1. Introduction

Non-invasive spectroscopic methods such as Fourier transformed infrared (FTIR) spectroscopy are powerful techniques that enable the detailed and distinct characterization of a variety of chemical and biological specimens. Hardware implementations such as the focal-plane-array (FPA) technique [1,2] can considerably reduce data acquisition times in Fourier transformed approaches and allow a faster data throughput than conventional single-point measurements. However, the long wavelengths of infrared light result in limited spatial resolution due to the diffraction limit which may also cause blending of spectral lines. In contrast, modern scanning-based methods, such as photothermal-induced imaging (PTIR) [3], scattering-type scanning near-field optical microscopy (s-SNOM) [4], photoinduced force microscopy (PiFM) [5] and tip-enhanced Raman spectroscopy (TERS) [6] provide higher spatial resolution than standard IR spectroscopy by circumventing the diffraction limit. Scanning-based methods, which significantly reduce the blending of spectral lines from phase mixtures, have been shown to improve the application of clustering; in certain cases, clustering cannot be applied without scanning-based methods [7]. As infrared signatures of biological samples have complex but quite similar characteristics, the in-depth analysis of these signatures requires a representative ensemble consisting of a large set of spectral data. However, scanning-based imaging requires a serial recording of the spectra, which severely restricts the size of the dataset compared to methods using array detectors. The often limited spectral irradiance of the IR source available in scanning installations also demands additional data acquisition time to reach an adequate signal-to-noise ratio at each image pixel. Moreover, the increased acquisition time in scanning-based methods leads to stability issues regarding the instrumental setup and the sample itself. This inherent trade-off between imaging quality and data acquisition time challenges the practicability of scanning methods and raised considerable interest in mathematical and statistical techniques for reduced sampling. Examples of such techniques include the use of discrete frequency infrared (DFIR) imaging, where only individual and sparse wavelengths are collected by using a tunable quantum cascade laser (QCL) source [8,9]. FTIR spectroscopy using s-SNOM (nano-FTIR) and asymmetric interferogram analysis supported by pattern recognition techniques is also examined in [10]. Another reduced sampling approach has been proposed by applying a rotating-frame transformation [11]. This technique has been shown to increase the compression rate the smaller the spectral window size containing all features and the broader these features are. The nano-FTIR approach developed in [12] combines interferometric bandpass sampling and interferogram recording in the form of a single synthetic hologram to reconstruct the full spectroscopic data. Alternatively, compressed sensing tools [13] can be exploited for the rapid acquisition and reconstruction of signals that have a sparse representation with respect to some basis. Such reduced signal sources can be compressed because of their redundancy without substantial loss of information. The imposed sparsity allows for a significant reduction of the sampling rate, i.e. in the sub-Nyquist regime, for efficient data acquisition, while the information content of the data is retained. Compressed sensing has previously been applied to FTIR [14] spectroscopy and nano-FTIR [15] hyperspectral imaging.

In this work, an alternative compressed scanning-based FTIR spectroscopy approach is presented to cope with possible non-sparse spectra containing highly resolved and widely distributed features in the Fourier representation. In particular, a low-rank matrix recovery [16,17] is applied that couples the spatial domain and the interferometer axis by reconstructing the most significant factorized basis elements. An additional incorporation of spatial smoothness using a Tikhonov regularization enhances the treatment of the inverse problem and an L-curve criterion adapts the additional parameter.

We demonstrate this method using experimental data previously measured using an FPA detector for a pilot study of Leishmania strains [18]. The datacube acquired in parallel is reduced by a factor of $20$ by randomly selecting sample points along the interferometer axis and the focal image plane, simulating the use of scanning-type approaches. Given this $5\%$ of data, the signal is adequately recovered using the reconstruction procedure and compared to the original interferometer data. Subsequently, a cluster analysis is applied in the context of bioanalytical fingerprinting to demonstrate a possible area of application for our compressed FTIR spectroscopy approach. For this purpose, principal component analysis (PCA) is implemented, thus enabling differentiation of biological specimens [19]. Similar procedures are applied for instance in the context of spectromicroscopic imaging of tumor tissues [20,21].

