0.7 Generalizations of the basic multiresolution wavelet system  (Page 12/28)

Tian and Wells [link] , [link] have constructed biorthogonal wavelet systems with both zero scaling function and wavelet moments. Closed form solutionsfor these biorthogonal coiflets have been found. They have approximation properties similar to the coiflets, and the filter coefficients are dyadicrationals as are the splines. The filter coefficients for these biorthogonal Coiflets are listed in [link] . Some members of this family are also in the spline family described earlier.

Lifting construction of biorthogonal systems

We have introduced several families of biorthogonal systems and their design methods. There is another method called a lifting scheme , which is very simple and general. It has a long history

[link] , [link] , [link] , [link] , [link] , [link] , and has been systematically developed recently [link] , [link] . The key idea is tobuild complicated biorthogonal systems using simple and invertible stages. The first stage does nothing but to separate even and oddsamples, and it is easily invertible. The structure is shown in [link] , and is called the lazy wavelet transform in [link] .

After splitting the data into two parts, we can predict one part from the other, and keep only the prediction error, as in [link] . We can reconstruct the data by recomputing the prediction and then add back theprediction. In [link] , $s$ and $t$ are prediction filters.

By concatenating simple stages, we can implement the forward and inverse wavelet transforms as in [link] . It is also called the ladder structure , and the reason for the name is clear from the figure. Clearly, the system is invertible, and thusbiorthogonal. Moreover, it has been shown the orthogonal wavelet systems can also be implemented using lifting [link] . The advantages of lifting are numerous:

• Lifting steps can be calculated inplace. As seen in [link] , the prediction outputs based on one channel of the data can be added to or subtracted from the data in other channels,and the results can be saved in the same place in the second channel. No auxiliary memory is needed.
• The predictors $s$ and $t$ do not have to be linear. Nonlinear operations like the medium filter or rounding can be used, and the systemremains invertible. This allows a very simple generalization to nonlinear wavelet transform or nonlinear multiresolution analysis.
• The design of biorthogonal systems boils down to the design of the predictors. This may lead to simple approaches that do not relay on theFourier transform [link] , and can be generalized to irregular samples or manifolds.
• For biorthogonal systems, the lifting implementations require less numerical operations than direct implementations [link] . For orthogonal cases,the lifting schemes have the computational complexity similar to the lattice factorizations, which is almost half of the directimplementation.