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Abstract
We demonstrate the applicability of using interpolated inverse discrete wavelet transforms as a general tool for modeling additive or multiplicative background or error signals in spectra. Additionally, we propose an unsupervised way of estimating the optimal wavelet basis along with the model parameters. We apply the method to experimental Raman spectra of phthalocyanine blue, aniline black, naphthol red, pigment yellow 150, and pigment red 264 pigments to remove their additive background and to CARS spectra of adenosine phosphate, fructose, glucose, and sucrose to remove their multiplicative background signals.
Published by Optica Publishing Group under the terms of the Creative Commons Attribution 4.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.
1. Introduction
Raman spectroscopy is a powerful technique for molecule identification and analysis. As Raman spectroscopy and its variations, such as coherent anti-Stokes Raman scattering (CARS), surface-enhanced Raman spectroscopy (SERS), or surface-enhanced resonance Raman spectroscopy (SERRS), are noninvasive and nondestructive, they are particularly suitable for biomedical applications amongst many others [1,2].
However, the spectra are often accompanied by some form of a background signal, obfuscating the Raman signal arising from molecular vibrations. Approaches to correct the background include asymmetric least squares [3,4], polynomial fits [5], and wavelet-based methods [6,7]. Regardless of their popularity, the above methods can suffer from some drawbacks. Manual polynomial fitting requires selection of cross-over points which is time-consuming and is clearly not an automatic approach. Asymmetric least squares uses regularization uniformly across the whole spectrum which can result in correct smoothing of the fitted background in parts while oversmoothing in undesirable parts of the spectrum. For a more extensive review, see [8].
In this study, we concentrate on interpolated discrete wavelet transforms previously considered in [9] as an extension of the wavelet prism method [10] to correct CARS multiplicative errors and also to amend some of the classical drawbacks mentioned above. Wavelet methods do not require manual selection of points and their ability to model features locally circumvents problems with standard regularization approaches such as with asymmetric least squares methods. What we call an interpolated inverse wavelet transform is a modification of the standard way of constructing an original signal via inverting a wavelet transform. The wavelet transform constructs a time-frequency domain representation of a function originally in the time domain using a small number of levels of coefficients commonly called the detail and approximation coefficients. This transformation can be inverted via the inverse wavelet transform where the coefficients levels are stitched together to reconstruct the original signal. While flexible, this allows only for a discrete number of different backgrounds to be modelled. We have extended this approach by interpolating between the individual inverse wavelet transform reconstruction steps, allowing for a continuous model for the spectral background.
We demonstrate the interpolation of wavelet reconstruction levels as a viable way of modeling background processes for general spectra and propose a criterion for automatically choosing the wavelet basis and the associated model parameters. The model is parameterized with only one or two parameters, enabling applications where high-dimensional parameters could be infeasible such as in Bayesian inference, see for example [9,11]
The rest of the manuscript is structured as follows. We formulate the interpolated wavelet transform models for additive and multiplicative backgrounds and the optimization scheme in Sections 2 and 3. Experimental details are documented in Section 4 and we present results for the experimental Raman and CARS spectra in Section 5. We close out the study with conclusions in Section 6.
2. Interpolated wavelet background model for additive backgrounds
We model a measured spectrum with an additive background $B(\nu _k, c, p, \mathcal {M})$ as
where $\nu _k$ denotes the measurement location, $c$ a constant offset of the background, $p$ the interpolated wavelet detail level, and $\mathcal {M}$ a chosen wavelet basis which can be considered as a model selection problem. We model the background as in [9], with an interpolation between the discrete wavelet reconstruction levels $D_j(\nu )$ given asWe optimize the parameter $p$ and model $\mathcal {M}$ as follows. The optimal parameter $\hat {p}$ and model $\hat {\mathcal {M}}$ for additive Raman spectra are ones which minimize
3. Interpolated wavelet background model for multiplicative backgrounds
For multiplicative spectra, such as in the case of CARS, we model a measurement spectrum as
where $z(\nu _k)$ is a measurement at $\nu _k$, $S(\nu _k)$ denotes the error-free spectrum and $\varepsilon _{\rm {m}}(\nu _k; c, p)$ is a modulating background or error function. The expression in Eq. 4 can be made additive by taking the logarithm: which enables us to model the modulating error function similarly to the approach in Section 2:We propose the following approach for optimizing the CARS model parameters. The optimal parameter $\hat {p}$ and model $\hat {\mathcal {M}}$ for CARS spectra are the ones which minimize
4. Experimental details
In the following Sections, we describe the experimental samples used for the numerical examples.
