Spectrum data reconstruction method and apparatus for reconstructing spectrally resolved spectrum data

Abstract

A spectrum data reconstruction method for reconstructing spectrally resolved spectrum data (S) of a measuring light field from sensor data (Fτ) provided by an interferometric autocorrelation measurement of interferogram data created by pairwise superimposing delayed versions of the measuring light field, said delayed versions of the measuring light field having different mutual delays (τ), comprises the steps of providing a spectral system matrix (Rτω), wherein the sensor data (Fτ) are determined by applying the spectral system matrix (Rτω) to the spectrum data (Sω) to be obtained, said spectral system matrix (Rτω) being created in dependency on the delays (τ) and instrument parameters of an spectrometer apparatus employed for the interferometric autocorrelation measurement, providing an initial prior estimate (µ S prior ) and an initial prior covariance (Σ S prior ) of the spectrum data (S), calculating a mean estimate (µ S ) and a covariance (Σ S ) of the spectrum data (Sω) by applying a Bayesian inference computation employing the sensor data (Fτ), the spectral system matrix (Rτω), the initial prior estimate (µ S prior ) the initial prior covariance (Σ S prior ) and the initial prior covariance (Σ S prior ), and output of the mean estimate (µs) and optionally the covariance (Σ S ) of the spectrum data (Sω) as the spectrum data (S) to be reconstructed. Furthermore, an interferometric autocorrelation measurement method, a spectrum data reconstruction apparatus and an interferometric autocorrelation measurement apparatus are described.

Description

[0001] Spectrum data reconstruction method and apparatus for reconstructing spectrally resolved spectrum data Field of the inventionThe invention relates to a spectrum data reconstruction method for reconstructing spectrally re-solved spectrum data of a measuring light field, an interferometric autocorrelation measurement method including the spectrum data reconstruction method, a spectrum data reconstruction ap-paratus for reconstructing spectrally resolved spectrum data of a measuring light field, and an in-terferometric autocorrelation measurement apparatus including the spectrum data reconstruc-tion apparatus. Applications of the invention are available e.g., in the field of spectroscopy, e.g.,Fourier Transform Spectroscopy. Technical background In the present specification, reference is made to the following prior art illustrating technical background of the invention and related techniques:[1] P. Jacquinot. New developments in interference spectroscopy. Reports on Progress in Phys-ics, 23(1):267, 1960;[2] Claude E. Shannon. Communication in the presence of noise. Proceedings of the IRE,37(1):10–21, 1949;[2] P. Fellgett. The theory of infrared sensitivities and its application to investigations of stellarradiation in the near infra-red. Ph. D. thesis, Univ. of Cambridge, 1958;[4] P. Jacquinot. New developments in interference spectroscopy. Reports on Progress in Phys-ics, 23(1):267, 1954;[5] Peter R. Griffiths et al. Fourier transform infrared spectrometry. John Wiley & Sons, 2007;[6] Siegfried Wartewig et al. Ir and raman spectroscopy: fundamental processing. Advanceddrug delivery reviews, 57(8):1144–1170, 2005;[7] David WT Griffith et al. Ftir remote sensing of biomass burning emissions of co2, co, ch4,ch2o, no, no2, nh3, and n2o. Global biomass burning: Atmospheric, climatic, and biospheric implications, pages 230–239, 2012;[8] Matthew J Baker et al. Using fourier transform ir spectroscopy to analyze biological materi-als. Nature protocols, 9(8):1771–1791, 2014;[9] David A Naylor et al. Interpolation of aliased spectra. Optics express, 15(20):12941–12957,2007;

[0010] Edwin T Jaynes et al. Probability theory: The logic of science. Cambridge university press,2003;

[0011] Devinderjit Sivia et al. Data analysis: a Bayesian tutorial. OUP Oxford, 2006;

[0012] Phil Gregory. Bayesian logical data analysis for the physical sciences: a comparative ap-proach with Mathematica® support. Cambridge University Press, 2005; and

[0013] Stefan Schmuck et al. Fourier spectroscopy: A Bayesian way. International Journal of Spec-troscopy, 2017:1–29, 2017.It is generally known that Fourier Transform Spectroscopy (FTS) is a well-established technique forobtaining high-resolution spectra across a wide range of the electromagnetic spectrum. The core principle of FTS is the measurement of the interference pattern (interferogram) produced by com- bining a light beam with a delayed version of itself. This interference pattern is then Fourier trans-formed to obtain the spectrum of the light source, as described in the seminal work by Jacquinot(1960) [1]. Traditional FTS relies on the measurement of light intensity as a function of optical path difference (delay), with uniform sampling of the interferogram at intervals determined bythe Nyquist-Shannon sampling theorem [2]. The spectrum is then obtained by applying the Fou-rier transform to convert the time-domain interferogram to a frequency-domain spectrum. FTS has found widespread application in infrared spectroscopy, where it is commonly known as Fourier Transform Infrared Spectroscopy (FTIR). The popularity of FTIR is largely due to two key advantages. The first is Fellgett’s advantage, also known as the multiplex advantage, which arises from the fact that all wavelengths are measured simultaneously, leading to a significant improve- ment in signal-to-noise ratio compared to scanning dispersive techniques [3]. The second is Jacquinot’s advantage, or the throughput advantage, which allows FTS instruments to achieve much higher optical throughput than dispersive spectrometers, as they don’t require narrow slits for high resolution [4]. These advantages have made FTIR a powerful tool in a wide range of applications. In chemistry and materials science, FTIR is used for molecular structure determination and polymer characteri- zation [5]. The pharmaceutical industry employs FTIR for quality control and counterfeit detection [6]. Environmental scientists use FTIR for monitoring atmospheric pollutants [7], while in the bio- medical field, it has applications in diagnostics and tissue imaging [8]. Fourier Transform Spectroscopy principles have been extended to enable imaging spectroscopy, contributing significantly to hyperspectral imaging. In imaging FTS, a separate FTS measurement is performed for each pixel on a camera sensor. This is typically achieved by coupling an imaging sys- tem to an interferometer. As the optical path difference is varied, a series of images is captured. The intensity variations of each pixel across this series form an interferogram, which is then Fou- rier transformed to obtain a spectrum for each spatial point.This approach results in a three-dimensional data cube (^, ^, ^) where ^ and ^ are spatial coordi-nates and ^ represents wavelength. Imaging FTS maintains the high spectral resolution character-istic of FTS while providing spatial information. It has found applications in remote sensing for at- mospheric studies and Earth observation, biomedical imaging for tissue analysis, and industrial quality control for mapping chemical compositions. The technique’s flexibility in spectral range and resolution makes it adaptable to various scenarios, from broad spectral surveys to detailed studies of narrow spectral features. While highly effective, the traditional FTS approach has limitations. It doesn’t easily incorporate prior knowledge about the spectrum, requires uniform sampling, and provides no information about the uncertainty in the spectral estimate. These limitations have been recognized in the lit- erature, with efforts to address them through various computational techniques [9]. Overcoming these limitations could significantly enhance the performance of FTS in all the aforementioned ap- plications, potentially enabling faster measurements, higher resolution, or improved performance in challenging measurement conditions. Bayesian inference is a generally known statistical method that updates the probability of a hy- pothesis as more evidence becomes available. Bayesian inference is based on Bayes’ theorem, which relates the conditional and marginal probabilities of random events

