Practical Global Backprojection-Convolution in Transmission Cone-beam Computed Tomography

Abstract: Global backprojection-convolution (GBC) is a recently developed theory for exact reconstruction in transmission cone-beam computed tomography (CBCT). It is the first exact inversion theory that applies when the X-ray source points comprise a multidimensional `source locus' $X \subset \mathbb R^3$. Theoretically, GBC is computationally highly expedient due to its structure, but producing a practical computational implementation poses a significant challenge because the method is uniquely vulnerable to four sources of discretisation error: (1) accurate discretisation of a multidimensional locus requires more points than for a 1-dimensional locus, (2) the convolution kernel has infinite range and so the backprojected volume must be of infinite size, (3) the discrete convolution kernel cannot be computed in closed form, and (4) aliasing artefacts in the backprojection are enormously magnified by the convolution step. In this article, we propose an assortment of strategies to mitigate the discretisation errors, at the level of the symbolic algorithm. As a prototype, we deploy the concept on the case where $X$ is a cylinder. The resulting algorithm is evaluated through a series of reconstructions of the 3D Shepp-Logan phantom. As additional validation, we also briefly present a reconstruction from a real experimental dataset.