4.1 Raman samples
The Raman spectra were obtained from a free online database of Raman spectra of pigments used in modern and contemporary art (The standard Pigments Checker v.5) [16]. All the spectra used here were corrupted by fluorescence interferences resulting in relatively large background components to the Raman lineshapes.
4.2 CARS samples
The sugar samples used in the multiplex CARS spectroscopy were equimolar aqueous solutions of D-fructose, D-glucose, and their disaccharide combination, sucrose ($\alpha$-D-glucopyranosyl-(1$\to$2)-$\beta$-D-fructofuranoside). For sample preparation, the sugar samples were dissolved in buffer solutions (50 mM HEPES, pH=7) at equal molar concentrations of 500 mM [17]. The adenosine phosphate sample was an equimolar mixture of adenosine mono-, di-, and triphosphate (AMP, ADP, ATP) in water in water for a total concentration of 500 mM [13]. The adenine ring vibrations [18] are found at identical frequencies for either for AMP, ADP or ATP around 1350 cm$^{-1}$. The phosphate vibrations between 900 and 1100 cm$^{-1}$ can be used to discriminate between the different nucleotides [19]. The tri-phosphate group of ATP shows a strong resonance at 1123 cm$^{-1}$, whereas the monophosphate resonance of AMP is found at 979 cm$^{-1}$. For ADP a broadened resonance is found in between at 1100 cm$^{-1}$.
4.3 Multiplex CARS spectroscopy
All CARS spectra used to validate our methodology were recorded using a multiplex CARS spectrometer, the detailed description of which can be found elsewhere [19,20]. In brief, a 10-ps and an 80-fs mode-locked Ti:sapphire lasers were electronically synchronized and used to provide the narrowband pump/probe and broadband Stokes laser pulses in the multiplex CARS process. The center wavelengths of the pump/probe and Stokes pulses were 710 nm. The Stokes laser was tunable between 750 and 950 nm. The sugar spectra were probed within a wavenumber range from 700 to 1250 cm$^{-1}$, and the AMP/ADP/ATP spectrum within a range from 900 to 1700 cm$^{-1}$. The linear and parallel polarized pump/probe and Stokes beams were made collinear and focused with an achromatic lens into a tandem cuvette. The latter could be translated perpendicular to the optical axis to perform measurements in either of its two compartments, providing a multiplex CARS spectrum of the sample and of a non-resonant reference under near-identical experimental conditions. Typical average powers used at the sample were 95 mW (75 mW, in case of AMT/ADP/ATP) and 25 mW (105 mW) for the pump/probe and Stokes laser, respectively. The anti-Stokes signal was collected and collimated by a second achromatic lens in the forward-scattering geometry, spectrally filtered by short-pass and notch filters, and focused into a spectrometer equipped with a CCD camera. The acquisition time per CARS spectrum was 200 ms for sugar spectra and 800 ms for the AMP/ADP/ATP spectrum.
5. Results and discussion
We apply the method to experimental Raman spectra of phthalocyanine blue, aniline black, and naphthol red for additive background removal and to experimental CARS spectra of adenosine phosphate, fructose, glucose, and sucrose for multiplicative background correction. The results are presented in Figs. 1–9, respectively, showing the experimental measurements, the estimated additive or multiplicative background, and the corrected spectrum.
Fig. 1. In blue, data for phthalocyanine blue. In red, corrected spectrum $f(\nu )$ with the estimated additive background shown in black. The resulting background was obtained using $\hat {\mathcal {M}} =$ symlet 5 wavelet with $\hat {p} = 10.663$.
Fig. 2. In blue, data for aniline black. In red, corrected spectrum $f(\nu )$ with the estimated additive background shown in black. The resulting background was obtained using $\hat {\mathcal {M}} =$ symlet 5 wavelet with $\hat {p} = 11.882$.
Fig. 3. In blue, data for naphthol red. In red, corrected spectrum $f(\nu )$ with the estimated additive background shown in black. The resulting background was obtained using $\hat {\mathcal {M}} =$ symlet 5 wavelet with $\hat {p} = 11.591$.
Fig. 4. In blue, data for pigment yellow 150. In red, corrected spectrum $f(\nu )$ with the estimated additive background shown in black. The resulting background was obtained using $\hat {\mathcal {M}} =$ Daubechies 22 wavelet with $\hat {p} = 7.756$.
Fig. 5. In blue, data for pigment red 264. In red, corrected spectrum $f(\nu )$ with the estimated additive background shown in black. The resulting background was obtained using $\hat {\mathcal {M}} =$ symlet 11 wavelet with $\hat {p} = 10.555$.
