Spectral X-ray CT for Fast NDT Using Discrete Tomography

Abstract

We present progress in fast, high-resolution imaging, material classification, and fault detection using hyperspectral X-ray measurements. Classical X-ray CT approaches rely on data from many projection angles, resulting in long acquisition and reconstruction times. Additionally, conventional CT cannot distinguish between materials with similar densities. However, in additive manufacturing, the majority of materials used are known a priori. This knowledge allows to vastly reduce the data collected and increase the accuracy of fault detection. In this context, we propose an imaging method for non-destructive testing of materials based on the combination of spectral X-ray CT and discrete tomography. We explore the use of spectral X-ray attenuation models and measurements to recover the characteristic functions of materials in heterogeneous media with piece-wise uniform composition. We show by means of numerical simulation that using spectral measurements from a small number of angles, our approach can alleviate the typical deterioration of spatial resolution and the appearance of streaking artifacts.

Learned Spectral Computed Tomography

“…The Poisson log-likelihood is sometimes approximated by a weighted Euclidean distance, but it should be noted that this is only a good approximation for high photon counts [20]. For both the KL and the weighted Euclidean distance, the misfit terms are differentiable, but generally nonconvex due to the non-linearity [22], [10]. This poses a problem to optimisation methods, as they may fail depending on initialisation and may require tuning of algorithmic parameters [9], small step sizes with many iterations [22] or indeed solving additional subproblems [10].…”

“…For both the KL and the weighted Euclidean distance, the misfit terms are differentiable, but generally nonconvex due to the non-linearity [22], [10]. This poses a problem to optimisation methods, as they may fail depending on initialisation and may require tuning of algorithmic parameters [9], small step sizes with many iterations [22] or indeed solving additional subproblems [10]. A further complexity arises due to the addition of hand-crafted regularisation terms which aim to encode some prior information, such as smoothness or sparsity promoting, on the projections β.…”

“…Although the typical requirement for material decomposition is N b ≥ N , the above constraints can be used to decompose N + 1 materials using N b = N energy bins, as well as to constrain and increase the accuracy of the solution by eliminating a linear dependency. A special case is when q ki are restricted to take binary values, which can be used to cast the problem into the discrete tomography setting [22]. We note that these constraints will typically need to be enforced explicitly in the solution of the resulting optimisation problems via e.g.…”

“…We note that these constraints will typically need to be enforced explicitly in the solution of the resulting optimisation problems via e.g. proximal operators [22], which increases the computational complexity. Contrary to this, data-driven models such as the one proposed here are able to implicitly enforce any constraints by using training data which adhere to them.…”

“…The unmixing problem poses a greater challenge as it is non-linear and generally non-convex, which may cause convex optimisation methods to fail to converge to a global minimum if the initial guess is not optimal. Nevertheless, a variant of ADMM with nonlinear operator constraint [24] has shown in practice very good performance for a case of 3 materials [22], albeit with high computational costs. Formulations based on the Bregman distance provide a more robust alternative, but require the solution of subproblems which can increase the computational cost even further [9].…”