Extended Multiplicative Signal Correction for Infrared Microspectroscopy of Heterogeneous Samples with Cylindrical Domains

Abstract

Optical scattering corrections are invoked to computationally distinguish between scattering and absorption contributions to recorded data in infrared (IR) microscopy, with a goal to obtain an absorption spectrum that is relatively free of the effects of sample morphology. Here, we present a modification of the extended multiplicative signal correction (EMSC) approach that allows for spectral recovery from fibers and cylindrical domains in heterogeneous samples. The developed theoretical approach is based on exact Mie theory for infinite cylinders. Although rigorous Mie theory implies utilization of comprehensive and time-consuming calculations, we propose to change the workflow of the original EMSC algorithm to minimize extensive calculations for each recorded spectrum at each iteration step. This makes the modified EMSC approach practical for routine use. First, we tested our approach using synthetic data derived from a rigorous model of scattering from cylinders in an IR microscope. Second, we applied the approach to Fourier transform IR (FT-IR) microspectroscopy data recorded from filamentous fungal and cellulose samples with pronounced fiber-like shapes. While the corrected spectra show greatly reduced baseline offsets and consistency, strongly absorbing regions of the spectrum require further refinement. The modified EMSC algorithm broadly mitigates the effects of scattering, offering a practical approach to more consistent and accurate spectra from cylindrical objects or heterogeneous samples with cylindrical domains.

Keywords

Extended multiplicative signal correction mid-infrared microspectroscopy fibers scattering

Introduction

Fourier transform infrared (FT-IR) microscopy and spectroscopic imaging¹ are commonly used to understand molecular composition in a variety of samples. Recent applications include examples from biology,^2–6 materials science,^7–9 forensics,^10,11 agriculture,^12,13 and medicine.^14–25 Although FT-IR microspectroscopy includes the exceptional molecular selectivity of fundamental absorption modes, high spectral accuracy from interferometric measurements, and robust signal-to-noise (S/N) ratios, spectral distortions²⁶ arise from scattering at interfaces²⁷ that can complicate the analysis of recorded data. For analytical measurements of samples whose sizes are comparable to the wavelength of impinging electromagnetic radiation, in practice, scattering from the object dominates the total recorded attenuation.

There is now an emerging understanding of the physics underlying spectral distortions associated with scattering of IR radiation from samples with spherical,^28,29 cylindrical,³⁰ thin film,^31–37 or more complex geometry.^38–40 The forward problem of predicting IR spectra, given the shape and properties of the sample, is now generally well described for a variety of cases. Various theoretical solutions of the inverse problem, namely recovering the absorption of the sample as well as its morphologic properties, for the case of single spheres^41–43 and cylinders³⁰ as well as for thin film samples^44,45 have been proposed. A modification of the experimental setup for minimization of recorded scattering has been also developed.⁴⁶ Theoretical approaches require either the knowledge of the particle’s size,³⁰ or multiple⁴¹ or even quite accurate^42,43 measurements of IR spectrum of the intact object. Though, in most cases, a single measurement of the sample with a number of spherical or cylindrical inclusions like overlapping or quite densely packaged cells²⁸ or fibers⁴⁷ without clear knowledge of the object size will likely occur. Therefore, for most cases, the inverse problem remains a challenging task.

A post-acquisition computational approach to address this need has been proposed. Termed extended multiplicative signal correction (EMSC),⁴⁸ this technique has been developed extensively during the last decade^49–54 for samples with spherical inclusions. It was shown that so-called dispersion artefact,²⁸ or effect of scattering from interfaces in which profile of the real part of the refractive index affects the recorded absorbance, can be mitigated with this algorithm. Extended multiplicative signal correction-corrected data with minimized contribution of undesirable scattering make it possible to adequately analyze IR spectroscopic data acquired from biological cells and tissues.^23,55 The applicability of EMSC thus far, however, is limited to objects with spherical shape. While the approach is effective, it is also limiting since the spherical symmetry may obscure effects that arise in other geometries. For example, many objects commonly measured by IR microscopy are cylindrical in shape (e.g., fibers, single crystals, and fungi) but an ideal cylinder as a scattering shape for EMSC has not been reported. Although the forward and inverse problems for a single intact fiber with known size are well-known,³⁰ a general formulation of the EMSC algorithm for cylinders and, particularly, its application to large data sets without a priori knowledge of cylinder size or refractive index is needed. Here we seek to fulfill this need by first theoretically extending the EMSC model to cylindrical geometries. Second, and in conjunction, we seek to formulate a computational strategy to work with large imaging data sets. Finally, we evaluate the efficacy of the developed approach for different samples.

