An algebraic structure of orthogonal wavelet space
In this paper the authors study the space of compactly supported orthonormal wavelets over real numbers. The wavelet subspace \(WM(2,g)\) of rank 2 (corresponding to 2-band wavelets) and genus \(g\) (corresponding to scaling filters of length \(2g\)) is mapped one-to-one to the \((g-1)\) dimensional real torus. Because of the degenerate phenomenon of the paraunitary matrix, this parametrization is not onto. Instead there exists an onto mapping from the torus to the closure of the wavelet space \(\overline{WM(2,g)}:=WM(2,g)\cup WM(2,g-2)\cup\cdots\). By utilizing the factorization theory, the authors present a fast implementation of the discrete wavelet transform (DWT). In general the computational complexity of a rank \(m\) orthogonal DWT is \(O(m^2g)\). By starting with a given scaling filter, the authors construct additional \((m-1)\) wavelet filters so that DWT is implemented in \(O(mg)\) operations.
scientific article; zbMATH DE number 1471672
scientific article