Existence, uniqueness, and efficiency of numerically unbiased attenuation pathlength estimators for photon counting detectors at low count rates.

Background: The first step in computed tomography (CT) reconstruction is to estimate attenuation pathlength. Usually, this is done with a logarithm transformation, which is the direct solution to the Beer-Lambert Law. At low signals, however, the logarithm estimator is biased. Bias arises both from the curvature of the logarithm and from the possibility of detecting zero counts, so a data substitution strategy may be employed to avoid the singularity of the logarithm. Recent progress has been made by Li et al. [IEEE Trans Med Img 42:6, 2023] to modify the logarithm estimator to eliminate curvature bias, but the optimal strategy for mitigating bias from the singularity remains unknown.

Purpose: The purpose of this study was to use numerical techniques to construct unbiased attenuation pathlength estimators that are alternatives to the logarithm estimator, and to study the uniqueness and optimality of possible solutions, assuming a photon counting detector.

Methods: Formally, an attenuation pathlength estimator is a mapping from integer detector counts to real pathlength values. We constrain our focus to only the small signal inputs that are problematic for the logarithm estimator, which we define as inputs of <100 counts, and we consider estimators that use only a single input and that are not informed by adjacent measurements (e.g., adaptive smoothing). The set of all possible pathlength estimators can then be represented as points in a 100-dimensional vector space. Within this vector space, we use optimization to select the estimator that (1) minimizes mean squared error and (2) is unbiased. We define "unbiased" as satisfying the numerical condition that the maximum bias be less than 0.001 across a continuum of 1000 object thicknesses that span the desired operating range. Because the objective function is convex and the constraints are affine, optimization is tractable and guaranteed to converge to the global minimum. We further examine the nullspace of the constraint matrix to understand the uniqueness of possible solutions, and we compare the results to the Cramér-Rao bound of the variance.

Results: We first show that an unbiased attenuation pathlength estimator does not exist if very low mean detector signals (equivalently, very thick objects) are permitted. It is necessary to select a minimum mean detector signal for which unbiased behavior is desired. If we select two counts, the optimal estimator is similar to Li's estimator. If we select one count, the optimal estimator becomes non-monotonic. The oscillations cause the unbiased estimator to be noise amplifying. The nullspace of the constraint matrix is high-dimensional, so that unbiased solutions are not unique. The Cramér-Rao bound of the variance matches well with the expected scaling law and cannot be attained.

Conclusion: If arbitrarily thick objects are permitted, an unbiased attenuation pathlength estimator does not exist. If the maximum thickness is restricted, an unbiased estimator exists but is not unique. An optimal estimator can be selected that minimizes variance, but a bias-variance tradeoff exists where a larger domain of unbiased behavior requires increased variance.