Theory

Extended Multiplicative Signal Correction

We closely follow the formalism for EMSC thoroughly discussed in the references.^48,56–58 According to this algorithm, the recorded signal ${\vec{Z}}_{Raw}$ can be decomposed as follows:

{\vec{Z}}_{Raw} = c + m \vec{\tilde{}} ν + h {\vec{Z}}_{Ref} + \sum_{i = 1}^{7} {\vec{g}}_{i} p_{i} + \vec{E}

(1)

where c is the constant offset, m is the gradient of the sloping baseline,

\vec{\tilde{}} ν

is the wavevector (inverse wavelength

λ^{- 1}

), h is the multiplicative scaling factor for reference signal

{\vec{Z}}_{Ref}

, p_i and

{\vec{g}}_{i}

are loadings and controlling parameters, respectively, for the scattering contribution, and, finally,

\vec{E}

is an unmodelled residual information.

The recorded spectrum can be corrected by finding values of c, m and h using a least squares linear regression model. The Mie-type scattering contribution is contained in loadings p_i which are calculated from decomposition of simulated scattering efficiency $Q_{sca}$ curves using a non-mean-centered principal component analysis (PCA). Scattering efficiency $Q_{sca}$ is calculated at each iteration step for each recorded signal ${\vec{Z}}_{Raw}$ . Usually, up to 1000 permutations of $Q_{sca}$ are required for an adequate estimation of p_i which arise from variations of particle size and refractive index.⁵⁰ Utilization of the appropriate expression for $Q_{sca}$ makes it possible to adjust the EMSC approach to particles with various shapes. Conceptually, using the scattering efficiency $Q_{sca}$ in Eq. 1 rather than the originally proposed⁵⁰ extinction efficiency $Q_{ext}$ is a different yet reliable way to implement the EMSC.⁵¹ Though, in Bassan et al.⁵⁰ it is assumed that under van de Hulst approximation,⁵⁹ the scatterer is non-absorbing and, as a consequence, $Q_{ext} = Q_{sca}$ .

Scattering from Cylindrical Objects

In most cases, the longitudinal dimension (i.e., length) of natural or synthetic fibers is much larger than the transverse (diameter) dimension.^60,61 Further, the image pixel size of a typical IR microspectroscopy setup is usually comparable to the diameter of a fiber. Consequently, the ends of the fiber have little influence on any pixel towards the center and fibers can be generally approximated with an infinite cylinder for the development of a scattering model. In the case of unpolarized illumination, the scattering efficiency of an infinite cylinder embedded in a host medium with refractive index n_h can be found as:

Q_{sca} = \frac{1}{2} [Q_{sca, ∥} + Q_{sca, ⊥}]

(2)

where

Q_{sca, ∥} = \frac{2}{x} [| b_{0, ∥} | 2 + 2 \sum_{l = 1}^{\infty} (| b_{l, ∥} | 2 + | a_{l, ∥} | 2)]

(3)

Q_{sca, ⊥} = \frac{2}{x} [| a_{0, ⊥} | 2 + 2 \sum_{l = 1}^{\infty} (| a_{l, ⊥} | 2 + | b_{l, ⊥} | 2)]

(4)

are longitudinal and transverse scattering efficiencies per unit length of an infinite cylinder. Here

x = 2 π n_{h} ν r

is a dimensionless size parameter and l is the order of a spherical harmonic. Coefficients a_l and b_l in Eqs. 3 and 4 for illumination parallel ∥ and orthogonal ⊥ to the cylinder axis of symmetry are given in the Appendix. While we do not discuss polarized illumination and oriented fibers in this study focused on method development, the approach presented here can be extended to those two cases as well by considering cylinder and molecular orientation with respect to beam polarization.

