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

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 ECC 2 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.

Epipolar Consistency in Transmission Imaging.

This paper presents the derivation of the Epipolar Consistency Conditions (ECC) between two X-ray images from the Beer-Lambert law of X-ray attenuation and the Epipolar Geometry of two pinhole cameras, using Grangeat's theorem. We motivate the use of Oriented Projective Geometry to express redundant line integrals in projection images and define a consistency metric, which can be used, for instance, to estimate patient motion directly from a set of X-ray images. We describe in detail the mathematical tools to implement an algorithm to compute the Epipolar Consistency Metric and investigate its properties with detailed random studies on both artificial and real FD-CT data. A set of six reference projections of the CT scan of a fish were used to evaluate accuracy and precision of compensating for random disturbances of the ground truth projection matrix using an optimization of the consistency metric. In addition, we use three X-ray images of a pumpkin to prove applicability to real data. We conclude, that the metric might have potential in applications related to the estimation of projection geometry. By expression of redundancy between two arbitrary projection views, we in fact support any device or acquisition trajectory which uses a cone-beam geometry. We discuss certain geometric situations, where the ECC provide the ability to correct 3D motion, without the need for 3D reconstruction.