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Infrared hyperspectral imaging is a powerful approach in the field of materials and life sciences. However, the extension to modern sub-diffraction nanoimaging still remains a highly inefficient technique, as it acquires data via inherent sequential schemes. Here, we introduce the mathematical technique of low-rank matrix reconstruction to the sub-diffraction scheme of atomic force microscopy-based infrared spectroscopy (AFM-IR), for efficient hyperspectral infrared nanoimaging. To demonstrate its application potential, we chose the trypanosomatid unicellular parasites Leishmania species as a realistic target of biological importance. The mid-infrared spectral fingerprint window covering the spectral range from 1300 to 1900 cm$^{-1}$ was chosen and a step width of 220 nm was applied for nanoimaging of single parasites. Multivariate statistics approaches such as hierarchical cluster analysis (HCA) were used for extracting the chemically distinct spatial locations. Subsequently, we randomly selected only 5% from an originally gathered data cube of 134 (x) $\times$ 50 (y) $\times$ 148 (spectral) AFM-IR measurements and reconstructed the full data set by low-rank matrix recovery. The technique is evaluated by showing agreement in the cluster regions between full and reconstructed data cubes. We conclude that the corresponding measurement time of more than 14 hours can be reduced to less than 1 hour. These findings can significantly boost the practical applicability of hyperspectral nanoimaging in both academic and industrial settings involving nano- and bio-materials.

The Kalman filter is an established tool for the analysis of dynamic systems with normally distributed noise, and it has been successfully applied in numerous application areas. It provides sequentially calculated estimates of the system states along with a corresponding covariance matrix. For nonlinear systems, the extended Kalman filter is often used which is derived from the Kalman filter by linearization around the current estimate. A key issue in metrology is the evaluation of the uncertainty associated with the Kalman filter state estimates. The "Guide to the Expression of Uncertainty in Measurements" (GUM) and its supplements serve as the de facto standard for uncertainty evaluation in metrology. We explore the relationship between the covariance matrix produced by the Kalman filter and a GUM-compliant uncertainty analysis. In addition, also the results of a Bayesian analysis are considered. For the case of linear systems with known system matrices, we show that all three approaches are compatible. When the system matrices are not precisely known, however, or when the system is nonlinear, this equivalence breaks down and different results can be reached then. Though for precisely known nonlinear systems the result of the extended Kalman filter still corresponds to the linearized uncertainty propagation of GUM. The extended Kalman filter can suffer from linearization and convergence errors. These disadvantages can be avoided to some extent by applying Monte Carlo procedures, and we propose such a method which is GUM-compliant and can also be applied online during the estimation. We illustrate all procedures in terms of a two-dimensional dynamic system and compare the results with those obtained by particle filtering, which has been proposed for the approximate calculation of a Bayesian solution. Finally, we give some recommendations based on our findings., Comment: 27 pages, 5 figures

We report on recent progress achieved in calculating various few-nucleon low-energy observables from effective field theory. Our discussion includes scattering and bound states in the 2N, 3N and 4N systems and isospin violating effects in the 2N system. We also establish a link between the nucleon-nucleon potential derived in chiral effective field theory and various modern high-precision potentials., Comment: 12 pp, uses aipproc style files, 4 figures, contribution to the conference on "Mesons and Light Nuclei", Prag, July 2001, to appear in the proceedings