Utilization of Eqs. 2–4 may be computationally expensive. Thus, GPU-enabled calculations⁵¹ or implementation of simplified analytical solutions like the original⁵⁰ or modified⁵⁴ van de Hulst approximation,⁵⁹ which have been successfully applied for spherical samples, are alternative ways to estimate $Q_{sca}$ . It should be understood, however, that in the case of cylinder, there is no adequate analytical approximation which is valid for a wide range of x. The Rayleigh approximation⁵⁹ is valid only for $x << 1$ , which is usually not the case for micron-sized objects under mid-IR illumination. Therefore, the use of Eqs. 2–4 is inevitable and the optimization of the EMSC performance becomes a crucial problem to solve.

Numerical Results

Scattering from Spheres and Cylinders

We start with a comparative analysis of scattering from single spherical and cylindrical particles. Although it is intuitive that the scattering patterns are different for these two shapes, it is worthwhile to discuss briefly the main features of each and the major differences between these two cases. The main purpose of this comparative analysis is to show the necessity of introduction of modified Resonant Mie Scattering (RMieS)/EMSC approach for the case of fibers. As a reference material, we chose polymethyl methacrylate (PMMA) whose experimentally measured spectra agree well with simulated ones.^40,41,43 Real and imaginary parts of the refractive index of PMMA are shown in Fig. 1a, while a schematic representation of illumination and scattering from the sphere and cylinder is shown in Fig. 1b and 1c.

Figure 1.

(a) Real, n, and imaginary, k, parts of the refractive index of bulk PMMA, and schematic representation of illumination and scattering from sphere (b) and cylinder (c) with radius r. (d–f) Scattering efficiencies $Q_{sca}$ for spherical and cylindrical PMMA particles for various radii r. Cylinders are illuminated with unpolarized illumination under the normal incidence, $η = 90^{\circ}$ . (g–i) Scattering efficiencies $Q_{sca}$ for PMMA cylinders with different radii r illuminated under various angles ζ.

Figure 1 explicitly shows that PMMA spheres and cylinders with the same radius r scatter electromagnetic waves in a different manner. For example, the scattering efficiency of a cylinder may be smaller, greater, or comparable with a sphere in $ν > 2000 {cm}^{- 1}$ spectral region, depending on the size of the particles. However, for sufficiently large particles, the scattering pattern is qualitatively, and in some cases quantitatively, the same in the fingerprint region, $ν < 2000 {cm}^{- 1}$ . This coincident correspondence may explain the success of the EMSC approach in handling materials that are not spherical as well; however, the differences demonstrate that the geometry must be considered for precise mitigation of scattering effects.

Unlike the sphere, the fiber is an object with rotational symmetry, thus $Q_{sca}$ also depends on the polarization of the incident illumination and spatial orientation of the fiber with respect to it. As an approximation when polarizers are not specifically used, IR spectroscopy utilizes unpolarized illumination. Thus, only the angle of incidence ζ gives rise to the additional degree of freedom which has to be included to permutations of $Q_{sca}$ along with refractive index n_p and radius r. For polarized electromagnetic wave and/or molecularly oriented cylindrical samples, further modifications may be added to these permutations. Figure 1 shows that the scattering pattern of a cylinder strongly depends on the diameter as well. For relatively small cylinders ( $r = 1 μ m$ ), the scattering pattern is very similar for small wavenumbers (longer wavelengths compared to the radius) and varies quantitatively for different ζ. In contrast, for a relatively large ( $r = 5 μ m$ ) cylinder, the variation is both quantitative and qualitative for different angles of incidence. Therefore, from analysis of Fig. 1, one can generally expect completely different spectral distortions for fibers compared to spherical samples.

Optimization of Extended Multiplicative Signal Correction Algorithm for Mie Scatterers