[0010] . Prior approaches to spectral analysis using Bayesian methods have been limited in scope, e.g. infering single fre-quencies of periodic signals

[0012] . More extensive modeling of spectral parameters was done bySchmuck and Svensson (2007)

[0013] , but was focused on finding the best hyperparameters for a specific type of interferometer realization. Most importantly they relied on specialized assump- tions not generalizable to broader spectroscopic applications, like a Brownian bridge prior, whosestructure does not harness the effectiveness of prior knowledge incorporation within the Bayes-ian framework. Objective of the invention It is an objective of the invention to provide an improved spectrum data reconstruction method for reconstructing spectrally resolved spectrum data of a measuring light field, interferometric au-tocorrelation measurement method, spectrum data reconstruction apparatus for reconstructingspectrally resolved spectrum data of a measuring light field and / or interferometric autocorrela- tion measurement apparatus, being capable of avoiding limitations and disadvantages of conven-tional techniques. In particular, reconstructing spectrally resolved spectrum data of a measuringlight field is to be improved in terms of reconstruction precision, reconstruction speed, recon- struction resolution, capability of introducing prior knowledge about the spectrum data, allowing uniform or non-uniform sampling, and / or providing information about the uncertainty in the spectral estimate. Summary of the inventionThis objective is solved by a spectrum data reconstruction method for reconstructing spectrallyresolved spectrum data of a measuring light field, an interferometric autocorrelation measure-ment method, a spectrum data reconstruction apparatus for reconstructing spectrally resolvedspectrum data of a measuring light field and / or an interferometric autocorrelation measurement apparatus comprising the features of the independent claims. Advantageous embodiments and applications of the invention are defined in the dependent claims. According to a first general aspect of the invention, the above objective is solved by a spectrum data reconstruction method for reconstructing spectrally resolved spectrum data (S) (in particular: computing a spectral estimate) of a measuring light field from sensor data (Fτ) provided by an in- terferometric autocorrelation measurement of interferogram data (interferograms) each being created by pairwise superimposing delayed versions of the measuring light field, said delayed ver-sions of the measuring light field having different mutual delays (τ). The term spectrum data re-fers to a data set representing a spectrum, e.g., an absorption, transmission and / or emission radi-ation spectrum, of a sample to be investigated. The measuring light field is a light field created byan interaction of the sample with measuring light.According to the invention, a spectral system matrix (Rτω) is provided, wherein the sensor data (Fτ)are determined by applying the spectral system matrix (Rτω) to the spectrum data (Sω) to be ob-tained, said spectral system matrix (Rτω) being created in dependency on the delays (τ) and instru-ment parameters of a spectrometer apparatus employed for the interferometric autocorrelationmeasurement. Furthermore, according to the invention, an initial prior estimate ^µ^^^^^^ ^ and an initial prior co-varianceof the spectrum data (S) are provided. This prior information may be providede.g., on the basis of earlier comparable reference measurements, numerical simulations and / or by initial default estimates.Furthermore, according to the invention, a mean estimate (µ^) and a covariance (Σ^ ) of thespectrum data (Sω) are calculated by applying a Bayesian inference computation employing the sensor data (Fτ), the spectral system matrix (Rτω), the initial prior estimate ^µ^^^^^^ ^ the initial priorcovarianceand the initial prior covariance and the mean estimate (µ^) and op-tionally the covariance (Σ^ ) of the spectrum data (Sω) are output as the spectrum data (S) to bereconstructed. Output of both of the mean estimate (µ^) and the covariance (Σ^ ) may be pre-ferred for providing all available informa^on on the measurement. Alterna^vely, output of themean estimate (µ^) may be sufficient fir characterizing the sample to be investigated.According to a second general aspect of the invention, the above objective is solved by an inter- ferometric autocorrelation measurement method, comprising the steps of providing a measuring light field, measuring sensor data (Fτ) by an interferometric autocorrelation measurement of in- terferogram data created by pairwise superimposing delayed versions of the measuring light field, said delayed versions of the light field having different mutual delays (τ), and applying the spec-trum data reconstruction method according to the first general aspect of the invention, or an em-bodiment thereof, to the sensor data (Fτ). According to a third general aspect of the invention, the above objective is solved by a spectrum data reconstruction apparatus being configured for reconstructing spectrally resolved spectrum data (S) of a measuring light field from sensor data (Fτ) provided by an interferometric autocorre- lation measurement of interferogram data created by pairwise superimposing delayed versions of the measuring light field, said delayed versions of the measuring light field having different mu- tual delays (τ).According to the invention, the spectrum data reconstruction apparatus includes a data pro-cessing device being configured for providing a spectral system matrix (Rτω), wherein the sensor data (Fτ) are determined by applying the spectral system matrix (Rτω) to the spectrum data (Sω) tobe obtained, said spectral system matrix (Rτω) being created in dependency on the delays (τ) andinstrument parameters of an spectrometer apparatus employed for the interferometric autocor- relation measurement, providing an initial prior estimate ^µ^^^^^^ ^ and an initial prior covarianceof the spectrum data (S), calculating a mean estimate (µ^) and a covariance (Σ^ ) of thespectrum data (Sω) by applying a Bayesian inference computation employing the sensor data (Fτ), the spectral system matrix (Rτω), the initial prior estimate ^µ^^^^^^ ^ the initial prior covarianceand the initial prior covariance and output of the mean estimate (µ^) of thespectrum data (Sω) and optionally the covariance (Σ^ ) as the spectrum data (S) to be recon-structed. Preferably, the spectrum data reconstruction apparatus may be configured for executing the spectrum data reconstruction method according to the first general aspect of the invention or an embodiment thereof.According to a fourth general aspect of the invention, the above objective is solved by an interfer-ometric autocorrelation measurement apparatus being configured for measuring spectrally re-solved spectrum data (S) of a measuring light field, comprising a measuring device including a sen-sor device being arranged for providing sensor data (Fτ) provided by an interferometric autocorre- lation measurement of interferogram data created by pairwise superimposing delayed versions of the measuring light field, said delayed versions of the measuring light field having different mu- tual delays (τ), and the spectrum data reconstruction apparatus according to the third general as- pect of the invention or an embodiment thereof. Further independent subjects of the invention comprise a computer-implemented device pro- grammed to perform the method according to the first general aspect of the invention or an em- bodiment thereof, and a computer program product which, when loaded into a computer-imple-mented device, executes the method according to the first general aspect of the invention, or anembodiment thereof. The computer-implemented device may be integrated in a measuring de-vice or may be provided as a separate computer unit. Advantageously, the inventors have found a computational method for enhancing efficiency in spectroscopic applications, particularly in the domain of FTS. This method, also termed as Bayes- ian Autocorrelation Spectroscopy (BAS), applies principles of Bayesian inference and information theory to the process of spectral reconstruction from interferometric measurements. With the invention, the above limitations of conventional techniques are addressed by applying Bayesian inference. The inventors have found that the combination of calcula^ng spectrally re- solved image data via solving the inverse problem through applying the regularized inversion (ap- plying the spectral system matrix) with applying Bayesian methods provides mul^ple advantages.In par^cular, the inven^on allows incorporating prior knowledge about the spectrum, handlingnon-uniform sampling naturally, providing quantitative measures of uncertainty in the spectralestimate, and enabling adaptive sampling strategies based on information theory

