Lifting schemes for the wavelet transform of discrete signals (Q2782695)

The authors consider the space \(C^N\) of all \(N\)-periodic signals, \(N\) even, and a biorthogonal discrete wavelet basis \(\{\phi, \tilde\phi, \psi, \tilde\psi\}\). For such wavelets they construct two lifting algorithms for decomposition and reconstruction of discrete signals in this biorthogonal wavelet basis. These algorithms use the Discrete Fourier Transforms DFT of the vectors of signal's coefficients. This results hold for interpolating wavelets, i.e., for wavelets that satisfy: \(\text{DFT} (\phi) = 1 + U\), where \(U\) is an \(N\)-periodic real function with the property that \(U(j+N) = -U(j)\). If \(N= 2^t\), a multiresolution scheme, based on the lifting algorithm, is constructed and used to define a decomposition of \(C^N\) into a direct sum of spaces \(V_t, W_t, W_{t-1}, \ldots, W_1\), where \(V_n = V_{n+1} + W_{n+1}\) and \(V_{n+1} \subset V_n\), \(W_{n+1} \subset W_n\). This decomposition is different from the standard multiresolution analysis because it does not allow the refinement equation. Finally, this approach is used to define a new family of biorthogonal discrete \(N\)-periodic wavelets.