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Information-optimal measurement: From fixed sampling protocols to adaptive spectroscopy

J. Schroeder, S. Howard, C. Eberle, J. Esslinger, N. Leopold-Kerschbaumer, K. V. Kepesidis, A. Döpp

TL;DR

This paper reframes spectral measurement as an information-theoretic, Bayesian decision problem, showing that Nyquist sampling is uniquely optimal only under complete ignorance and that prior information enables adaptive, uncertainty-aware data acquisition. The authors formalize Bayesian Autocorrelation Spectroscopy (BAS), deriving how measurement delays can be chosen to maximize information gain and demonstrating real-time uncertainty propagation. They validate BAS across three domains—molecular fingerprinting via FTIR, optical vortex characterization with spectral covariance propagation, and hyperspectral imaging via RGB-enabled fusion—showing accelerated measurements and robust uncertainty quantification. The work suggests a broad paradigm shift toward information-optimal instruments, with potential extensions to richer priors, multi-stream sensing, and objective-driven information metrics.

Abstract

All measurements of continuous signals rely on taking discrete snapshots, with the Nyquist-Shannon theorem dictating sampling paradigms. We present a broader framework of information-optimal measurement, showing that traditional sampling is optimal only when we are entirely ignorant about the system under investigation. This insight unlocks methods that efficiently leverage prior information to overcome long-held fundamental sampling limitations. We demonstrate this for optical spectroscopy - vital to research and medicine - and show how adaptively selected measurements yield higher information in medical blood analysis, optical metrology, and hyperspectral imaging. Through our rigorous statistical framework, performance never falls below conventional sampling while providing complete uncertainty quantification in real time. This establishes a new paradigm where measurement devices operate as information-optimal agents, fundamentally changing how scientific instruments collect and process data.

Information-optimal measurement: From fixed sampling protocols to adaptive spectroscopy

TL;DR

This paper reframes spectral measurement as an information-theoretic, Bayesian decision problem, showing that Nyquist sampling is uniquely optimal only under complete ignorance and that prior information enables adaptive, uncertainty-aware data acquisition. The authors formalize Bayesian Autocorrelation Spectroscopy (BAS), deriving how measurement delays can be chosen to maximize information gain and demonstrating real-time uncertainty propagation. They validate BAS across three domains—molecular fingerprinting via FTIR, optical vortex characterization with spectral covariance propagation, and hyperspectral imaging via RGB-enabled fusion—showing accelerated measurements and robust uncertainty quantification. The work suggests a broad paradigm shift toward information-optimal instruments, with potential extensions to richer priors, multi-stream sensing, and objective-driven information metrics.

Abstract

All measurements of continuous signals rely on taking discrete snapshots, with the Nyquist-Shannon theorem dictating sampling paradigms. We present a broader framework of information-optimal measurement, showing that traditional sampling is optimal only when we are entirely ignorant about the system under investigation. This insight unlocks methods that efficiently leverage prior information to overcome long-held fundamental sampling limitations. We demonstrate this for optical spectroscopy - vital to research and medicine - and show how adaptively selected measurements yield higher information in medical blood analysis, optical metrology, and hyperspectral imaging. Through our rigorous statistical framework, performance never falls below conventional sampling while providing complete uncertainty quantification in real time. This establishes a new paradigm where measurement devices operate as information-optimal agents, fundamentally changing how scientific instruments collect and process data.

Paper Structure

This paper contains 26 sections, 57 equations, 3 figures.

Table of Contents

  1. Bayesian Autocorrelation Spectroscopy
  2. Molecular fingerprinting
  3. Optical vortices
  4. Hyperspectral imaging
  5. Conclusions and Outlook
  6. Materials and Methods
  7. Supplementary Text
  8. Bayesian inference model
  9. General linear Gaussian inference
  10. Limited rank update through finite measurements
  11. Information optimal sampling
  12. Specifications leading to classical sampling
  13. Finite Dimensionality of the Parameter Space
  14. Modeling the Constrained Signal: From Gaussian Processes to a Finite Trigonometric Basis
  15. Uninformed Prior and D-Optimality
  16. ...and 11 more sections

Figures (3)

  • Figure 1: BAS reconstruction of a biological FTIR absorbance spectrum. (A) The prior distribution (orange band) is derived from a set of blood-based infrared spectra exhibits small variance, indicating strong constraints on possible spectra. BAS adaptive estimation (red) closely matches the true spectrum (gray) across the full wavenumber range. Zoom panels highlight key spectral regions showing detailed agreement between reconstruction and ground truth. (B) SNR evolution with measurement count shows BAS adaptive sampling (solid red) reaching a high SNR fast and continuously refining the estimate, followed by BAS with prior knowledge and fixed Nyquist sampling (dashed red) lacking this refinement, while uninformed reconstruction (dotted) performs significantly worse. Wavenumber-resolved SNR maps reveal sampling dynamics, with adaptive BAS exploring spectral regions most efficiently. Right panel shows the reconstructed absorbance spectrum for reference.
  • Figure 2: Wavelength-resolved phase reconstruction of an optical vortex beam. Three-dimensional visualization of the beam's phase structure across wavelengths ($750-850\nm$), showing the characteristic azimuthal phase variation. The zoom panel shows the azimuthal phase profiles extracted at 780 (design wavelength) and 840 with uncertainty bands derived from covariance propagation through the full BAS spectral reconstruction and phase retrieval chain. The observed wavelength dependence reflects the chromatic behavior of the spiral phase plate.
  • Figure 3: Spectral resolution enhancement of an RGB camera using a liquid crystal retarder. (A) Device overview, consisting of a liquid crystal retarder for delay control between two polarization components and an RGB camera with a colour and polarization filter pixel array, which constitute a compact common-path interferometer with three measurement channels. The inset shows the quantum efficiency of the colour filters. (B) Evolution of the BAS estimate with uncertainty bands as a function of iteration, showing progressive refinement which already improves upon the three-channel resolution after only few measurements.