Although there are various optimizations and improvements of the EMSC algorithm for Mie scattering in terms of speed⁵¹ and stability,⁵⁴ there is at least one possible improvement that has not been addressed so far. The original RMieS/EMSC algorithm⁵⁰ generates around $\approx 1000$ permutations of $Q_{sca}$ at each iteration step for each given data set which arises from various combinations of size, mean value of the refractive index, and its variation according to Kramers–Kroning relations. The use of the van de Hulst analytic solution for $Q_{sca}$ does not significantly affect the performance of RMieS/EMSC in terms of speed and accuracy; however, the use of the EMSC extended for cylindrical particles requires significant computational resources because of the lack of a simple and accurate approximation for $Q_{sca}$ . GPU-enabled calculations of the Bessel functions which enter Eqs. 3 and 4 may enhance the performance as shown in Bassan et al.⁵¹ We propose that there is a way to avoid extensive calculations in this case. From Eqs. 3 and 4, along with expressions for expansion coefficients given in the Appendix, one can conclude that the only two parameters which contribute to $Q_{sca}$ , the refractive index n_p and size parameter $x = 2 π n_{h} ν r$ , describe possible permutations which arise from variations of ν and r. Thus, various combinations of ν and r may yield the same values of x for fixed n_h which is considered to be unity. Therefore, the large number of permutations of three variables can be shrunk to a smaller number (approximately by the order of magnitude) while retaining the accuracy of the method. One more usually overlooked practical aspect in implementation is that these permutations are repeatedly calculated for almost the same set of x and n_p during iterations. Thus, it might be more practical and convenient to calculate the possible set of $Q_{sca}$ before launching a RMieS/EMSC procedure. For each iteration step then, we propose to either find these values in pre-calculated data or perform an interpolation instead of calculating $Q_{sca}$ . This approach, as detailed in the flow chart shown in Fig. 2, will be less computationally expensive compared to straightforward simulation of $Q_{sca}$ with Eqs. 2–4 and is a significant, enabling step for large imaging data sets.
Figure 2.
Flow chart illustrating the iterative procedure of the EMSC algorithm. Red boxes indicate the modifications to the original RMieS/EMSC procedure⁵⁰ proposed in this work.

To demonstrate the utility of this concept, we have simulated the comprehensive $Q_{sca}$ data set for a number of values of x and n_p that span the vast majority of possible parameters of polymeric and biological samples. The endmembers of our database comprise $x_{min} = 0.5$ and $x_{max} = 25$ , which correspond to $(ν = 800 {cm}^{- 1}, r = 1 μ m)$ and $(ν = 4000 {cm}^{- 1}, r = 10 μ m)$ , respectively. Thus, for commonly utilized bandwidth for IR spectral analyses, this size parameter covers the vast majority of combinations of frequency and radii of fibers. Simultaneously, the minimum and maximum real refractive index of the particle has been set to $n_{p, min} = 1$ and $n_{p, max} = 2.1$ , respectively, which is once again spans the vast majority of polymeric and biological samples.

Figure 3a shows the calculated $Q_{sca}$ on a fine grid of x and n_p (1577 × 232 points, respectively), while Fig. 3b shows $Q_{sca}$ on the same range and grid of x and n_p but spline-interpolated from coarsely simulated data (122 × 23 points). It can be seen that interpolation of $Q_{sca}$ reproduces most of the essential features of the actual curves. To get a deeper insight, we provide the cross-sections of $Q_{sca}$ for fixed refractive index n_p and size parameter x, in Fig. 3c and 3d, respectively. The negligible discrepancy between precisely calculated scattering efficiency and obtained via interpolation arises only in $n_{p} > 1.7$ range for fixed x, while interpolated curves agree well with precise $Q_{sca}$ for fixed n_p. Given that most real objects are not perfect cylinders, and even for perfect geometries the error will be small, we anticipate that the algorithm will be more than adequate in accuracy and a significant increase in computing speed for most applications.
Figure 3.
(a) Accurate calculation with fine mesh and (b) spline interpolation from coarse data for scattering efficiency $Q_{sca}$ as a function of refractive index n_p and size parameter x for non-absorbing cylinder under unpolarized incident illumination impinging under $ζ = 45^{\circ}$ angle, and corresponding efficiency for fixed refractive index (c) and size parameter (d). Solid (red and blue) lines correspond to exact solution while dashed (green and yellow) lines represent interpolated data.

Verification of Extended Multiplicative Signal Correction on Simulated Data

Finally, to verify the proposed modification of the EMSC procedure and to demonstrate its validity, we apply the developed algorithm to synthetic data. Figure 4a shows absorbance of a PMMA cylinder with r = 5 µm calculated with the rigorous forward model from Davis et al.³⁰ The condenser and detection optics are composed of Schwarzschild objectives with numerical apertures ${NA}_{c} = {NA}_{d} = 0.65$ and an obscuration with numerical apertures NA_cob = NA_dob = 0.23.
Figure 4.
(a) Absorbance of PMMA cylinder with r = 5µm calculated with forward model from Davis et al.³⁰ (b) Ideal scattering-free absorbance calculated with Eq. 5 and used as $Z_{Ref}$ in Eq. 1. (c) EMSC-corrected absorbance and (d) absorbance calculated with Eq. 5 but for k recovered via iterative algorithm from Davis et al.³⁰ (e, f) Absolute errors between corrected and ideal absorbance for EMSC and iterative algorithm, respectively.

