An Algebraic Structure of Orthogonal Wavelet Space

In this paper we study the algebraic structure of the space of compactly supported orthonormal wavelets over real numbers. Based on the parameterization of wavelet space, one can define a parameter mapping from the wavelet space of rank 2 (or 2-band, scale factor of 2) and genus gto the (g−1) dimensional real torus (the products of unit circles). By the uniqueness and exactness of factorization, this mapping is well defined and one-to-one. Thus we can equip the rank 2 orthogonal wavelet space with an algebraic structure of the torus. Because of the degenerate phenomenon of the paraunitary matrix, the parameterization map is not onto. However, there exists an onto mapping from the torus to the closure of the wavelet space. And with such mapping, a more complete parameterization is obtained. By utilizing the factorization theory, we present a fast implementation of discrete wavelet transform (DWT). In general, the computational complexity of a rank morthogonal DWT is O(m²g). In this paper we start with a given scaling filter and construct additional (m−1) wavelet filters so that the DWT can be implemented in O(mg). With a fixed scaling filter, the approximation order, the orthogonality, and the smoothness remain unchanged; thus our fast DWT implementation is quite general.