The proof-of-principle study in this article shows that acquisition times in scanning-based FTIR applications can be significantly reduced, paving the way for the high-resolution measurement of a broad class of chemical mappings that is not limited by assumptions on sparsity or simplicity of feature distribution. The developed and applied compressed FTIR approach is, to the best of our knowledge, the first compression and reconstruction method for non-sparse spectra admitting a broad-band background and multi-peaks encountered for instance in biological systems. An open-source Python package allows for the low-rank reconstruction approach described above to be adapted to various applications [22].

The remainder of this article is structured as follows. In Section 2, the data acquisition and numerical low-rank approximation methods used in this study are presented. Section 3 demonstrates the reconstruction quality, first in terms of the interferometer data, then by showing stability under the computation of absorbance and finally by providing valuable cluster information in a bioanalytical fingerprinting application. Finally, Section 4 gives some conclusions and an outlook to potential future research.

2. Methods

2.1 Data

The data from a previous pilot study of Leishmania strains [18] are used for a proof-of-principle analysis of the proposed low-rank matrix reconstruction procedure. The corresponding FTIR microspectroscopical datasets were obtained from four different strains of the humanpathogenic species Leishmania infantum (’IMT151’), L.donovani (’BD09’), L.tropica (’LCR-L830’), and L. tarentolae (’Ltar’) as non-pathogenic species.

FTIR absorbance spectra, $A(\tilde {\nu })$, have been acquired in transflection mode where the beam is transmitted through the Leishmania films and reflected from the specimen slide. The collected spectrum approximates the transmission spectrum $T(\tilde {\nu })$ as a function of the wavenumber $\tilde {\nu }$ of the films, while the features in the spectrum are the considered absorption bands. The measurement was performed on a Vertex 80v FTIR spectrometer (Bruker Optics GmbH) to which an FTIR Hyperion 3000 microscope was coupled. Under standard measurement conditions, the spectrum is calculated by the instrument software from the detected intensity $I^{\mathrm{raw}}(t)$ as a function of the interferometer path difference $t$. Here, we will utilize the interferometer trace $I^{\mathrm{raw}}(t)$ directly, since the compression has to be performed before the Fourier transform. The corresponding interferogram trace is recorded for different locations $\vec {r}=(x,y)$ on the film for which $I^{\mathrm{raw}}_{x,y}(t)$ is measured in parallel using a FPA detector with 128$\times$128 pixel elements. The illumination of the sample films was conducted with a Cassegrain objective at a 15x magnification, where the size of 2.9 $\mu$m on the sample corresponds to the dimension of a single pixel.

Data acquisition was performed to obtain discrete digitized interferograms $I^{\mathrm{raw}}(t)$ with $N_t = 3554$ equidistant points ($\Delta t \approx 1.27 \mu$m). The sensitivities vary between the different FPA-elements, for which all pixels have been normalized by the median of the interferogram $I_{x,y}(t)$ at each pixel individually.

2.2 Low-rank reconstruction

The concept of low-rankness relies on the existence of a factorization that approximates the dataset by a sum of only a few rank-one matrices. Let $X_{ij}, i=1, \ldots ,N_{\vec {r}}, j=1, \ldots , N_t$, denote the sought matrix of measurements, where the rows are the spatial position of the measurements and the columns are the interferometer position. Assuming that measurements of $X$ are taken only at a subset $\Omega \subset \{1, \ldots , N_{\vec {r}}\}\times \{1,\ldots , N_t \}$, one can define the available reduced data matrix $X_\Omega$ comprising the subsampled measurements and zeros otherwise. Since a matrix factorization approach is pursued in the following, the actual reconstruction of the whole matrix $X$ can now be posed as an inverse problem which reads: given $X_\Omega$ and some desired rank $r\in \mathbb {N}$, find $U\in \mathbb {R}^{N_{\vec {r}}, r}$ and $V\in \mathbb {R}^{N_t, r}$ such that

Numerically, an alternating algorithm [27] is employed to estimate $U$ and $V$ iteratively together with an L-curve criterion [28] to select the regularization parameter $\lambda >0$; see Algorithm 1 for details on the implementation.