Scattering-free absorbance calculated in accordance with the Beer’s law^30,41 has been chosen as the reference spectrum $Z_{Ref}$ for the EMSC:
$A = \frac{4 π \tilde{ν} {kd}_{eff}}{log e (10)}$
(5)
where $d_{eff} = 4 r / 3$ is the effective thickness of the sample.⁵³

It can be seen from comparing Fig. 4a and 4b that the recorded signal is significantly different from the ideal one given by Eq. 5, which is explained by the effects of scattering. To mitigate scattering, we implemented the EMSC procedure in the following manner: we limited the database with $x \in [3, 17]$ and $n_{p} \in [1.2, 1.6]$ and used 10 iterations. It can be seen from Fig. 4c that the effects of scattering are removed and the corrected absorbance coincides with the ideal case from Fig. 4b. However, as it might be expected, it is difficult to recover the signal at strongly absorbing bands near $ν = 1728 {cm}^{- 1}$ and $ν = 1152 {cm}^{- 1}$ . For comparison, Fig. 4d also provides the absorbance calculated with Eq. 5 but for k recovered via iterative algorithm from Davis et al.³⁰ It can be seen from Fig. 4e and 4f that both EMSCs adapted for fibers, and the iterative algorithm from Davis et al.³⁰ provides reliable results with acceptable errors. However, the EMSC procedure can be easily applied for large data sets, while the formalism from Davis et al.³⁰ is only applicable to pixels located in the middle of fiber since it assumes a perfect cylindrical object. Thus, while the results of the EMSC algorithm are not as precise as an exact model, the EMSC algorithm is more generally applicable without need for constraints of geometry. We note that the approximations within the EMSC approach may not allow a perfect recovery, especially for strongly absorbing vibrational modes, but are considerably improved compared to the recorded data in Fig. 4a.

Experimental Methods

Samples

Two types of fibrous samples, fungi and cellulose fibers, were prepared to test the modified EMSC approach proposed in this study. For fungal samples, a filamentous fungal strain Fusarium spp. with pronounced cylinder-like shapes^62,63 were grown in petri dishes on potato dextrose agar (PDA) media incubated at 28 ℃ and 80% relative humidity. Fungal filaments (hyphae) that grew above the PDA surface were carefully picked and deposited on the top of low-E glass slides (Kevley Technologies) in such a way that the curved surfaces of these filaments lay horizontally on the low-E slides, shown in Fig. 5a. Similarly, cellulose samples were prepared by laying fibers from cellulose pad strips (MilliporeSigma, CFSP 173000) on a 4 mm thick rectangular ${BaF}_{2}$ window (Spectral Systems LLC.).
Figure 5.
(a) Optical reflection microscopy image of filamentous structures hyphae from fungal strain Fusarium spp. onto a reflective low-E glass slide for IR imaging. The lower and left dense parts show hyphae mixed with PDA media. Scale bar: 100 µm. (b) Cellulose fibers images acquired in the transmission mode. Scale bar: 100 µm. (c, d) Recorded absorbance at $\vec{} = 1653 {cm}^{- 1}$ and $\to = 1059 {cm}^{- 1}$ . (e, f) The same data after applying EMSC adapted for fibers.

Data Acquisition

A Cary 620 FT-IR Microscope (Agilent Technologies) imaging system was used to collect FT-IR imaging data at 4 cm^–1 spectral resolution (truncated to the detector sensitive range of 3800 cm^–1 to 900 cm^–1 wavenumbers) and 5.5 × 5.5 µm² per pixel spatial resolution using 15 × (NA = 0.62) magnification objective in standard definition mode. For each image, spectra were collected using identical spectral and spatial parameters with 128 spectral co-additions for background and 64 co-additions for the sample. In total, 16 384 spectra per image were collected in a 704 × 704 µm² (128 × 128 pixel²) area. For both types of samples, similar parameters were used for FT-IR data collection except the fungal sample was imaged in transflection mode while the cellulose sample was imaged in transmission mode. The optical microscopy image for fungi shown in Fig. 5a was captured using the same FT-IR microscopy setup in reflection mode, while the optical image for cellulose sample shown in Fig. 5b was collected on Zeiss Axio Imager M2 microscopy system with the 10 × (NA = 0.3) objective in transmission mode. We note that the choice of different substrates both for transflection and transmission modes might affect the recorded signal; however, for sufficiently large samples, this effect is almost negligible for the transflection regime,³⁴ and one could expect the same trend for transmission mode.

