Spline-based regularisation for discrete FBP reconstruction

Abstract

In this paper, we show that tomographic images are degraded by the unsuitable discretisation of continuous schemes, and the non-trivial null space in the case of angular sampling. Usually these two types of degradations are not studied separately. However, discretisation can be performed properly, while the null space is irreducible. For this reason, we study the relationships between continuous and discrete versions of a direct reconstruction method (FBP). They are characterized by an interpolation / sampling kernel, called the Pixel Intensity Distribution Model (PIDM). By defining the latter as B-spline functions, the existence and the uniqueness of the solution is guaranteed. It follows that projections must be oversampled. We test the robustness of this exact solution (for an infinite number of projections) by decreasing the number of angles. PIDM results are much better then FBP ones, showing that FBP reconstructed images are degraded not only by the null space, but also by unsuitable discretisation.

We also analyze the influence of degradations induced by an imaging device (mechanical instability and blur) and by projection noise in SPECT. Discretisation-related degradations depend on projection sampling. For this reason, proper oversampling is achieved when the corresponding degradations are negligible in comparison to the ones induced by the imaging device.

Our algorithm is constrained by the amount of information input to the system and controlled by the number of projection angles and the PIDM order. Optimal values of these parameters could be found for a predefined task using the ROC curve methodology.