[0011] . BAS is em-ployed for implementing the FTS process as a sequential Bayesian update (one update step ormultiple update steps), providing a probabilistic framework for spectral estimation. In particular,the inventors, employing Gaussian assumptions for both the prior and likelihood distributions inBayesian inference, have found the formulation of closed-form update equations for the meanand covariance of the spectral estimate, allowing an efficient numerical calculation, in particular wit a step-wise refinement of the spectral estimate. As a further substantial advantage, the invention can be applied to existing FTS setups to improve their performance without requiring hardware modifications. It also provides a foundation for de- veloping new, more efficient spectroscopic instruments by enabling information-theoretic optimi- zation of sampling strategies. The combination of FTS principles with Bayesian inference forms the foundation for this inven- tion, aiming to enhance spectral resolution and measurement efficiency in spectroscopic applica- tions. This approach builds upon recent advancements in Bayesian spectral analysis

[0012] , and ex-tends them to the specific context of Fourier transform spectroscopy. By doing so, it is capable ofextending the boundaries of what is achievable in spectroscopic measurements across a wide range of scientific and industrial applications.Advantageously, the mean estimate (µ^) providing the (most) probable spectrum and the covari-ance (Σ^ ) providing informa^on about the uncertainty and how this uncertainty correlates be-tween different spectral channels of the spectrum data are calculated. The covariance represents an error estimate of the reconstruction, resulting in a substantial advantage over conventional techniques. According to a preferred embodiment of the invention, the spectrum data reconstruction methodmay include further steps of repeatedly calculating an updated mean estimate (µ^) and an up-dated covariance (Σ^ ) of the spectrum data (Sω) by applying the Bayesian inference computation,wherein for each update calculat ( )ion a previously calculated mean estimate µ^ and covariance(Σ ) of the spectrum data (S ) are employed as a current prior estimate ^µ ^^^^^^ ω ^ ^ and a currentprior covariance and after termination of the repeatedly updating steps, output of theupdated mean estimate (µ ) and optionally the updated covariance (Σ ) of the spectrum data^ ^(S ) as the spectrum data (S) to be reconstructed.ωAdvantageously, the closed-form update equations for the mean and covariance of the spectralestimate allow for the continuous refinement of the spectral estimate, e.g., as new measure- ments are taken, incorporating both the new data and the existing uncertainty in the estimate.Termination of the repeatedly updating steps may be set in dependency on particular applicationconditions by a predetermined termination criterion, e.g., by comparing the covariance with a predetermined reference value or testing the number of iterations. According to a particularly preferred embodiment of the invention, the spectrum data reconstruc-tion method may include further steps of calculating multiple values of an information gain forpotential delay values using the calculated covariance (Σ ) of the spectrum data (S ), and outputω^nextof a selected delay value (τ ) which provides a maximum value of the information gain. In terms of the interferometric autocorrelation measurement method of the invention, the multi- ple values of the information gain may be calculated for potential delay values using the calcu-lated covariance (Σ ) of the spectrum data (S ), and the selected delay value (τ ) which pro-ω next^vides the maximum value of the information gain, is employed for measuring subsequent sensor τ data (F). Accordingly, this embodiment of the invention provides a framework for quantifying the infor- mation content of each measurement. This enables an adaptive sampling strategy, where the next measurement delay can be chosen to maximize the expected information gain. Such strate- gies have the advantageous potential to significantly reduce the number of measurements re- quired for a given spectral resolution, thereby improving the efficiency of spectroscopic tech- niques. According to a further advantageous embodiment of the invention, the spectrum data reconstruc-tion method may include a further step of calculating a Fourier transform of the sensor data (Fτ)for obtaining a Fourier estimate ^µ^^^^^^^^ ^ of the spectrum data (Sω), wherein the mean estimate(µ^) and optionally the covariance (Σ^ ) of the spectrum data (Sω) is calculating by applying theBayesian inference computation further employing the Fourier estimate^µ^^^^^^^^ ^. Advanta-geously, the Fourier transform of the sensor data (Fτ) provides an additional prior information im- proving the result of the Bayesian inference computation Advantageously, multiple variants are available for providing the initial prior estimate ^µ^^^^^^^ and the initial prior covariance One or more of the variants may be selected depending on the application conditions of the invention. According to preferred variants, a reference data-base of relevant spectra, data of previously measured spectra, e.g., of similar samples, and / ormeasuring light source characteristics, e.g., for absorption experiments, may be employed as sources of the prior information.According to a further preferred embodiment of the spectrum data reconstruction method, thestep of providing the initial prior estimate ^µ^^^^^^ ^ may include providing multiple differentmodel prior estimates a step of prior model averaging the model prior estimates comprising at least one of calculating weights for each model prior estimate based on a Leave-One-Out Cross-Validation (LOO-CV) score, and calculating weighted average of spectral estimates from the model prior estimates ^µ^^^^^^^^^^^.The invention can be applied for reconstructing spectrally resolved spectrum data of a measuringlight field collected with a conventional spectrometer setup, like a FTS device, in particular includ-ing a single photosensor element. Advantageously, the application of the invention may be ex-tended to imaging measurements. Accordingly, the sensor data (Fτ) may comprise spatially