Biorthogonal and semiorthogonal wavelet bases

Much of the literature on wavelets tends to be biased towards discussions on orthogonal wavelets because they are convenient and simple to implement. However, we feel that is necessary to make the reader aware that wavelets need not be orthogonal and that wavelets with other properties can be quite useful too. In this section we discuss biorthogonal wavelets as one alternative to orthogonal wavelets. We direct the reader to [1,7,10,14,15] for more information on other kinds of wavelets. 

Briefly, when using orthogonal compactly supported wavelets it is not straightforward to obtain a wavelet which has symmetrical properties [7,12] and allows for an exact reconstruction. That is of course with the exception of the trivial Haar wavelet. Biorthogonal wavelets relax the assumptions of orthogonality, and allow for a perfect reconstruction with symmetrical wavelets. 

Property Symmetry Asymmetry Nearly symmetrical Orthogonal Compact support Explicit expression Scaling function 

Here we have used the original wavelets for representing the functions, and the dual wavelets for computing the coefficients in the expansion of the 

There are two sets of scaling and wavelet defining equations for each pair of basis functions 

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