An Algebraic Structure of Orthogonal Wavelet Space 1

In this paper we study the algebraic structure of the space of compactly supported orthonormal wavelets over real numbers. Based on the parametrization of wavelet space, one can deene a parameter mapping from the wavelet space of rank 2 (or 2-band, scale factor of 2) and genus g to the (g ? 1) dimensional real torus (the products of unit circles). By the uniqueness and exactness of factorization, this mapping is well-deened 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 parametrization 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 parametrization 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 m orthogonal DWT is O(m 2 g). In this paper we start with a given scaling lter and construct additional (m?1) wavelet lters so that the DWT can be implemented in O(mg). With a xed scaling lter, the approximation order, the orthogonality, and the smoothness remain unchanged, thus our fast DWT implementation is quite general.