re- solved sensor data measured with multiple sensor elements (pixels) of a two-dimensional photo- sensor, and the spectrally resolved spectrum data (S) may be separately reconstructed for each of the sensor elements.In summary, the present invention is capable of enhancing the capabilities of e.g., Fourier Trans-form Spectroscopy (FTS) through the application of Bayesian inference methods. The inventionallows to incorporate prior knowledge about the spectrum into the reconstruction process, allow-ing for more robust results in challenging measurement conditions, provides quantitative uncer-tainty estimates for spectral reconstructions, enhancing the reliability of spectroscopic measure-ments, enables efficient non-uniform and adaptive sampling strategies to reduce measurementtime and data acquisition requirements, and develops a flexible framework that can be applied toboth existing FTS setups and guide the design of new, more efficient spectroscopic instruments.By these advantages, the invention substantially advances the field of spectroscopy, offering im-proved performance and new capabilities across a wide range of scientific and industrial applica- tions that rely on high-resolution spectral analysis. Brief description of the drawings Further advantages and details of the invention are described in the following with reference to the attached drawings, which schematically show in:Figures 1 to 3: flowcharts illustrating embodiments of quan^fying a probability of the spectrallyresolved image data; andFigure 4: features of embodiments of the interferometric autocorrelation measurementapparatus including the spectrum data reconstruction apparatus of the invention. Detailed description of preferred embodiments of the invention Features of preferred embodiments of the invention are described in the following with particular reference to the data processing and mathematical background thereof. Details of designing the measuring part of the interferometric autocorrelation measurement apparatus for providing the light field sensor data and / or controlling components of the interferometric autocorrelationmeasurement apparatus, like an autocorrelation device (e.g., an interferometer), imaging op^csand / or a photosensor device, are not described as far as they are known per se from conven^onal techniques, e.g., from FTS techniques. As an alterna^ve to employing the inven^on for processing light field sensor data collected with FTIR spectrometry, other measuring principles may be used for providing the light field sensor data, e.g., as described in the European patent applicationEP 24198926.8 ("Optical imaging apparatus and spectral imaging method for creating spectrally and spatially resolved image data", not published on the priority day of the present specification).In par^cular with regard to the collec^on of light field sensor data and the relation between aspectral range of interest and a spectral resolution of interest and the delay and the interval ofdelay variation employed in an autocorrelation measurement, the application EP 24198926.8 isintroduced to the present specification by reference. Calculations are implemented based on dis-cretized sensed data as known per se from conven^onal numerical techniques. Exemplary reference is made to embodiments, wherein a series of updates of the mean es^mateand covariance are provided for itera^vely improving the reconstruc^on result. It is emphasizedthat the inven^on is not restricted to these mul^ple updates embodiment, but rather can be im- plemented with a single update only, e.g., if a par^cularly fast reconstruc^on is intended, the re- sult of the single update reconstruc^on is sufficient and / or the prior match the result quite well. Mathematical formulationIn traditional FTS, the measured interferogram ^(^) for a given optical path difference ^ is givenby: is the spectrum to be estimated. The spectrum can be derived by applying the Fouriertransform to the interferogram: This conventional approach imposes limitations on the spectral range and resolution: where Δ^ is the delay step size and ^^^^ is the maximum delay. For applying the Fourier transform^(^), has to be sampled in regular intervals with a fixed step size.The inventive application of BAS is a probabilistic model that relates the measured interferogramto the underlying spectrum. The measured signal for a given delay ^ is expressed as: is the spectrum to be estimated, ^(^) represents the spectral efficiency of the detec-tor, and ^(^) is additive noise. Consideration of the additive noise is an optional feature of the in-ventive reconstruction. Depending on the application the n-term can be neglected. This formulation generalizes the traditional FTS equation by explicitly including the detector effi- ciency ^(^) and the noise term ^(^). This can be expressed as a linear system: ^= ^^ + ^ (5)where ^ is the vector of measured intensities (measured sensor data), ^ is the discretized spec-trum, ^ is the spectral system matrix with elements = [1 + cos(^^^^)]^(^^), and ^ is thenoise vector. Bayesian Inference Framework BAS employs a Bayesian inference framework to estimate the spectrum. The inventive key idea isto treat the spectrum ^(^) as a random variable and update assumptions (beliefs) about it asmeasurements are collected. This is formalized using Bayes’ theorem: is the posterior probability of the spectrum given the measurements, ^(^|^) is the likelihood of the measurements given the spectrum, is the prior probability of the spectrum, is the evidence (a normalization factor that is not needed to infer anything about ^).Practically, the spectrum data reconstruction method is directed on calculating the moments ofthis distribution, specifically the mean estimate ^^^^^^^^^^ ^ , which represents the most probable spectrum is, and the covariance Σ^^^^^^^^^^, which represents the uncertainty and how this uncer- tainty correlates between different spectral channels. Updating scheme To derive practical update equations, preferably Gaussian assumptions for both the prior and like-lihood distributions may be made. It is assumed that the prior distribution of the spectrum isGaussian with mean and covariance ^(^) = ^(^prior, ^prior^ ^ ) Similarly, it is assumed that the likelihood is Gaussian, with