Data Analysis

Before the modified EMSC algorithm was applied, a minimum noise fraction (MNF)⁶⁴ preprocessing was performed on the collected FT-IR data for noise reduction using Environment for Visualization (ENVI)-Interactive Data Language (IDL) version 4.8. The MNF corrected data were processed with the modified RMieS/EMSC procedure adapted and optimized for cylinder-shape samples from original RMieS/EMSC Matlab code.⁶⁵ The modified algorithm was implemented in Matlab (Mathworks, Inc., Version 2018a) environment and tested on a desktop computer equipped with an Intel Core i7-6920HQ CPU @ 2.90 GHz processor. The pre-calculation of $Q_{sca}$ database on a fine grid with 4 cm^–1 spectral resolution takes up to several minutes, while the interpolation procedure using default Matlab functions requires < 50 ms for each iteration, depending on the number of permutations. Post-processing of 1 pixel of data requires ∼ 500 ms per iteration; thus, as might be expected, PCA takes most of the computational time.

Results and Discussion

As mentioned previously, the main utility of the EMSC algorithm lies in the generalized analysis of spectra from complex samples in an image. We demonstrate the applicability of our approach to two complex samples, namely fungi and fibrous cellulose. Fungi have been analyzed extensively by IR microscopy^66–72 and previous attempts at correcting data used the formulation of the EMSC that did not take into account the actual shape of the sample. Here, we apply the cylindrical version developed in this study, which is more suitable for the morphology.

The individual hyphae diameters are in the approximate range of 2–6.5 µm. However, in most of the cases, a collection of hyphae forms a bundle with diameters up to 25 µm. For given spatial resolution, each hyphae within the bundle might be considered as a single fiber. The diameter of each individual cellulose fiber is in the range of 6–20 µm. The size parameters shown in Fig. 3 cover all of these possible variations for the whole spectral range; therefore, we have used this pre-calculated set of $Q_{sca}$ for the EMSC procedure. As a reference spectrum, we have chosen the data recorded from the densest parts of the sample both for fungi and cellulose, assuming that the relative scattering contribution is minimal in these cases.⁴⁰ Each pixel of the measured data has been processed via EMSC with 10 iterations.

The images before and after correction, shown in Fig. 5c and 5e for fungi and in Fig. 5d and 5f for cellulose, demonstrate changes in several regions. The recorded data become more consistent after application of the scattering correction and increased contrast is clearly seen for regions where fiber-like samples are located. We note that fibers not in the plane of focus within these samples are blurred and may be difficult to visualize in the IR images, compared to the optical microscopy image. This is especially true of the samples on reflective slides (e.g., fungi here). In both cases, the images reflect the corrections in the underlying spectra. As expected from our results on synthetic data, a major manifestation in the correction is the mitigation of scattering-induced baseline offsets in individual spectra, as can be seen in Fig. 6. Examining spectra from the respective images shows that, indeed, EMSC successfully reduces the effect of scattering as seen in spectra gathered from samples with pronounced cylinder shapes in both cases. The question of precisely recovering band shapes is more complex and there are several confounding factors. First, the downside of wide and easy applicability to entire images is that pixels may be addressed that do not capture the entire scattering profile of a fiber. Second, in complex samples of the type seen in the two examples here, multiple scattering and focusing may also play a role that cannot be captured by EMSC. It should be recognized that there are no methods available to entirely correct spectra from arbitrary number of these effects. We note that the success of our implementation does include a mitigation of scattering induced baseline variations and accurate appearing spectra.
Figure 6.
Recorded (top) and corrected (bottom) absorbance for fungi (left) and cellulose (right) samples. Spectra are recorded from 30 pixels containing samples with a pronounced cylinder-like shape.