Fig. 6. In blue, a measured CARS spectrum for adenosine phosphate. In red, corrected spectrum $S(\nu )$ with the estimated multiplicative background shown in black. The resulting background was obtained using $\hat {\mathcal {M}} =$ Daubechies 10 wavelet with $\hat {p} = 12.652$.
Fig. 7. In blue, a measured CARS spectrum for fructose. In red, corrected spectrum $S(\nu )$ with the estimated multiplicative background shown in black. The resulting background was obtained using $\hat {\mathcal {M}} =$ coiflet 3 wavelet with $\hat {p} = 13.402$.
Fig. 8. In blue, a measured CARS spectrum for glucose. In red, corrected spectrum $S(\nu )$ with the estimated multiplicative background shown in black. The resulting background was obtained using $\hat {\mathcal {M}} =$ Daubechies 5 wavelet with $\hat {p} = 13.616$.
Fig. 9. In blue, a measured CARS spectrum for sucrose. In red, corrected spectrum $S(\nu )$ with the estimated multiplicative background shown in black. The resulting background was obtained using $\hat {\mathcal {M}} =$ Daubechies 35 wavelet with $\hat {p} = 11.702$.
For the wavelet models $\mathcal {M}$, Daubechies 5–35, symlet 5–25, and Coiflet 1–5 wavelets were used. This totals 57 different wavelet bases.
Results for the phthalocyanine blue, aniline black, and naphthol red show smooth backgrounds for the Raman spectra, albeit with some boundary effects for the naphthol red background. Similar boundary effects are visible in the results for pigment red 264 in Fig. 5. Figure 4 shows the background removal for the Raman spectrum for pigment yellow 150. The small and practically constant background is successfully removed and no overfitting is observable. The optimized wavelet bases were symlet 5, symlet, symlet 5, Daubechies 22, symlet 11 for the phthalocyanine blue, aniline black, naphthol red, pigment yellow 150, and pigment red 264 Raman spectra.
For the CARS spectra, the results are similar to the results presented in [9]. The results for the adenosine phosphate, fructose, and glucose CARS spectrum in Figs. 6–8 show different slowly varying background signals with roughly similar frequency contents which can also be approximately inferred from the obtained wavelet interpolation parameters which are of similar magnitude, $\hat {p} = 12.652, 13.402, 13.616$, respectively. The sucrose CARS results in Fig. 9 show a more linear background which is also evident in the lower wavelet interpolation parameter value, $\hat {p} = 11.702$, implying lower frequency components in comparison to the other three cases. The optimized wavelet basises for the adenosine phosphate, fructose, glucose and sucrose CARS spectra were Daubechies 38, symlet 12, symlet 14, and Daubechies 31.
The wavelet background model shows flexible behavior. For the phthalocyanine blue Raman sample in Fig. 1, the estimated background is practically constant. In Figs. 5 and 9 for the pigment red 264 and sucrose, the behaviour resembles a linear fit. With phthalocyanine blue, aniline black, and naphthol red Raman samples, as well as, the adenosine phosphate, fructose, glucose CARS samples, the model is able to capture varying degrees of oscillating features.
6. Conclusions
In this study, we proposed an interpolated wavelet-based background model as a tool for modeling background functions for spectra. We successfully optimize the background parameter and the wavelet basis using an optimization scheme based on minimizing areas of the corrected Raman spectra for both Raman and CARS measurements. The model can model a flexible family of background functions while being parameterized according with only two parameters for Raman spectra with additive backgrounds and one parameter for CARS spectra with multiplicative backgrounds. We apply the proposed method to experimental Raman spectra of aniline black and phthalocyanine blue and to experimental CARS spectra of adenosine phosphate, fructose, glucose, and sucrose. Results for CARS spectra agree with previously obtained results.
Funding
Academy of Finland (327734).
Disclosures
The authors declare no conflicts of interest.
Data Availability
Data and software underlying the results presented in this paper are publicly available in the corresponding author’s personal repository on GitHub or from the authors upon reasonable request.
