Practical design of perfect-translation-invariant real-valued discrete wavelet transform (Q2922068)

This paper is about the construction of a real-valued tight wavelet frame with the perfect translation invariance property (PTI) and the design of an approximate real-valued tight wavelet frame with the PTI property, having fast decomposition and a reconstruction algorithm.NEWLINENEWLINEConsider \(\{g_n = g(\cdot-pn): n\in\mathbb{Z}\}\) for some \(p>0\). Define \(\mathcal{W}_p^g f:=\sum_{n}\langle f, g_n \rangle g_n\). Then, \(\mathcal{W}_p^g\) is said to be having the PTI property if \([\mathcal{W}_p^g f](\cdot-t) = [\mathcal{W}_p^g f(\cdot-t)]\) for all \(t\in\mathbb{R}\). A necessary condition for \(\mathcal{W}_p^g\) to have the PTI property is if \(g\) satifies (1) \(|\hat g(\omega)|\leq C(1+|\omega|^2)^{-1/2-\epsilon}\) and (2) \(\hat g(\omega)\overline{\hat g(\omega-2\pi k/p)}=0\), \(k\neq0, k\in\mathbb{Z}\).NEWLINENEWLINEUsing Meyer-type scaling and wavelet functions \(\phi,\psi\), the paper then constructs a wavelet tight frame having the PTI property \(\{\psi_{j,k}:=\sqrt{2}^{j/2}\psi(\sqrt{2}^j\cdot-\frac{1+\sqrt{2}}{2}k): j,k\in\mathbb{Z}\}\) with dilation factor \(\sqrt{2}\). However, for such a type of wavelet tight frames, the computational cost is too large for the decomposition and reconstruction algorithm for a digital signal \(\{f_n\}_{n\in\mathbb{Z}}\).NEWLINENEWLINEBy relaxing the tightness condition, the paper then turns to the construction of an approximate tight wavelet frame having the PTI property. \(\{g_n\}\) is a tight frame if \(f=\frac1A\sum_{n}\langle f, g_n \rangle g_n\) for all \(L^2\) function \(f\) for some \(A>0\); while \(\{g_n\}\) is an approximate tight frame if \(f=\sum_{n}r_n\langle f, g_n \rangle g_n\) for all \(L^2\) function \(f\) and some wights \(\{r_n\}\) independent of \(f\). Using again the Meyer-type scaling and wavelet functions, the paper constructs an approximate tight frame generated from \(\phi_{j,k}=\sqrt{2}^{j/2}\phi(\sqrt{2}^j\cdot-\frac14 k)\) and \(\psi_{j,k}=\sqrt{2}^{j/2}\psi(\sqrt{2}^j\cdot-p_j k)\), where \(p_{2n}=\frac54\) and \(p_{2n+1}=\frac{7}{4\sqrt{2}}\). The paper then uses the two scale relation of \(V_0=W_{-1}+W_{-2}+V_{-2}\) to deduce filters \(h^1\), \(h^2\), and \(g^0\), \(g^{1/4}\), \(g^{2/4}\), \(g^{3/4}\). Afast decomposition and reconstruction for digital signals can then be computed efficiently using these six filter convolution.