If this condition is weakened you may end up with biorthogonal wavelets.

In biorthogonal wavelets, separate decomposition and reconstruction filters are defined.

Designing biorthogonal wavelets allows more degrees of freedom than orthogonal wavelets.

For example, JPEG 2000 is an image compression standard that uses biorthogonal wavelets.

Cohen-Daubechies-Feauveau wavelet are the historically first family of biorthogonal wavelets, which was made popular by Ingrid Daubechies.

A biorthogonal wavelet is a wavelet where the associated wavelet transform is invertible but not necessarily orthogonal.

René Puchinger's detailed introduction to wavelets with coverage of multiresolution analysis of orthogonal and biorthogonal wavelets.

The Cohen-Daubechies-Feauveau wavelet and other biorthogonal wavelets have been used to compress fingerprint scans for the FBI.

Cohen-Daubechies-Feauveau wavelet (Sometimes referred to as CDF N/P or Daubechies biorthogonal wavelets)