Use of Quillen-Suslin theorem for laurent polynomials in wavelet filter bank design

In this chapter we give an overview of a method recently developed for designing wavelet filter banks via the Quillen-Suslin Theorem for Laurent polynomials. In this method, the Quillen-Suslin Theorem is used to transform vectors with Laurent polynomial entries to other vectors with Laurent polynomial entries so that the matrix analysis tools that were not readily available for the vectors before the transformation can now be employed. As a result, a powerful and general method for designing non-redundant wavelet filter banks is obtained. In particular, the vanishing moments of the resulting wavelet filter banks can be controlled in a very simple way, which is especially advantageous compared to other existing methods for the multi-dimensional cases.