Wavelet prism decomposition analysis applied to CARS spectroscopy: a tool for accurate and quantitative extraction of resonant vibrational responses

Abstract

We propose an approach, based on wavelet prism decomposition analysis, for correcting experimental artefacts in a coherent anti-Stokes Raman scattering (CARS) spectrum. This method allows estimating and eliminating a slowly varying modulation error function in the measured normalized CARS spectrum and yields a corrected CARS line-shape. The main advantage of the approach is that the spectral phase and amplitude corrections are avoided in the retrieved Raman line-shape spectrum, thus significantly simplifying the quantitative reconstruction of the sample's Raman response from a normalized CARS spectrum in the presence of experimental artefacts. Moreover, the approach obviates the need for assumptions about the modulation error distribution and the chemical composition of the specimens under study. The method is quantitatively validated on normalized CARS spectra recorded for equimolar aqueous solutions of D-fructose, D-glucose, and their disaccharide combination sucrose.

Bayesian Quantification for Coherent Anti-Stokes Raman Scattering Spectroscopy

“…We model CARS spectral measurements with an additive error model given as where y k denotes a measurement that has been discretized with spectral sampling resolution h > 0 at a wavenumber location ν k = kh with , f (ν k ; p , θ ) is the CARS spectrum model with parameter p controlling the baseline and parameters θ for the Voigt line shape, and with measurement error with known variance. For the spectrum, we use a parameter-wise separable model where p is the interpolated discrete wavelet transform (DWT) detail level, ε m (ν; p ) is the modulating error function, and S (ν; θ ) is the error-corrected CARS signal, similar to the representation used in ref ( 22 ). The signal S can further be represented as where the exponential part corresponds to the non-Raman part with A J practically constant (see ref ( 22 ) for details), is the Hilbert transform, and where * denotes convolution.…”

“…For the spectrum, we use a parameter-wise separable model where p is the interpolated discrete wavelet transform (DWT) detail level, ε m (ν; p ) is the modulating error function, and S (ν; θ ) is the error-corrected CARS signal, similar to the representation used in ref ( 22 ). The signal S can further be represented as where the exponential part corresponds to the non-Raman part with A J practically constant (see ref ( 22 ) for details), is the Hilbert transform, and where * denotes convolution. N stands for the number of line shapes, with each line shape having θ n ≔ ( a n , ν n , σ n , γ n ) T parameters standing for the amplitude, location, scale of the Gaussian shape, and scale of the Lorentzian shape, respectively.…”

“…Instead of the wavelet prism method, 22 we model the modulating error function as where p ∈ [1, J ], and β = p – ⌊ p ⌋, i.e., as an interpolation between the discrete wavelet reconstruction levels D j to have a continuous model for the background in contrast to the wavelet prism method where only the discrete wavelet reconstruction levels are used. With the above, we can have an unnormalized posterior formulated as where is the vector of observations given via eq 1 , is the parameter vector ( θ 1 , ..., θ N ) T for the N Voigt peaks, represents the likelihood distribution of the forward model, and π 0 ( p , θ ) denotes prior distributions for some or all of the model parameters p and θ .…”

“…Recently, a procedure based on the wavelet prism decomposition algorithm was proposed to address this issue. 22 …”

“…Using the Hilbert transform, we construct the modulus of the resonant part of the CARS spectrum. A nonresonant part, estimated from the data, 22 is added to obtain an error-free CARS spectrum, which is finally modulated with a slowly varying error function. Next, we describe the numerical algorithms used for statistical inference and line narrowing, along with our Bayesian prior distributions.…”