The Hilbert Transform of B-Spline Wavelets

• Published in: IEEE signal processing letters Vol. 28; pp. 693 - 697
• Main Authors: Yu, Bo, Yang, Xiuzhu
• Format: Journal Article
• Language: English
• Published: New York IEEE 2021; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Asymptotes; B spline functions; B-spline wavelets; Digital imaging; hilbert transform; Hilbert transformation; Image processing; Integral equations; Mathematical model; Polynomials; Signal processing; Splines (mathematics); Symmetry; Transforms; Wavelet transforms
• ISSN: 1070-9908; 1558-2361
• DOI: 10.1109/LSP.2021.3069122

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 <inline-formula><tex-math notation="LaTeX">m</tex-math></inline-formula> 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 <inline-formula><tex-math notation="LaTeX">m</tex-math></inline-formula> 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 <inline-formula><tex-math notation="LaTeX">m=3</tex-math></inline-formula> and <inline-formula><tex-math notation="LaTeX">m=4</tex-math></inline-formula> are given and the graphs of these two B-spline wavelets as well as their Hilbert transforms are presented. These two cases of <inline-formula><tex-math notation="LaTeX">m=3</tex-math></inline-formula> and <inline-formula><tex-math notation="LaTeX">m=4</tex-math></inline-formula> provide two Hilbert transform pairs of wavelets, which can be used in digital image processing.