Note that, due to the alternating optimization, the update steps

3. Results and discussion

We applied the proposed low-rank approach to parts of the data from a previous pilot study of Leishmania [18]. Specifically, $5\%$ of the original data is selected by uniformly sampling each datacube in the focal plane $(x, y)$ and in the interferometer domain $t$. Application of the low-rank matrix reconstruction then yields an estimate of the whole dataset. Since the complete dataset is known, the potential of our proposed approach can be assessed. In doing so, we consider in particular the application to bioanalytical fingerprinting.

3.1 Low-rank reconstruced interferometer data and imaging

Applying Algorithm 1 to each of the subsampled Leishmania strains yields an approximation to the original full data that, due to the availability of the original full dataset, can be assessed in terms of its reconstruction quality. In Fig. 1, the sample interferogram of the L. tarentolae sample film taken from pixel position (50, 50) is highlighted in a). The center burst region is magnified in c) to allow the interferogram’s courses between the original (blue) and reconstructed (red) intensities to be followed. Additionally, the $5\%$ sub-sampled data utilized is shown for the corresponding pixel in the left column of Fig. 1 as an example. Note that, even though the center burst region is not observed directly, an adequate reconstruction of the signal is achieved. Comparing the original interferogram data for $t=1715 \Delta t$ in b) with its reconstructed counterpart in d), we notice a successful reconstruction, which can be quantified by the nearly identical pixel intensities and the adequately reproduced structural patterns of the sample film. The expected smoothing effect of a low-rank approximation that removes small random noise [31], and thereby making certain features more apparent is clearly observed in our experiments outside the center burst region. In the shown pixel domain at the center burst, a removal of systematic stripes can be observed in the upper part of the image. The scattered artifacts observed can be explained by the relatively small interferometer position region around $1680\Delta t$ to $1720\Delta t$, in which the intensity changes drastically. This results in an under-determined region when taking samples uniformly from the datacube. An adaptive or data-driven sampling strategy can improve this result, which is a subject to future research. For the examples considered, a maximum rank of $20$, a convergence tolerance $\tau =0.01$ and $20$ log-equidistant regularization nodes in $[10^{-4}, 10^{-1}]$ were chosen. The resulting accuracy is measured in terms of a local and global convergence criteria (cf. Algorithm 1, $\mathrm {res}_L$ and $\mathrm {res}_G$).

 

Fig. 1. Sample interferogram of the L. tarentolae film is shown for pixel (50, 50) in a) and the corresponding spatial image of interferometer position $1715\Delta t$ in b). The interferometer domain around the center burst is magnified in c) and the pixel domain of interest that is subsequently analyzed in the bioanalytical fingerprinting application is shown as a red rectangle in the pixel domain reconstruction in d).

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In addition to the results shown for $5\%$ sub-sampled data of the L. tarentolae film, Table 1 contains results obtained for other compression rates. More precisely, the root mean squared error of reconstructions obtained for $1\%, 5\%, 15\%$ and $50\%$ of sub-sampled data are given. It can be observed that taking additional data into account results in better approximations.

Table 1. Comparison of reconstruction results for varying amounts of available data in terms of the root mean squared error (RMSE).

View Table

3.2 Univariate consideration of low-rank reconstructed mid-IR signatures

The low-rank approximation was performed directly on the interferometer data. However, the relevant quantity of the spectroscopic analysis (e.g., bioanalytical fingerprinting) is based on the absorbance, $A(\tilde {\nu })$. Therefore, the quality of the reconstruction is evaluated for the absorbance which is obtained via the following procedure. First, a measurement is performed on the sample substrate without any sample (’background spectrum’); then, a second measurement is conducted on the slide with the film (’sample spectrum’). In the following, the data acquired is referred to as film data and background data. In order to remain close to a potential compressed measurement situation the low-rank approximation has been carried out on both measurements individually. The absorbance is calculated as

Prior to the Fourier transform, the interferogram data is processed by methods commonly used in FTIR Spectroscopy [32]. For apodization, the Blackmann-Harris-3-Term filter was applied. Zero-filling was performed using a factor of two, resulting in a spectral resolution of $\approx 1.1\,\mathrm {cm}^{-1}$.

