The Hilbert transform of B-spline wavelets

Both the Hilbert transform and the B-spline wavelets are important tools in signal processing, which makes a study of relation between these two subjects be of practical significance. In particular, because the B-spline wavelets have good properties including vanishing moments, symmetry, compact support and so on, we focus on the Hilbert transform of B-spline wavelets in this letter. For this purpose, the B-spline wavelets of order $m$ is described in a piecewise polynomial form firstly. An important property of the Pascal triangle transform is then explored. Based on these results, an explicit form of the Hilbert transform of B-spline wavelet of order $m$ is established. Furthermore, the vanishing moments, symmetry and asymptote behavior of the Hilbert transform of B-spline wavelets are also discussed. To demonstrate the effectiveness of these results, two examples in the case of $m=3$ and $m=4$ are given and the graphs of these two B-spline wavelets as well as their Hilbert transforms are presented. These two cases of $m=3$ and $m=4$ provide two Hilbert transform pairs of wavelets, which can be used in digital image processing.