0.7 Generalizations of the basic multiresolution wavelet system  (Page 10/28)

where the series converge strongly.

Moreover, the ${\psi }_{j,k}$ and ${\tilde{\psi }}_{j,k}$ constitute two Riesz bases, with

if and only if

This theorem tells us that under some technical conditions, we can expand functions using the wavelets and reconstruct using their duals.The multiresolution formulations in Chapter: A multiresolution formulation of Wavelet Systems can be revised as

where

If [link] holds, we have

where

Although $W_j$ is not the orthogonal complement to $V_j$ in $V_{j+1}$ as before, the dual space ${\tilde{W}}_j$ plays the much needed role. Thus we have four sets of spaces that form twohierarchies to span $L^2\left(\mathbf{R}\right)$ .

In Section: Further Properties of the Scaling Function and Wavelet , we have a list of properties of the scaling function and wavelet that do not require orthogonality. The results forregularity and moments in  Chapter: Regularity, Moments, and Wavelet System Design can also be generalized to the biorthogonal systems.

Comparisons of orthogonal and biorthogonal wavelets

The biorthogonal wavelet systems generalize the classical orthogonal wavelet systems. They are more flexible and generally easy todesign. The differences between the orthogonal and biorthogonal wavelet systems can be summarized as follows.

• The orthogonal wavelets filter and scaling filter must be of the same length, and the length must be even. This restriction has been greatlyrelaxed for biorthogonal systems.
• Symmetric wavelets and scaling functions are possible in the framework of biorthogonal wavelets. Actually, this is one of the mainreasons to choose biorthogonal wavelets over the orthogonal ones.
• Parseval's theorem no longer holds in biorthogonal wavelet systems; i.e., the norm of the coefficients is not the same as the norm ofthe functions being spanned. This is one of the main disadvantages of using the biorthogonal systems. Many design efforts have been devoted tomaking the systems near orthogonal, so that the norms are close.
• In a biorthogonal system, if we switch the roles of the primary and the dual, the overall system is still sound. Thus we can choose the bestarrangement for our application. For example, in image compression, we would like to use the smoother one of the pair to reconstruct thecoded image to get better visual appearance.
• In statistical signal processing, white Gaussian noise remains white afterorthogonal transforms. If the transforms are nonorthogonal, the noise becomes correlated or colored. Thus, when biorthogonal wavelets are usedin estimation and detection, we might need to adjust the algorithm to better address the colored noise.

Example families of biorthogonal systems

Because biorthogonal wavelet systems are very flexible, there are a wide variety of approaches to design different biorthogonal systems. The key is todesign a pair of filters $h$ and $\tilde{h}$ that satisfy [link] and [link] and have other desirable characteristics. Here we review several families of biorthogonalwavelets and discuss their properties and design methods.