Optimal Frequency-Based Weighting for Spectral X-Ray Projection Imaging

Abstract

The purpose of this work is to derive a weighting scheme that maximizes the frequency-dependent ideal observer signal-difference-to-noise ratio, commonly referred to as detectability index or Hotelling-SDNR, for spectral X-ray projection imaging. Starting from basic statistical decision theory, optimal frequency-dependent weights are derived for a multiple-bin system and the Hotelling-SDNR calculated. A 28% increase in detectability index is found for high frequency objects when applying optimal frequency-dependent weights instead of pixel-based weights to a simplified model of a silicon detector, decreasing towards 0% for low frequency objects. Simulation results indicate a potentially large increase in detectability for high-frequency object imaging using silicon detectors, thus meriting further evaluations on a real system.

A framework for performance characterization of energy‐resolving photon‐counting detectors

“…To attain the performance limit given by , the model observer needs to take data from all the energy bins into account and give an optimal weight to each energy bin at each spatial frequency, which is difficult for a human observer. For any given task, however, frequency‐dependent optimal weighting of the energy bin images can be used to form a single image, for which an ideal linear observer attains the same performance for that task . This observation allows the following interpretation: is the maximum achievable detectability in a single image that is formed as a weighted sum of the bin images, where the weights are frequency‐dependent and optimized for the given detection task.…”

“…For any given task, however, frequency-dependent optimal weighting of the energy bin images can be used to form a single image, for which an ideal linear observer attains the same performance for that task. 35 This observation allows the following interpretation: d 02 is the maximum achievable detectability in a single image that is formed as a weighted sum of the bin images, where the weights are frequencydependent and optimized for the given detection task. Note that the optimization of the weights ensures that this detectability is always greater than (or equal to) the detectability in an image formed by simply adding the bin images together with equal weight (i.e., an image formed from all counts above the lowest threshold).…”

“…A good measure of the performance of an energy‐resolving detector should, therefore, take optimal weighting into account. Taking the spatial‐frequency dependence of the signal and noise into account when calculating optimal weights can improve detectability in an energy‐weighted image, either by calculating an optimized weight for each image or by letting the weights themselves be functions of spatial frequency . This is equivalent to applying different spatial filters to the different energy bin images before forming the weighted sum, an approach studied in the dual‐energy case by Richard and Siewerdsen …”

“…Taking the spatial-frequency dependence of the signal and noise into account when calculating optimal weights can improve detectability in an energy-weighted image, either by calculating an optimized weight for each image 34 or by letting the weights themselves be functions of spatial frequency. 35 This is equivalent to applying different spatial filters to the different energy bin images before forming the weighted sum, an approach studied in the dual-energy case by Richard and Siewerdsen. 29 Another way to utilize spectral x-ray measurements is to perform basis material decomposition, which builds on the observation that the linear attenuation coefficient of any substance in the human body can be approximated well as a linear combination of a small number of basis functions, typically two in the absence and three in the presence of a high-atomic number contrast agent with a K-edge in the diagnostic energy range.…”

Optimization of beam quality for photon-counting spectral computed tomography in head imaging: simulation study

“…[38][39][40] A comprehensive description of this theory can be found in the publication by Barrett and Myers 41 To apply the theory to our photon-counting spectral imaging system, we assume two hypotheses, H 0 and H 1 , representing the absence and presence of an imaging target, respectively. The task is then defined as deciding whether the imaging target is present or not.…”