Calibration‐free beam hardening reduction in x‐ray CBCT using the epipolar consistency condition and physical constraints

Abstract

Background The beam hardening effect is a typical source of artifacts in x‐ray cone beam computed tomography (CBCT). It causes streaks in reconstructions and corrupted Hounsfield units toward the center of objects, widely known as cupping artifacts. Purpose We present a novel efficient projection data‐based method for reduction of beam‐hardening artifacts and incorporate physical constraints on the shape of the compensation functions. The method is calibration‐free and requires no additional knowledge of the scanning setup. Method The mathematical model of the beam hardening effect caused by a single material is analyzed. We show that the effect of beam hardening on the resulting functions on the line integral measurements are monotonous and concave functions of the ideal data. This holds irrespective of any limiting assumptions on the energy dependency of the material, the detector response or properties of the x‐ray source. A regression model for the beam hardening effect respecting these theoretical restrictions is proposed. Subsequently, we present an efficient method to estimate the parameters of this model directly in projection domain using an epipolar consistency condition. Computational efficiency is achieved by exploiting the linearity of an intermediate function in the formulation of our optimization problem. Results Our evaluation shows that the proposed physically constrained ECC2 algorithm is effective even in challenging measured data scenarios with additional sources of inconsistency. Conclusions The combination of mathematical consistency condition and a compensation model that is based on the properties of x‐ray physics enables us to improve image quality of measured data retrospectively and to decrease the need for calibration in a data‐driven manner.

Calibration by differentiation – Self‐supervised calibration for X‐ray microscopy using a differentiable cone‐beam reconstruction operator

“…It can potentially converge towards physically implausible results if the hyperparameters such as the learning rate are not controlled carefully. This can be counteracted by introducing further constraints such as monotonicity and convexity of the polynomial 48 . However, PyTorch and comparable deep learning frameworks are primarily designed for unconstrained optimization of highly parametrized models and might not be perfectly suited for strongly constrained settings.…”

Comparative Evaluation of Cone-Beam Consistency Conditions for Mono-Material Beam Hardening Correction in Flat Detector Computed Tomography

“…It can be assumed that the second-degree polynomial for mono-material correction is convex and monotonically increasing in the interval 0, g max p [26]. Hence, α 2 ≥ 0 and the abscissa of the parabola's vertex is ≤ 0.…”

“…In clinical CT, the polynomial coefficients are estimated using homogeneous water phantoms during the scanner's calibration [22] [23]. Instead of tedious and recurrent calibrations, the optimal polynomial coefficients can be estimated using DCCs as described in [24] [25] [26]. The key idea here is to optimize the polynomials by minimizing the inconsistencies due to the violation of the linear forward model of projection generation caused by polychromatic X-ray attenuation.…”

Simple beam hardening correction method (2DCalBH) based on 2D linearization

“…However, the former method has been shown to not completely correct the dark bands (Jin et al 2015) while the latter method showed overcompensation for the dark bands in real studies (Schuller et al 2015). Recently, in the industrial field, two different methods proposed to use epipolar consistency conditions to reduce beam hardening artifacts (Würfl et al 2019, Würfl 2020). However, these methods produce a change in the soft-tissue texture and the authors are unsure about its performance in clinical CT.…”