Running Head : ALGEBRAIC STRUCTURE OF WAVELET SPACEContact

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, an orthogonal rank m DWT has an O(m 2) complexity. In this paper we starts with a given scaling lter and construct an additional (m?1) wavelet lters so that the DWT can be implemented in O(m). With a xed scaling lter, the approximation order, the orthogonality, and the smoothness remain unchanged, thus our fast DWT implementation is quite general. S 1 the unit circle in R 2 S m?1 the unit sphere in R m A a matrix with m rows A(z) the Laurent series or polyphase decomposition of A m the rank of a wavelet matrix, or the scale factor of a wavelet system g the genus of a wavelet matrix, which implies the matrix has size m mg; or the m ? band lter bank has length mg ~ A(z) the adjoint of A(z) A the Hermitian adjoint of A A the complex conjugate of A A t the transpose of A I m the m m identity matrix v a unit column vector V (z) primitive paraunitary matrix d the number of factor in paraunitary factorization k 1 the smallest index that A k 6 = 0 i;j the delta function ^ () the Fourier transform of (x) U a unitary matrix H the canonical Haar matrix (A) the characteristic Haar matrix of A det(A) the determinant of A H the Pollen product WM(2; g) the collection of all real orthogonal wavelet matrices of rank 2 and genus g WM(2) the collection of all real orthogonal wavelet matrices …