the measurements ^ normally distrib-uted around ^^ (our model’s prediction) with covariance^(^|^) = ^(^^, ^^) The inventors employ a key property of Gaussian distributions according to which the product of two Gaussians is also Gaussian. Given that both the prior and likelihood are Gaussian, the poste-rior distribution ^(^|^) will also be Gaussian. This property can be leveraged to derive updateequations, as follows. Generally, the product of two Gaussian distributions to a new Gaussian where: Applying this to the inventive spectrum data reconstruction, can be identified with (Σ^^^^^^)−1, This leads directly to the update equations: These equations can be rearranged to the more familiar Kalman filter form: ^newprio ^= (^ − ^^)^r ^ (14) where ^, known as the Kalman gain, is given by: ^= ^ prior^prior^^(^^^^ ^ + ^^)^^(15) These update equations provide a mechanism to optimally combine prior beliefs about the spec- trum with new measurement information. Note that this mathematical formulation does not require measurements at evenly spaced delaysbut allows arbitrary ones. The Kalman gain ^ determines how much it should be trusted the newmeasurement versus the prior belief. When the measurement noise Σ^ is large compared to theprior uncertainty , ^ will be small, and it will be trusted the prior more. Conversely, when ^is small compared to ^ will be large, and it will be trusted the new measurement more.this case the prior can be considered as uninformative, as the measurements dominate the esti-mate. In this limit ^ → ^−1 and thus^new^ = ^^^^ (16)Importantly, in the case of Nyquist sampling this is equivalent to the FFT solution used in FTS. Thismeans the update equation for the mean can be rewritten as This form allows for an informed estimate to be calculated from a regular FTS reconstruction with knowledge of only the measured delays (which are encoded in ^) but not the intensity values ^. Further, it is noted, that the individual steps of the invention (in particular updates, information gain calculation, etc., see below) can also be implemented using distributions other than Gaussian distributions. In these cases, mean estimates and covariances may be numerically approximated (e.g., with methods like Markov Chain Monte Carlo (MCMC), Laplace approximation, variational inference, or approximate Bayesian computation). Incorporation of prior knowledgeThe Bayesian framework allows for natural incorporation of prior knowledge about the spectrum.This prior knowledge can come from physical constraints, e.g., spectral information on the sourceof the measuring light source and / or spectral information on the photosensor device, previousmeasurements, and / or theoretical models. It is encoded in the initial mean covarianceE.g., the prior covariance could encode correlations between spectral components whichamount to a certain "smoothness" of the spectrum. The information encoded in the prior pro- vides a form of regularization on the solution. This makes the spectral estimate more robust to noise, as it combines the information from the measurements with our prior knowledge about the spectrum’s properties. However, a rigorous framework may be preferred to ensure that prior information does not lead to biases that deteriorate performance. The challenge lies in selecting or averaging over a set of candidate priors (henceforth called prior models) in a way that optimizes the spectral reconstruc- tion while avoiding overfitting. The orthodox Bayesian approach would consider a continuous distribution over possible prior models and marginalize them: ^(^|^) = ∫ ^(^|^, ^)^(^|^)^^ (18)where ^ is the spectrum, ^ is the data, and ^ represents a prior model. However, this integralmay be intractable to calculate analytically and computationally expensive to approximate using methods such as Markov Chain Monte Carlo (MCMC). If applicable, to make the problem tractable, a discrete set of candidate prior models {^1, ^2, ...,^^} is considered. While not strictly Bayesian, this approach is well motivated by practical consid-erations and computational feasibility. One might consider calculating ^(^^|^), which, for a uni- form prior over models, would be proportional to the evidence ^(^|^^). However, in the jointlyGaussian and linear inference case of the invention, this approach could favor the most unin-formative prior and potentially overfit to the data.To mitigate overfitting, assessing prior models using LeaveOne-Out Cross-Validation (LOO-CV) as ameasure of predictive accuracy may be employed in advantageous manner. The LOO-CV score fora given prior model ^^ is calculated as follows: ∫^(^^|^, ^^)^(^|^^^ , ^^)^^(19) where ^^ is the ^-th data point, ^−^ represents all data points except the ^-th, and we marginalizeover the signal ^. This integration over ^ is crucial as it accounts for the uncertainty in the signalestimation given the prior model and the data.To weigh the prior models, using a softmax function with a temperature parameter is proposed: where ^^ is the LOO-CV score for prior model ^^, and ^ is a temperature parameter. This ap-proach, while not standard in Bayesian analysis, offers a flexible and intuitive way to balance be-tween prior model selection and prior model averaging. As ^ → 0, the weights approach a hardselection of the best-performing prior model, while higher values of ^ lead to more uniformweighting across prior models.The set of candidate prior models may be constructed from priors inferred from previously seenor measured spectra, as well as an uninformative prior to maintain the limiting case of traditional Fourier transform spectroscopy. This combination has the advantage that the method can lever- age prior information when available while still performing robustly in novel scenarios. Prior model averaging, a more orthodox Bayesian approach, can be performed as: While prior model averaging can provide more robust predictions by incorporating uncertainty in model selection, it may not fully represent the continuous integral over "prior model space" that a true Bayesian approach would consider. The choice between averaging and selection may depend on the specific application and the diversity of the candidate prior model set.In conclusion, the inventive framework provides an advantageous