The Cascade Hilbert-Zero Decomposition: A Novel Method for Peaks Resolution and Its Application to Raman Spectra

Eugene B. Postnikov, Elena A. Lebedeva, Andrey Yu. Zyubin and Anastasia I. Lavrova
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Eugene B. Postnikov: Department of Theoretical Physics, Kursk State University, 305000 Kursk, Russia
Elena A. Lebedeva: Mathematics and Mechanics Faculty, Saint Petersburg State University, 198504 Saint Petersburg, Russia
Andrey Yu. Zyubin: Research an Educational Center “Fundamental and Applied Photonics. Nanophotonics”, Immanuel Kant Baltic Federal University, 236041 Kaliningrad, Russia
Anastasia I. Lavrova: Saint-Petersburg Research Institute of Phthisiopulmonology, 194064 Saint Petersburg, Russia

Mathematics, 2021, vol. 9, issue 21, 1-14

Abstract: Raman spectra of biological objects are sufficiently complex since they are comprised of wide diffusive spectral peaks over a noisy background. This makes the resolution of individual closely positioned components a complicated task. Here we propose a method for constructing an approximation of such systems by a series, respectively, to shifts of the Gaussian functions with different adjustable dispersions. It is based on the coordination of the Gaussian peaks’ location with the zeros of the signal’s Hilbert transform. The resolution of overlapping peaks is achieved by applying this procedure in a hierarchical cascade way, subsequently excluding peaks of each level of decomposition. Both the mathematical rationale for the localization of intervals, where the zero crossing of the Hilbert-transformed uni- and multimodal mixtures of Gaussians occurs, and the step-by-step outline of the numerical algorithm are provided and discussed. As a practical case study, we analyze results of the processing of a complicated Raman spectrum obtained from a strain of Mycobacterium tuberculosis . However, the proposed method can be applied to signals of different origins formed by overlapped localized pulses too.

Keywords: multiresolution; Hilbert transform; peak resolution; Raman spectra (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2021
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