Optimal Filter Selection for Spectrum Reconstruction of a Pulsed X-ray Source

Calibrating a polychromatic X-ray imaging system for use as a precise measuring instrument is essential: the lower the calibration accuracy, the greater the error in interpreting X-ray images. Estimating the source spectrum remains a crucial step in this calibration process. Standard methods using spectrometers and energy-dispersive detectors are readily available for continuous X-ray sources. However, such continuous sources are unsuitable for studying fast-moving objects due to severe motion blur. Pulsed X-ray sources are required instead, rendering these standard estimation methods technologically unfeasible. Instead, spectra can be reconstructed from integrated radiation intensities recorded by a detector after passing through various optical filter combinations. Because using all available combinations is generally impractical, the problem of optimal experimental design arises: selecting the best subset for a fixed number of measurements. In the literature, this problem has been addressed unsatisfactorily. The optimal set is typically sought assuming a simple nonparametric method that ignores prior knowledge of the spectrum and noise while unjustifiably using the matrix condition number as the optimality criterion. This work introduces an appropriate probabilistic model of detector signal formation alongside a user-customizable loss function. We derive a generalized reconstruction method that minimizes the expected value of this error via Bayesian inference. By analytically evaluating this criterion in a prior, pre-experimental form, we can use it directly as the optimality criterion for filter selection. We prove that the condition number is strictly inapplicable in this general case and demonstrate that our proposed experimental design yields significantly higher accuracy in source spectrum estimation for the same measurement count. These results are equally applicable to estimating emission, transmission, and reflection spectra across various spectral ranges.
The research was subsidized by the Ministry of Science and Education under projects FRC CSC RAS 124040200033-9 and IPTM RAS 075-00296-26-00.