pragmatic approach to incorpo-rating prior information in spectral reconstruction. By selecting and weighting a set of candidate prior models based on predictive accuracy, the strengths of Bayesian inference can be leveraged while mitigating the risks of overfitting and bias. This adaptive prior selection and averaging pro- cess ensures that the method can leverage prior information when it is beneficial while maintain- ing robustness against potentially biased or overly restrictive priors. It provides a data-driven mechanism to balance the influence of different prior models, allowing the method to adapt to the specific characteristics of each spectroscopic measurement scenario. Information-Theoretic Sampling Optimization A key advantage of BAS is its ability to optimize sampling strategies based on information theory. In traditional FTS, measurements are typically taken at uniform intervals determined by the Nyquist-Shannon sampling theorem. BAS, however, allows for non-uniform sampling by quantify- ing the information content of each potential measurement, as described in the following with reference to Figure 3. The information gain from a new measurement may be quantified using the concept of differen- tial entropy. For a multivariate Gaussian distribution, the differential entropy is given by: where ^ is the dimensionality of the distribution and |Σ| is the determinant of the covariance ma-trix. The information gain from a new measurement is then defined as the reduction in entropy:Information Gain = ^(prior) − ^(posterior) (23)This can be computed as: Information Gain =^ ^(log(det(^)) − log(det(^^))) (24)where ^ = ^^^^^ + ^^ is the predicted measurement covariance. This formulation allows for the selection of measurement delays that maximize information gain.The next measurement delay ^ can be chosen by maximizing the information gain:log(det(^^))) (25) Advantageously, this adaptive sampling strategy can significantly reduce the number of required measurements for a given spectral resolution, especially when the spectrum contains localized features or when certain spectral regions are of particular interest. Interestingly, under certain conditions (uniform prior uncertainty, uniform spectral efficiency, and isotropic measurement noise), the optimal sampling strategy according to this information-theo- retic criterion recovers the uniform sampling of traditional FTS. Specifically, the maxima of highest information gain occur at multiples of ^ / ^^^^, which is exactly the Nyquist sampling rate. For bandlimited signals, the optimal sampling rate becomes ^ / (^^ −^^), consistent with bandpass sampling theory. The BAS framework thus provides a unified treatment of sampling in FTS, encompassing both tra- ditional uniform sampling and more sophisticated adaptive strategies. By explicitly quantifying the information content of each measurement, BAS opens up new possibilities for efficient spectral estimation, particularly in scenarios where measurement time or data storage is at a premium.Practical embodiment 1: Enhancement of Existing FTS System with Non-Adaptive SamplingAccording to a preferred embodiment, the invention may implement Bayesian Autocorrelation Spectroscopy (BAS) as a post-processing enhancement to an existing Fourier Transform Spectros-copy (FTS) system, improving spectral reconstruction without hardware modifications of the FTSsystem. The FTS system provides an interferometric autocorrelation measurement apparatus 100,including a measuring device 10 with a sensor device and a spectrum data reconstruction appa-ratus 20 according to an embodiment of the invention, as schematically shown in Figure 4. Themeasuring device 10 comprises an existing FTS instrument including the sensor device and havingraw interferogram data export capability. The spectrum data reconstruction apparatus 20 is acomputer unit running a BAS algorithm software implementation of the spectrum data recon- struction method according to an embodiment of the invention. The spectrum data reconstruc-tion apparatus 20 may be integrated in control components of the measuring device 10, i.e., asoftware implementing the spectral data reconstruction may run in the measuring device 10.An interferometric autocorrelation measurement method according to an embodiment of the in-vention comprises the steps of providing a measuring light field, interacting of the measuring lightfield with a sample to be investigated and measuring sensor data by an interferometric autocorre-lation measurement of interferogram data with the measuring device 10 (FTS instrument).The data collection of the measuring step comprises collecting interferogram data using standarduniform sampling (sampling with uniform delay steps), recording delay values for each data pointand storing instrument parameters, like in particular spectral range, resolution, and detector effi-ciency curve. Subsequently, a data preprocessing is executed, including an import of raw interfer-ogram data and delay values, applying standard FTS corrections (e.g., phase correction, baselineremoval) and constructing a spectral system matrix R using delay values and detector efficiency.For providing an initial prior estimate ^^^^^^^and an initial prior covariance of the spectrumdata (step S1.0 in Figure 1 or step S2.1 in Figure 2)), a prior model construction is implemented asfollows.Candidate prior models are defined based on a database of relevant spectra of at least one com-parable sample (expected to have similar spectral properties like the sample to be investigated), previously measured spectra of similar samples, known light source characteristics for absorption experiments, and / or general knowledge about spectral correlations. Furthermore, the priormodel construction may include providing an uninformative prior for comparison and / or imple-menting a Leave-One-OutCross-Validation (LOO-CV) for prior model evaluation.The subsequent spectral reconstruction may be executed according to one of the following op-tions, including a sequential updating (option A, see Figure 1) or a one-step process using FFT re-construction (option B, see Figure 2).According to option A (sequential updating), ^^^^^^^and Σ^^^^^^are initialized based on the chosenprior model (step S1.0 in Figure 