Infrared scattering scanning near-field optical microscopy (IR s-SNOM) provides for spectroscopic imaging with nanometer spatial resolution, yet full spatio-spectral imaging is constrained by long measurement times. Here, we demonstrate the application of compressed sensing algorithms to achieve hyperspectral FTIR-based nano-imaging at an order of magnitude faster imaging speed to achieve the same spectral content compared to conventional approaches. At the example of the spectroscopy of a single vibrational resonance, we discuss the relationship of prior knowledge of sparseness of the employed Fourier base functions and sub-sampling. Compressed sensing nano-FTIR spectroscopy promises both rapid and sensitive chemical nano-imaging which is highly relevant in academic and industrial settings for fundamental and applied nano- and bio-materials research.

Bandpass correction in spectrometer measurements using monochromators is often necessary in order to obtain accurate measurement results. The classical approach of spectrometer bandpass correction is based on local polynomial approximations and the use of finite differences. Here we compare this approach with an extension of the Richardson-Lucy method, which is well known in image processing, but has not been applied to spectrum bandpass correction yet. Using an extensive simulation study and a practical example, we demonstrate the potential of the Richardson-Lucy method. In contrast to the classical approach, it is robust with respect to wavelength step size and measurement noise. In almost all cases the Richardson-Lucy method turns out to be superior to the classical approach both in terms of spectrum estimate and its associated uncertainties.

A Bayesian approach is proposed for the analysis of magnetic field fluctuation thermometry. The approach addresses the estimation of temperature from the measurement of a noise power spectrum as well as the analysis of previous calibration measurements. A key aspect is the reliable determination of uncertainties associated with the obtained temperature estimates, and the proposed approach naturally accounts for both the uncertainties in the calibration stage and the noise in the temperature measurement. Erlang distributions are employed to model the fluctuations of thermal noise power spectra and we show that such a procedure is justified in the light of the data. We describe in detail the Bayesian approach and briefly refer to Markov Chain Monte Carlo techniques used in the numerical calculation of the results. The MATLAB® software package we used for calculating our results is provided. The proposed approach is validated using magnetic field fluctuation power spectra recorded in the sub-kelvin region for which an independently determined reference temperature is available. As a result, the obtained temperature estimates were found to be fully consistent with the reference temperature.

The Bayesian approach to solving inverse problems relies on the choice of a prior. This critical ingredient allows the formulation of expert knowledge or physical constraints in a probabilistic fashion and plays an important role for the success of the inference. Recently, Bayesian inverse problems were solved using generative models as highly informative priors. Generative models are a popular tool in machine learning to generate data whose properties closely resemble those of a given database. Typically, the generated distribution of data is embedded in a low-dimensional manifold. For the inverse problem, a generative model is trained on a database that reflects the properties of the sought solution, such as typical structures of the tissue in the human brain in magnetic resonance (MR) imaging. The inference is carried out in the low-dimensional manifold determined by the generative model which strongly reduces the dimensionality of the inverse problem. However, this proceeding produces a posterior that admits no Lebesgue density in the actual variables and the accuracy reached can strongly depend on the quality of the generative model. For linear Gaussian models we explore an alternative Bayesian inference based on probabilistic generative models which is carried out in the original high-dimensional space. A Laplace approximation is employed to analytically derive the required prior probability density function induced by the generative model. Properties of the resulting inference are investigated. Specifically, we show that derived Bayes estimates are consistent, in contrast to the approach employing the low-dimensional manifold of the generative model. The MNIST data set is used to construct numerical experiments which confirm our theoretical findings. It is shown that the proposed approach can be advantageous when the information contained in the data is high and a simple heuristic is considered for the detection of this case. Finally, pros and cons of both approaches are discussed.

Near-field goniometric measurements are employed to determine the photometric characteristics of light sources, i.e., the spatial and angular distribution of the emitted light. To this end, a complex measurement system consisting of a goniometer and a CCD-based imaging photometer is employed. In order to gain insight into the measurement system and to enable characterization of the whole measurement setup, we propose to apply a computer model to conduct virtual experiments. Within the computer model, the current state of all parts of the virtual experiment can be easily controlled. The reliability of the computer model is demonstrated by a comparison to actual measurement results. As an example for the application of the virtual experiment, we present an analysis of the impact of axial malpositions of the goniometer and camera.