References
1. G. McNay, D. Eustace, W. E. Smith, K. Faulds, and D. Graham, “Surface-enhanced Raman scattering (SERS) and surface-enhanced resonance Raman scattering (SERRS): A review of applications,” Appl. Spectrosc. 65(8), 825–837 (2011). [CrossRef]
2. S. Li, Y. Li, R. Yi, L. Liu, and J. Qu, “Coherent anti-Stokes Raman scattering microscopy and its applications,” Front. Phys. 8, 1 (2020). [CrossRef]
3. H. F. M. Boelens, P. H. C. Eilers, and T. Hankemeier, “Sign constraints improve the detection of differences between complex spectral data sets: LC-IR as an example,” Anal. Chem. 77(24), 7998–8007 (2005). [CrossRef]
4. S. He, W. Zhang, L. Liu, Y. Huang, J. He, W. Xie, P. Wu, and C. Du, “Baseline correction for Raman spectra using an improved asymmetric least squares method,” Anal. Methods 6(12), 4402–4407 (2014). [CrossRef]
5. F. Gan, G. Ruan, and J. Mo, “Baseline correction by improved iterative polynomial fitting with automatic threshold,” Chemom. Intell. Lab. Syst. 82(1-2), 59–65 (2006). [CrossRef]
6. C. M. Galloway, E. C. L. Ru, and P. G. Etchegoin, “An iterative algorithm for background removal in spectroscopy by wavelet transforms,” Appl. Spectrosc. 63(12), 1370–1376 (2009). [CrossRef]
7. M. Chi, X. Han, Y. Xu, Y. Wang, F. Shu, W. Zhou, and Y. Wu, “An improved background-correction algorithm for Raman spectroscopy based on the wavelet transform,” Appl. Spectrosc. 73, 78–87 (2019). PMID: 30251877. [CrossRef]
8. K. H. Liland, T. Almøy, and B.-H. Mevik, “Optimal choice of baseline correction for multivariate calibration of spectra,” Appl. Spectrosc. 64(9), 1007–1016 (2010). [CrossRef]
9. T. Härkönen, L. Roininen, M. T. Moores, and E. M. Vartiainen, “Bayesian quantification for coherent anti-Stokes Raman scattering spectroscopy,” J. Phys. Chem. B 124(32), 7005–7012 (2020). [CrossRef]
10. Y. Kan, L. Lensu, G. Hehl, A. Volkmer, and E. M. Vartiainen, “Wavelet prism decomposition analysis applied to CARS spectroscopy: A tool for accurate and quantitative extraction of resonant vibrational responses,” Opt. Express 24(11), 11905–11916 (2016). [CrossRef]
11. M. Moores, K. Gracie, J. Carson, K. Faulds, D. Graham, and M. Girolami, “Bayesian modelling and quantification of Raman spectroscopy,” arXiv, arXiv.1604.07299 (2016). [CrossRef]
12. E. M. Vartiainen, “Phase retrieval approach for coherent anti-Stokes Raman scattering spectrum analysis,” J. Opt. Soc. Am. B 9(8), 1209–1214 (1992). [CrossRef]
13. E. M. Vartiainen, H. A. Rinia, M. Müller, and M. Bonn, “Direct extraction of Raman line-shapes from congested CARS spectra,” Opt. Express 14(8), 3622–3630 (2006). [CrossRef]
14. Y. Liu, Y. J. Lee, and M. T. Cicerone, “Broadband CARS spectral phase retrieval using a time-domain Kramers–Kronig transform,” Opt. Lett. 34(9), 1363–1365 (2009). [CrossRef]
15. M. T. Cicerone, K. A. Aamer, Y. J. Lee, and E. Vartiainen, “Maximum entropy and time-domain Kramers–Kronig phase retrieval approaches are functionally equivalent for CARS microspectroscopy,” J. Raman Spectrosc. 43(5), 637–643 (2012). [CrossRef]
16. “The standard Pigments Checker v.5,” https://chsopensource.org/pigments-checker/. Accessed: 2023-02-14.
17. M. Müller, H. A. Rinia, M. Bonn, E. M. Vartiainen, M. Lisker, and A. van Bel, “Quantitative multiplex CARS spectroscopy in congested spectral regions,” Multiphoton Microscopy in the Biomedical Sciences VII 6442, 21–29 (2007).
18. M. Mathlouthi and D. V. Luu, “Laser-Raman spectra of d-fructose in aqueous solution,” Carbohydr. Res. 78(2), 225–233 (1980). [CrossRef]
19. H. A. Rinia, M. Bonn, and M. Müller, “Quantitative multiplex CARS spectroscopy in congested spectral regions,” J. Phys. Chem. B 110(9), 4472–4479 (2006). [CrossRef]
20. M. Müller and J. M. Schins, “Imaging the thermodynamic state of lipid membranes with multiplex CARS microscopy,” J. Phys. Chem. B 106(14), 3715–3723 (2002). [CrossRef]