Figure 2 displays the original mid-infrared signatures of L. tarentolae (’Ltar’) given in blue together with the low-rank reconstructed fingerprints (red). Apart from slight intensity offsets, the original spectra are most widely congruent to the reconstructed spectra. The low-rank approximation is even capable of modeling the spectral background and correctly reflects the band shapes and positions, as well as their full-width-at-half-maximum (FWHM) to a great extent. A high conformity of band ratios is also observed.

 

Fig. 2. Spectra of a parasitic film containing L. tarentolae (’Ltar’) recorded at four points randomly selected from the $128 \times 128$ matrix. Spectral windows used for the PCA-based approach are highlighted in grey and correspond to $3000-2700$ cm$^{-1}$ and $1800-1500$ cm$^{-1}$, respectively.

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The selected pixels in Fig. 2 show the successful reconstruction of spectra that vary between the positions. The quality of the reconstruction differs between the spectra. For instance, a higher noise level of the reconstructed absorbance at pixel (114, 110) can be observed. We relate this observation to be the result of the following possible contributions: (i) a higher noise level of the background single channel interferogram and (ii) possible dirt or defect in the substrate. In general, an agreement of the computed absorbance data, which is a non-linear transformation of the intensity, can be assessed.

3.3 PCA on reconstructed datacubes of parasitic samples

The data selected in this work originally aim at bioanalytical fingerprinting employing PCA clustering. In the systematic pilot study of [18], PCA allowed a distinct identification and discrimination by unique mid-IR spectral fingerprints of Leishmania strains at the respective wavenumber windows, enabling successful segregation between information-rich and information-poor spectral components. With this goal in mind the reconstruction of the dataset of this pilot study is also evaluated by comparing the PCA clustering obtained from the full and the reconstructed datasets.

PCA is a standard tool in statistical data science when dealing with large datasets and is closely related to concepts such as proper orthogonal decomposition (POD), singular value decomposition (SVD) and the discrete Karhunen-Loève transform [33–35]. The general aim is to compute a linear transformation to a system of uncorrelated variables, such that most of the data variance is explained in decreasing order by the first dominant principal components (cf. Appendix A).

In preparation for the PCA, the acquired data is further processed in order to achieve the highest differentiation capability among the respective parasitic datasets in accordance with the procedure in [18]. To this end, the absorbance 1 of every measurement is smoothed along the spectral axis using the Savitzky-Golay algorithm with a five-point window and second-order polynomials. Additionally, the $2$$^{nd}$ derivative is derived to reveal curvature information that improves separability. Subsequently, in the spectral ranges $3000-2700$ cm$^{-1}$ and $1800-1500$ cm$^{-1}$, as shown in Fig. 2, 540 samples are selected in total. The spectra are chosen from 420 spatially connected pixels. Figure 1 shows the area chosen for the L. tarentolae film as an example, indicated by a red box.

The reduced spatial and spectral window effectively reduces each dataset to a $420$ by $540$ data matrix by vectorizing the $(x, y)$ coordinates.

In the following, we consider the four parasitic samples that comprise $4 \times 420$ fingerprints, perform a PCA on the low-rank reconstructed datacube and discuss the similarities of the resulting projections onto the variance-weighted space in terms of PCA scores.

The scatter plots shown in Fig. 3 illustrate the resulting PCA scores derived from the full dataset (column 1) and from the low-rank reconstructed data (column 2) projected onto the principal components obtained by applying PCA to the reconstructed data. The similarity of the results validates the accuracy of the reconstruction employed as well as the stability during data processing, i.e. equivalent cluster regions can be observed for both cases, even though the compressed FTIR approach takes only $5\%$ of the available data into account. We observe slightly larger and more scattered cluster regions for the reconstructed data, as can be seen for instance for IMT151 in the “low-rank” column of Fig. 3 compared to the original “full” column. The cluster orientations and positions coincide for the dominant components (PC1 vs. PC2) and also for less influencing coordinates (PC2 vs. PC3). For the sake of a complete presentation, the Table contained in Fig. 3 gives the percentage of data variance included in the first four principal components when PCA is applied to the reconstructed dataset. The resulting PCA loadings are also given in Fig. 3. We like to point out that a thorough cluster analysis is left out in this proof of concepts study. For a quantitative comparison of the resulting regions it is advisable to employ additional cluster detection algorithms, such as k-means or density based clustering. We refer to [36] for an overview.

