A study of modeling x-ray transmittance for material decomposition without contrast agents

This study concerns how to model x-ray transmittance, exp ( -- ∫ μ a ( r , E ) d r ), of the object using a small number of energy-dependent bases, which plays an important role for estimating basis line-integrals in photon counting detector (PCD)-based computed tomography (CT). Recently, we found that low-order polynomials can model the smooth x-ray transmittance, i.e. object without contrast agents, with sufficient accuracy, and developed a computationally efficient three-step estimator. The algorithm estimates the polynomial coefficients in the first step, estimates the basis line-integrals in the second step, and corrects for bias in the third step. We showed that the three-step estimator was ~1,500 times faster than conventional maximum likelihood (ML) estimator while it provided comparable bias and noise. The three-step estimator was derived based on the modeling of x-ray transmittance; thus, the accurate modeling of x-ray transmittance is an important issue. For this purpose, we introduce a modeling of the x-ray transmittance via dictionary learning approach. We show that the relative modeling error of dictionary learning-based approach is smaller than that of the low-order polynomials.