Design of double density wavelet filter banks

We look at the design of oversampled filter banks and the resulting framelets. The framelets we design can improve shift invariant properties over decimated wavelet transform. Shift invariance has applications in many areas particularly denoising and coding and compression. Our contribution here is on filter bank completion. We develop factorization methods to find wavelet filters from given scaling filters. We look at a special class of framelets from a filter bank perspective, in that we design double density filter banks (DDFB´s). We denote the z-transform of a sequence h(.) as H(z) and its Fourier transform as H^f(ω). Now, for the perfect reconstruction, i.e. Y(z) = X(z), it must be necessary that (1) H₀(z) H˜₀(z) + H₁(z) H˜₁(z) + H₂(z) H˜₂(z) = 2, (2) H₀(z) H˜₀(-z) + H₁(z) H˜_-1(z) + H₂(z) H˜_-2(z) = 0. Alternatively we can write the above perfect reconstruction conditions in the polyphase domain. The two polyphase matrices are given, where H˜(z) is a type 1 analysis polyphase matrix, and H(z) is the type 2 synthesis polyphase matrix, we can write the perfect reconstruction conditions as [H(z)]^T H˜(z) = I.