We compared our corrected spectra to those reported in other studies, though we caution that comparing single fibers is much easier than comparing spectra from complicated mixtures of cylindrical objects at varying density in an imaging plane, as attempted here. In general, the major spectra line shapes and peak positions agree well with typical spectra previously reported for both cases. For example, cellulose spectra for cotton fluff reported in the study by Zhbankov⁷³ show prominent features around 2950 cm^–1 and 3300 cm^–1, which are also observed in Fig. 6b and 6d. However, we note that the peak at 3300 cm^–1 for corrected spectra is sharper than that reported in the literature. While the scattering correction works well in removing baseline variations, the strongly absorbing region of the high wavenumber region does not agree as well. This observation is consistent with our synthetic data in which strongly absorbing regions of the PMMA spectrum were not easily corrected. Moreover, the same trend for peaks at the $ν > 3000 {cm}^{- 1}$ spectral region has already been observed in Bassan et al.⁵⁰ for EMSC of biological samples. The primary effect of baseline offset correction and the consistency of spectra from different parts of the sample is highly encouraging as there is no reason to expect any variation in the actual absorbance of individual spectra. The result underscores the major benefit of the EMSC algorithm and its application here in that the method corrects spectra from heterogeneous samples which is otherwise not possible. Figure 6c shows that our corrected fungi spectra also agree well with the comprehensive data recently reported in Kosa et al.⁷⁰ in broad terms. One could clearly observe the spectral features due to the presence of lipids (around 2900 cm^–1 and 1400 cm^–1), proteins (around 1600 cm^–1), sugars, and phosphates (around 1200–1100 cm^–1). Once again, however, precise comparisons between each filament’s true spectrum and the corrected one is difficult to verify in these samples. There is more variation in the corrected spectra of fungi as opposed to that in cellulose fibers. This might be expected from biological variations in composition and shape as there seems to be little other reason why the corrections would perform any differently from the cellulose case. The limitations in both these cases may arise from the limited extent of scattered light captured by the optical system at every pixel, multiple scattering from overlapped fibrous material, preparation or biologic variation, and variations in geometric parameters – all of which are to be expected in complex samples. Finally, the samples themselves are highly heterogeneous – from single fibers to almost a bulk-like composition. Despite these variations and complications, the utility of the EMSC algorithm in application is a major benefit while consistency of spectra and elimination of baseline offsets due to scattering are remarkable.

It is noteworthy that recently developed improvements (in terms of stability, convergence, and overall performance) of EMSC for spherical samples might be also applied for fibers, e.g., characterizing the sample by the effective optical path length instead of its radius⁵³ or optimizing the number of iterations and principal components used in Eq. 1.⁵⁴ These strategies require comprehensive benchmarks which are beyond the scope of this paper. Together, these advances and the rigorous basis of extension to fibers proposed here can greatly accelerate the development of better algorithms as well as further testing and use for IR imaging of heterogeneous samples containing cylindrical domains.

Conclusion

In this study, we have proposed a modified version of the extended multiplicative signal correction procedure for IR spectroscopy that specifically addresses cylindrical-shaped samples and domains in heterogeneous materials. In order to utilize rigorous Mie theory for calculation of the scattering efficiency of the infinite cylinder further, we have proposed to change the usual workflow of the EMSC algorithm. Namely, instead of estimating the large number of permutations of $Q_{sca}$ at each iteration step for each recorded curve, we propose calculating the set of possible $Q_{sca}$ before launching the EMSC procedure. Interestingly, this approach arises from rigorous theoretical analysis and allows us to decrease the number of variables entering the calculation of $Q_{sca}$ : from the usually utilized three (wavenumber ν, radius r, and refractive index n_p) to two (size parameter x and refractive index n_p), which further reduces the computational complexity of EMSC. We tested our method with synthetic data formulated from a rigorous forward model of fibers. The correction is shown to work well for most regions of the spectrum but less than optimal from a rigorous analytical solution of a perfectly shaped and centered (in the microscope) cylinder. Finally, we demonstrated the application of the modified and optimized EMSC algorithm to data recorded from polymers and fungi. The pronounced effects arising from scattering were successfully removed. Mitigating the effects of scattering from recorded absorption spectra improves the reliability of the analysis from fiber-shaped objects or cylindrical domains in samples and leads to improved analytical capability that is generally applicable in different applications spanning materials science, forensics, and biological domains.

Footnotes

Conflict of Interest

The authors report there are no conflicts of interest.

Funding

This work was supported in part by the Agilent thought leader award to RB.

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