Design of double density wavelet filter banks

We look at the design of oversampled filter banks and the resulting framelets. The undecimated wavelet transform is known for its shift invariant properties and has applications in areas such as denoising [7]. The framelets we will design, will have improved shift invariant properties over decimated wavelet transform. Shift invariance has applications in many areas particularly denoising [6], [10], [5] and coding and compression [11]. Our contribution here is on filter bank completion. We will develop factorization methods to find wavelet filters from given scaling filters. We will look at a special class of framelets from a filter bank perspective, in that we will design double density filter banks (DDFB’s) as shown in Figure 1. We denote the z-transform of a sequence as and its Fourier transform as . Using the basic multirate identities we obtain the following expression for . (1) Now, for the perfect reconstruction, i.e. , it must be necessary that