biorfilt

It is well known in the subband filtering community that if the same FIR filters are used for reconstruction and decomposition, then symmetry and exact reconstruction are incompatible (except with the Haar wavelet). Therefore, with biorthogonal filters, two wavelets are introduced instead of just one.

One wavelet, $$\tilde{\psi }$$, is used in the analysis, and the coefficients of a signal s are

The other wavelet, ψ, is used in the synthesis:

Furthermore, the two wavelets are related by duality in the following sense:
$${\displaystyle \int {\tilde{\psi }}_{j,k}}(x){\psi }_{j^{\prime },k^{\prime }}(x)dx=0$$ as soon as j ≠ j′ or k ≠ k′ and
$${\displaystyle \int {\tilde{\varphi }}_{0,k}}(x){\varphi }_{0,k^{\prime }}(x)dx=0$$ as soon as k ≠ k′.

It becomes apparent, as A. Cohen pointed out in his thesis (p. 110), that “the useful properties for analysis (e.g., oscillations, null moments) can be concentrated in the $$\tilde{\psi }$$ function; whereas, the interesting properties for synthesis (regularity) are assigned to the ψ function. The separation of these two tasks proves very useful.”

$$\tilde{\psi }$$ and ψ can have very different regularity properties, ψ being more regular than $$\tilde{\psi }$$.

The $$\tilde{\psi }$$, ψ, $$\tilde{\varphi }$$ and ϕ functions are zero outside a segment.