TY - JOUR

T1 - The Cascade Hilbert-Zero Decomposition: A Novel Method for Peaks Resolution and Its Application to Raman Spectra

AU - Постников, Евгений

AU - Лебедева, Елена Александровна

AU - Зюбин, Андрей

AU - Лаврова, Анастасия Игоревна

N1 - Publisher Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland.

PY - 2021/11/1

Y1 - 2021/11/1

N2 - 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.

AB - 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.

KW - Hilbert transform

KW - Multiresolution

KW - Peak resolution

KW - Raman spectra

KW - CELLS

KW - BACTERIA

KW - R-PEAKS

KW - IDENTIFICATION

KW - peak resolution

KW - SPECTROSCOPY

KW - multiresolution

KW - TRANSFORM

KW - SCATTERING

UR - http://www.scopus.com/inward/record.url?scp=85118769558&partnerID=8YFLogxK

UR - https://www.mendeley.com/catalogue/b7a5b8a6-3ba9-3b3b-a028-0dda10e0eb4d/

U2 - 10.3390/math9212802

DO - 10.3390/math9212802

M3 - Article

VL - 9

JO - Mathematics

JF - Mathematics

SN - 2227-7390

IS - 21

M1 - 2802

ER -