A Biorthogonal wavelet is a

wavelet A wavelet is a wave-like oscillation with an amplitude that begins at zero, increases or decreases, and then returns to zero one or more times. Wavelets are termed a "brief oscillation". A taxonomy of wavelets has been established, based on the num ...

where the associated

wavelet transform In mathematics, a wavelet series is a representation of a square-integrable ( real- or complex-valued) function by a certain orthonormal series generated by a wavelet. This article provides a formal, mathematical definition of an orthonormal ...

invertible In mathematics, the concept of an inverse element generalises the concepts of opposite () and reciprocal () of numbers. Given an operation denoted here , and an identity element denoted , if , one says that is a left inverse of , and that ...

but not necessarily

orthogonal In mathematics, orthogonality is the generalization of the geometric notion of '' perpendicularity''. By extension, orthogonality is also used to refer to the separation of specific features of a system. The term also has specialized meanings in ...

. 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 :

\sum_ a_n \tilde a_=2\cdot\delta_

. Then the wavelet sequences can be determined as

b_n=(-1)^n \tilde a_ \quad \quad (n=0,\dots,N-1)

\tilde b_n=(-1)^n a_ \quad \quad (n=0,\dots,N-1)

References

* {{signal-processing-stub