1). Based on the priors, the updated estimates are calculated foreach measurement point (steps S1.1 and S1.2 in Figure 1) based on equations (13) and (14) ac-cording to: Update Update For the next iteration, the estimates of the first calculation are set according to = ^^newand ^prior = ^new^ ^ as the updated prior input for the next calculation (step S1.4 in Figure 1).The updating iterations are terminated and final reconstruction is output based on a testing (eval-uating) of a predefined stopping condition (comprising e.g., a predefined uncertainty thresholdand / or a predefined maximum number of measurement or delay applied). If the stopping condi-tion is fulfilled, a final BAS update is performed and a final spectral estimate with uncertaintybounds is generated and output.According to option B (One-step process using FFT reconstruction), a standard Fourier transformis executed at first (step S2.0 in Figure 2) to obtain a Fourier estimate of the spectral data.Using the Fourier estimate and the priors (step S2.1 in Figure 2) and the spectral system matrix(step S2.2 in Figure 2), updated estimates of the spectral data are obtained (step S2.3 in Figure 2, see equation (17)) bycalculating and updating Prior model averaging is provided by calculating weights for each prior model based on LOO-CVscores and computing weighted average of spectral estimates from different prior models.Finally, uncertainty is quantified by extracting uncertainty estimates from diagonal elements ofΣ^^^^ , and the final spectral estimate with uncertainty bounds are presented as output of thespectrum data reconstruction method.Practical embodiment 2: Adaptive Sampling FTS SystemAccording to a further preferred embodiment, an FTS instrument design is employed, which incor-porates real-time adaptive sampling based on the BAS framework, wherein optimal delay valuesare dynamically selected to maximize information gain, as illustrated in Figure 3.Again, the FTS system provides an interferometric autocorrelation measurement apparatus 100,including a measuring device 10 with a sensor device and a spectrum data reconstruction appa-ratus 20 according to an embodiment of the invention, as schematically shown in Figure 4. Prefer-ably, the measuring device 10 is an interferometer FTS instrument which is adapted with a preci-sion delay control. The sensor device preferably may comprise a high-speed, low-noise detector.Furthermore, the measuring device 10 and / or the spectrum data reconstruction apparatus 20 in-cludes a real-time control system for rapid data processing and decision making. The spectrumdata reconstruction apparatus 20 is a computer unit running a BAS algorithm software implemen-tation of the spectrum data reconstruction method according to the embodiment of the inven- tion, including the adaptive sampling optimization.Furthermore, as noted above, the interferometric autocorrelation measurement method accord-ing to an embodiment of the invention comprises the steps of providing a measuring light field,interacting of the measuring light field with a sample to be investigated and measuring sensordata by an interferometric autocorrelation measurement of interferogram data with the measur-ing device 10 (FTS instrument). However, in this case the sensor data are not measured with fixeddelays, but with adaptively changed delays set as follows.Firstly, a system initialization is implemented, including defining a spectral range of interest andinitial spectral resolution target of interest. This may be done on the basis of prior knowledge onthe sample and the measuring device, in particular a measuring light source and a photosensordevice thereof. The spectral range of interest and the spectral resolution target of interest are re-lated to the delay and interval of delay variation as described in the European patent applicationEP 24198926.8 ("Optical imaging apparatus and spectral imaging method for creating spectrallyand spatially resolved image data", not published on the priority day of the present specification).Prior models (^^^^^^^, Σ^^^^^^) are initialized based on available information or uninformative prior,and an initial delay is set to zero (step S3.0 in Figure 3).On the basis of the system initialization, an adaptive sampling and real-time processing loop is im-plemented as follows.Firstly, information gain for potential delay values using current optimal estimate is calculated(step S3.1 in Figure 3) according toInformation Gain =^ ^ (log(^^^(^^Σ^^^^ + Σ^)) − log(^^^(Σ^))) (26)The delay value ^^^^^ is selected, which provides maximizing the information gain. The measuringdevice 10, e.g., the path length difference in an interferometer of an FTIR setup, is adjusted to theselected delay ^^^^^ . and the measurement (step S3.2 in Figure 3) is acquired at the selected delay^^^^^. On the basis of the current measurement, the spectral estimate and uncertainty are updated(step S3.3 in Figure 3) using the BAS equations for calculating the spectral data (step S3.4 in Figure3) as mentioned above with reference to option A of embodiment 1. Furthermore, the spectralsystem matrix R is continuously updated based on actual delay values.Subsequently, weights are recalculated for each prior model based on updated LOO-CV scores,and weighted average of spectral estimates are computed from different prior models. The up- dated weighted average is used for the next information gain calculation.As noted above, the termination and final reconstruction is based on the predefined stoppingcondition (comprising e.g., a predefined uncertainty threshold and / or a predefined maximumnumber of measurement or delay applied). The features of the invention disclosed in the above description, the drawings and the claims can be of significance both individually as well as in combination or sub-combination for the realiza- tion of the invention in its various embodiments. The invention is not restricted to the preferred embodiments described above. Rather a plurality of variants and derivatives is possible which also use the inventive concept and therefore fall within the scope of protection. In addition, the inven- tion also claims protection for the subject and features of the subclaims independently of the fea- tures and claims to which they refer.

