The Double Density Dwt

This chapter takes up the design of discrete wavelet transforms based on oversampled filter banks. In this case the wavelets form an overcomplete basis, or frame. In particular, we consider the design of systems that are analogous to Daubechies’ orthonormal wavelets — that is, the design of minimal length wavelet filters satisfying certain polynomial properties, but now in the oversampled case. The wavelets are constructed using maximally flat FIR filters in conjunction with extension methods for paraunitary matrices. Because there are more degrees of freedom in the design problem, the wavelets described in this chapter are much smoother than orthonormal wavelets of the same support. The oversampled dyadic DWT considered in this chapter is based on a single scaling function and two distinct wavelets. Having more wavelets than necessary gives a closer spacing between adjacent wavelets within the same scale. Like the dual-tree DWT, the oversampled DWT presented here is redundant by a factor of 2, independent of the number of levels. In comparison, the redundancy of the undecimated DWT grows with the number of levels.