Spline and Spline Wavelet Methods with Applications to Signal and Image Processing

Analytic and directional wavelet packets

“…The space Π[N/2] can be split into mutually orthogonal subspaces, which we denote by Π 0 [N/2] and Π 1 [N/2], in a way that is similar to the split of the space Π[N ]. Projection of a signal Y ∈ Π[N/2] onto these subspaces and the inverse operation are done using the analysis and synthesis p-filter banksH [2] are orthogonal to each other in the space Π[N/2] and their 2-sample shifts are mutually orthogonal…”

“…where µ = 0, 1. The modulation matrices of the p-filter bank H [2] arẽ M [2] [n] =M[2n], M [2] [n] = M[2n], (2.7)…”

“…The signal ψ 2r [2],ρ is a linear combination of 2-sample shifts of the discrete-spline WP ψ 2r [1],λ , therefore ψ 2r [2],ρ ∈ 2r S whose orthonormal bases are formed by 4-sample shifts ψ 2r [2],ρ [· − 4l] , l = 0, ..., N/4 − 1, of the signals ψ 2r [2],ρ , which are referred to as the second-level discrete-spline wavelet packets of order 2r. The orthogonal projection of a signal x ∈ Π[N ] onto the subspace 2r S ρ [2] is the signal…”

“…Practically, derivation of the wavelet packet transform coefficients y λ [1] , λ = 0, 1, from x and the inverse operation are implemented using Eq. (2.5), while the transform y λ [1] ←→ y ρ [2] are implemented similarly using the modulation matrices of the p-filter bank H [2] defined in Eq. (2.7).…”

“…(2.7). The second-level wavelet packets ψ 2r [2],ρ are derived from the first-level wavelet packets ψ 2r [1],λ by filtering the latter with the p-filters h µ [2] , λ, µ = 0, 1, ρ = µ, if λ = 0; 3 − µ, if λ = 1. . Figure 2.2 displays the second-level wavelet packets originating from discrete splines of orders 2, 6 and 10 and their DFTs.…”