Computation of Battle–Lemarie Wavelets Using an FFT-Based Algorithm

Abstract

Battle and Lemarie derived independently wavelets by orthonormalizing B-splines. The scaling function Φ _m (t) corresponding to Battle–Lemarie's wavelet ψ _m (t) is given by $\phi _m (t) = \sum\nolimits_{k \in {\text{Z}}} {\alpha _{m,k} B_m (t - k)}$, where B _m(t) is the mth-order central B-spline and the coefficients α _{m, k} satisfy $\sum\nolimits_{k \in {\text{Z}}} {\alpha _{m,k} e^{ - jk\omega } = {1 \mathord{\left/ {\vphantom {1 {\sqrt {\sum\nolimits_{k \in {\text{Z}}} {B_{2m} (k)e^{ - jk\omega } } } }}} \right. \kern-\nulldelimiterspace} {\sqrt {\sum\nolimits_{k \in {\text{Z}}} {B_{2m} (k)e^{ - jk\omega } } } }}}$. In this paper, we propose an FFT-based algorithm for computing the expansion coefficients α _{m, k} and the two-scale relations of the scaling functions and wavelets. The algorithm is very simple and it can be easily implemented. Moreover, the expansion coefficients can be efficiently and accurately obtained via multiple sets of FFT computations. The computational approach presented in this paper here is noniterative and is more efficient than the matrix approach recently proposed in the literature.