# Ideal observer estimation for binary tasks with stochastic object models

## TL;DR

This paper introduces a new ideal observer formulation that uses stochastic object models to optimize data acquisition for binary tasks.

## Contribution

The novel approach formulates intrinsic and extrinsic class separability to evaluate data acquisition efficiency.

## Key findings

- The extrinsic likelihood ratio is the expectation of the intrinsic likelihood ratio under the posterior PDF.
- The new ideal observer was successfully applied to dual-energy CT projection domain material decomposition.
- Performance results aligned with physics predictions, showing practical applicability.

## Abstract

Objective. We propose a new formulation for ideal observers (IOs) that incorporate stochastic object models (SOMs) for data acquisition optimization. Approach. A data acquisition system is considered as a (possibly nonlinear) discrete-to-discrete mapping from a finite-dimensional object space, \documentclass[12pt]{minimal}
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$x \in R^{n_d}$\end{document}x∈Rnd, to a finite-dimensional measurement space, \documentclass[12pt]{minimal}
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$y \in R^{m}$\end{document}y∈Rm. For binary tasks, the two underlying SOMs, \documentclass[12pt]{minimal}
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$H_0$\end{document}H0 and H1, are specified by two probability density functions (PDFs) \documentclass[12pt]{minimal}
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$p_{0}(x)$\end{document}p0(x), \documentclass[12pt]{minimal}
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$p_{1}(x)$\end{document}p1(x). This leads to the notion of intrinsic likelihood ratio (LR) \documentclass[12pt]{minimal}
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$\Lambda_{I} (x) = p_{1}(x)/p_{0}(x)$\end{document}ΛI(x)=p1(x)/p0(x) and intrinsic class separability (ICS), the latter quantifies the population separability that is independent of data acquisition. With respect to ICS, the IO employs the ‘extrinsic’ LR \documentclass[12pt]{minimal}
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$\Lambda(y) = {\mathrm{pr}} (y|H_1)/{\mathrm{pr}} (y|H_0)$\end{document}Λ(y)=pr(y|H1)/pr(y|H0) of the data and quantifies the extrinsic class separability (ECS). The difference between ICS and ECS measures the efficiency of data acquisition. We show that the extrinsic LR \documentclass[12pt]{minimal}
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$\Lambda (y)$\end{document}Λ(y) is the expectation of the intrinsic LR \documentclass[12pt]{minimal}
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$\Lambda_{I} (x)$\end{document}ΛI(x), where the expectation is with respect to the posterior PDF \documentclass[12pt]{minimal}
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${\mathrm{pr}} (x|y, H_0)$\end{document}pr(x|y,H0) under \documentclass[12pt]{minimal}
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$H_0$\end{document}H0. Main results. We use two examples, one to clarify the new IO and the second to demonstrate its potential for real world applications. Specifically, we apply the new IO to spectral optimization in dual-energy CT projection domain material decomposition (pMD), for which SOMs are used to describe variability of basis material line integrals. The performance rank orders obtained by IO agree with physics predictions. Significance. The main computation in the new IO involves sampling from the posterior PDF \documentclass[12pt]{minimal}
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${\mathrm{pr}} (x|y, H_0)$\end{document}pr(x|y,H0), which are similar to (fully) Bayesian reconstruction. Thus our IO computation is amenable to standard techniques already familiar to CT researchers. The example of dual-energy pMD serves as a prototype for other spectral optimization problems, e.g., for photon counting CT or multi-energy CT with multi-layer detectors.