Dynamic multi-source X-ray tomography using a spacetime level set method

Abstract

A novel variant of the level set method is introduced for dynamic X-ray tomography. The target is allowed to change in time while being imaged by one or several source–detector pairs at a relatively high frame-rate. The algorithmic approach is motivated by the results in [22], showing that the modified level set method can tolerate highly incomplete projection data in stationary tomography. Furthermore, defining the level set function in spacetime enforces temporal continuity in the dynamic tomography context considered here. The tomographic reconstruction is found as a minimizer of a nonlinear functional. The functional contains a regularization term penalizing the L2 norms of up to n derivatives of the reconstruction. The case n=1 is shown to be equivalent to a convex Tikhonov problem that has a unique minimizer. For n≥2 the existence of a minimizer is proved under certain assumptions on the signal-to-noise ratio and the size of the regularization parameter. Numerical examples with both simulated and measured dynamic X-ray data are included, and the proposed method is found to yield reconstructions superior to standard methods such as FBP or non-negativity constrained Tikhonov regularization and favorably comparable to those of total variation regularization. Furthermore, the methodology can be adapted to a wide range of measurement arrangements with one or more X-ray sources.