 

Fig. 3. Score scatter plots of the principal components (PC1 to PC3) for the original dataset (column 1) and for the low-rank reconstructed dataset (column 2) as the data is projected onto the coordinate system obtained from applying PCA to the reconstructed dataset. Column 3 and column 4 represent the first three computed loadings of the PCA for the selected spectral windows and the table (lower right) gives the percentage of variance captured by the first four principal components.

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4. Conclusion

A low-rank matrix reconstruction approach has been developed for compressed scanning-based FTIR spectroscopy. The approach was explored in terms of its application to artificially reduced data from a previous pilot study of Leishmania strains. Specifically, reconstruction of the whole dataset was shown to be possible when randomly selecting only 5% of the data. Furthermore, derived results such as bioanalytical fingerprinting could also be successfully carried out from the reduced dataset. We conclude that the resources required for measurements in scanning-based FTIR spectroscopy can be significantly reduced by means of the proposed approach. This can turn these methods into a practical alternative to FTIR spectroscopy and imaging methods, such as spectromicroscopy approaches.

Future research could extend the scope of this proof-of-principle study by applying the proposed approach to other scenarios in FTIR spectroscopy or spectromicroscopy. Furthermore, alternative smoothness and regularity assumptions could be explored in connection with the proposed approach, as well as data-driven sampling strategies and more elaborate ways to choose the rank parameter.

A. Principal components analysis

Mathematically, the $n$ by $p$ data matrix $X$ with $n$ denoting the observations and $p$ representing the features, i.e. the considered absorbance spectra from Section 2 is analyzed by performing an eigendecomposition of the covariance matrix

$$ X^{T} X = U \Sigma U^{T}, \quad \textrm{with}\quad U, \Sigma\in\mathbb{R}^{p, p}. $$

The component $U$, containing the eigenvectors of $X^{T} X$, is orthogonal, as the covariance is a real symmetric matrix and the diagonal matrix $\Sigma$ comprises the singular values of $X$, which are assumed to be in decreasing order. If there is an $r_{\textrm {PCA}}\ll p$, such that all singular values $\Sigma _{k, k}$ with $k>r_{\textrm {PCA}}$ are negligible, PCA can be employed as a model reduction tool. Projecting the original data onto the eigenbasis spanned by the first $r_{\textrm {PCA}}$ columns of $U$ yields the so-called PCA scores

$$ S_{i, j} = \sum_{k=1}^{p} X_{i, k}U_{k, j} \quad \textrm{for}\quad i=1,\ldots, n, j=1, \ldots r_{\textrm{PCA}}, $$

as visualized in Fig. 3. These PCA scores can be used for identifying individual characteristics of underlying dataset and cluster analysis.

Besides the known limitations of defining only an orthogonal transformation [37], the simplicity of the algorithm in contrast to more elaborated clustering algorithms renders it the method of choice in this article.

B. Visualization of reconstruction quality

Beside the already mentioned comparisons of the original dataset and the reconstructed low-rank approximation, we provide a visual course through the datacube. For this, a video is provided as visualization that compares the dataset L. tarentolae for the selected range $t=1600\Delta t, \ldots , 1800\Delta t$ with its low-rank reconstruction using $5\%$ of sub-sampled data. Figure 4 shows a snapshot of the video.

 

Fig. 4. Performance of the reconstruction visualized by a video that sweeps through the interferometer axis (see Visualization 1). Snapshot taken at $t=1639\Delta t$.

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Funding

European Metrology Programme for Innovation and Research (18HLT06RaCHy); Deutsche Forschungsgemeinschaft (EL 492/1-1, RU 420/13-1).

Acknowledgments

We thank Ferenc Borondics, Samuel Johnson and Markus Raschke for fruitful discussions.

Disclosures

The authors declare no conflicts of interest.

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