and biorthogonal wavelets. Based on the matrix decomposition results of biorthogonal wavelets, it can be shown that the difference between biorthogonal and orthogonal wavelets is, a pseudo identity matrix pair and an invertible constant matrix components in the biorthogonal factorization, which in the orthogonal setting will be the identity matrix and a unitary constant matrix. Thus by setting the pseudo identity matrix pair to be the identity matrix and using the Schmidt orthogonalization method on the invertible matrix, it is very straightforward to convert a biorthogonal wavelet pair into an orthogonal wavelet.