Biorthogonal \(M\)-channel compactly supported wavelets
This paper presents a class of \(M\)-channel subband coding schemes with perfect reconstruction based on box spline filters. For the compactly supported scaling function \(\varphi\in L^2(R)\) and integer \(M\geq 2\), the \(M\)-band refinement equation reads as: \(\widehat{\varphi}(x)=m_0({x\over M}) \widehat{\varphi}({x\over M})\), \(m_0(0)=1\), where \(\widehat{\varphi}\) is the Fourier transform of \(\varphi\) and \(m_0\) the low-pass filter. Then there are \(M-1\) compactly supported othogonal wavelets, \(\Psi_l\), \(1\leq l\leq M-1\), connected to \(\varphi\) through \(\widehat{\Psi}_l(Mx)=m_l(x)\widehat{\varphi}(x)\), with \(m_l\) the high-pass filters. The biorthogonal case is defined by \(2M\) filters: \((m_0,m_1,\ldots,m_{M-1})\) for the analysis block, and \((\widetilde{m_0},\widetilde{m_1},\ldots,\widetilde{m_{M-1}})\) for the synthesis block. The perfect reconstruction condition turns into a matrix equality \(M_s\overline{M}_a^T=I\), where the entries of \(M_s,M_a\) are constructed from the \(2M\) filters (equation 5). The author solves this system when \(\varphi\) is a box spline. He also studies the critical exponent of \(\widetilde{m_0}\) obtaining the Sobolev and Hölder regularity of the reconstruction scaling function.
scientific article; zbMATH DE number 1471469