Claims

Claims1. Spectrum data reconstruction method for reconstructing spectrally resolved spectrumdata (S) of a measuring light field from sensor data (Fτ) provided by an interferometric autocorre-lation measurement of interferogram data created by pairwise superimposing delayed versions ofthe measuring light field, said delayed versions of the measuring light field having different mu- tual delays (τ),characterized by the steps of- providing a spectral system matrix (Rτω), wherein the sensor data (Fτ) are determined by applyingthe spectral system matrix (Rτω) to the spectrum data (Sω) to be obtained, said spectral system ma- trix (Rτω) being created in dependency on the delays (τ) and instrument parameters of an spec- trometer apparatus employed for the interferometric autocorrelation measurement,- providing an initial prior estimate ^µ ^^^^^^ ^ and an initial prior covarianceof the spec-trum data (S),- calculating a mean estimate (µ^) and a covariance (Σ^ ) of the spectrum data (Sω) by applying aBayesian inference computation employing the sensor data (Fτ), the spectral system matrix (Rτω), the initial prior estimate ^µ^^^^^^ the initial prior covarianceand the initial prior covari-ance- output of the mean estimate (µ^) and optionally the covariance (Σ^ ) of the spectrum data (Sω)as the spectrum data (S) to be reconstructed.

2. Spectrum data reconstruction method according to claim 1, including further steps of- repeatedly calculating an updated mean estimate (µ^) and an updated covariance (Σ^ ) of thespectrum data (Sω) by applying the Bayesian inference computation, wherein for each update cal-culation a previously calculated mean estimate (µ^) and covariance (Σ^ ) of the spectrum data(S ) are employed as a current prior estimateand a current prior covarianceand- after termination of the repeatedly updating steps, output of the updated mean estimate (µ^)and optionally the updated covariance (Σ^ ) of the spectrum data (Sω) as the spectrum data (S) tobe reconstructed.

3. Spectrum data reconstruction method according to one of the foregoing claims, includingfurther steps of- calculating multiple values of an information gain for potential delay values using the calculatedcovariance (Σ^ ) of the spectrum data (Sω), and- output of a selected delay value (τnext) which provides a maximum value of the information gain.

4. Spectrum data reconstruction method according to claim 1, including a further step of- calculating a Fourier transform of the sensor data (Fτ) for obtaining a Fourier estimate ^µ^^^^^^^^^ of the spectrum data (Sω), wherein- the mean estimate (µ^) of the spectrum data (Sω) is calculated by applying the Bayesian infer-ence computation further employing the Fourier estimate^µ^^^^^^^^ ^.

5. Spectrum data reconstruction method according to one of the foregoing claims, wherein- the initial prior estimate ^µ ^^^^^^ ^ and the initial prior covarianceare provided based onat least one of a reference database of relevant spectra, data of previously measured spectra,e.g., of similar samples, and measuring light source characteristics, e.g., for absorption experi-ments.

6. Spectrum data reconstruction method according to one of the foregoing claims, whereinthe step of providing the initial prior estimate ^µ^^^^^^ ^ includes- providing multiple different model prior estimates ^µ ^^^^^^^^^^ ^, and- a step of prior model averaging the model prior estimates ^µ^^^^^^^^^^^, comprising at least one of calculating weights for each model prior estimate ^µ^^^^^^^^^^ ^ based on a Leave-One-Out Cross-Vali-dation (LOO-CV) score, and calculating weighted average of spectral estimates from the model prior estimates7. Spectrum data reconstruction method according to one of the foregoing claims, wherein- the sensor data (Fτ) comprise spatially resolved sensor data measured with multiple sensor ele-ments (pixels) of a two-dimensional photosensor, and- the spectrally resolved spectrum data (S) are separately reconstructed for each of the sensor ele-ments.

28. Interferometric autocorrelation measurement method, comprising the steps of- providing a measuring light field,- measuring sensor data (Fτ) by an interferometric autocorrelation measurement of interferogramdata created by pairwise superimposing delayed versions of the measuring light field, said delayed versions of the light field having different mutual delays (τ), and- applying the spectrum data reconstruction method according to one of the foregoing claims tothe sensor data (Fτ).

9. Interferometric autocorrelation measurement method according to claim 8, wherein the spectrum data reconstruction method according to claim 3 is employed, including the steps of- calculating the multiple values of the information gain for potential delay values using the calcu-lated covariance (Σ^ ) of the spectrum data (Sω),- employing the selected delay value (τnext) which provides the maximum value of the informationgain, for measuring subsequent sensor data (Fτ).

10. Spectrum data reconstruction apparatus being configured for reconstructing spectrallyresolved spectrum data (S) of a measuring light field from sensor data (Fτ) provided by an interfer- ometric autocorrelation measurement of interferogram data created by pairwise superimposing delayed versions of the measuring light field, said delayed versions of the measuring light field having different mutual delays (τ),characterized by a data processing device being configured for- providing a spectral system matrix (Rτω), wherein the sensor data (Fτ) are determined by applyingthe spectral system matrix (Rτω) to the spectrum data (Sω) to be obtained, said spectral system ma- trix (Rτω) being created in dependency on the delays (τ) and instrument parameters of an spec- trometer apparatus employed for the interferometric autocorrelation measurement,- providing an initial prior estimate ^µ ^^^^^^ ^ and an initial prior covarianceof the spec-trum data (S),- calculating a mean estimate (µ^) and a covariance (Σ^ ) of the spectrum data (Sω) by applying aBayesian inference computation employing the sensor data (Fτ), the spectral system matrix (Rτω), the initial prior estimate ^µ^^^^^^ the initial prior covarianceand the initial prior covari-ance (Σ^^^^^^), and- output of the mean estimate (µ^) and optionally the covariance (Σ^ ) of the spectrum data (Sω)as the spectrum data (S) to be reconstructed.

311. Interferometric autocorrelation measurement apparatus (100) being configured for meas-uring spectrally resolved spectrum data (S) of a measuring light field, comprising- a measuring device (10) including a sensor device being arranged for providing sensor data (Fτ)provided by an interferometric autocorrelation measurement of interferogram data created by pairwise superimposing delayed versions of the measuring light field, said delayed versions of themeasuring light field having different mutual delays (τ), and- the spectrum data reconstruction apparatus (20) according to claim 10.

12. Computer-implemented device programmed to perform the method according to one of the claims 1 to 7.

13. Computer program product which, when loaded into a computer-implemented device, executes the method according to one of claims 1 to 7. 4

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