Estimation of Basis Line-Integrals in a Spectral Distortion-Modeled Photon Counting Detector Using Low-Order Polynomial Approximation of X-ray Transmittance

Photon counting detector (PCD)-based computed tomography exploits spectral information from a transmitted x-ray spectrum to estimate basis line-integrals. The recorded spectrum, however, is distorted and deviates from the transmitted spectrum due to spectral response effect (SRE). Therefore, the SRE needs to be compensated for when estimating basis line-integrals. One approach is to incorporate the SRE model with an incident spectrum into the PCD measurement model and the other approach is to perform a calibration process that inherently includes both the SRE and the incident spectrum. A maximum likelihood estimator can be used to the former approach, which guarantees asymptotic optimality; however, a heavy computational burden is a concern. Calibration-based estimators are a form of the latter approach. They can be very efficient; however, a heuristic calibration process needs to be addressed. In this paper, we propose a computationally efficient three-step estimator for the former approach using a low-order polynomial approximation of x-ray transmittance. The low-order polynomial approximation can change the original non-linear estimation method to a two-step linearized approach followed by an iterative bias correction step. We show that the calibration process is required only for the bias correction step and prove that it converges to the unbiased solution under practical assumptions. Extensive simulation studies validate the proposed method and show that the estimation results are comparable to those of the ML estimator while the computational time is reduced substantially.