Analysis of beam hardening streaks in tomography

Abstract

The mathematical foundation of X-ray CT is based on the assumption that by measuring the attenuation of X-rays passing through an object, one can recover the integrals of the attenuation coefficient $\mu(x)$ along a sufficiently rich family of lines $L$, $\int_L \mu(x) \text{d} x$. This assumption is inaccurate because the energy spectrum of an X-ray beam in a typical CT scanner is wide. At the same time, the X-ray attenuation coefficient of most materials is energy-dependent, and this dependence varies among materials. Thus, reconstruction from X-ray CT data is a nonlinear problem. If the nonlinear nature of CT data is ignored and a conventional linear reconstruction formula is used, which is frequently the case, the resulting image contains beam-hardening artifacts such as streaks. In this work, we describe the nonlinearity of CT data using the conventional model accepted by all CT practitioners. Our main result is the characterization of streak artifacts caused by nonlinearity. We also obtain an explicit expression for the leading singular behavior of the artifacts. Finally, a numerical experiment is conducted to validate the theoretical results.