Impact of the non-negativity constraint in model-based iterative reconstruction from CT data

Model-based iterative reconstruction is a promising approach to achieve dose reduction without affecting image quality in diagnostic x-ray computed tomography (CT). In the problem formulation, it is common to enforce non-negative values to accommodate the physical non-negativity of x-ray attenuation. Using this a priori information is believed to be beneficial in terms of image quality and convergence speed. However, enforcing non-negativity imposes limitations on the problem formulation and the choice of optimization algorithm. For these reasons, it is critical to understand the value of the non-negativity constraint. In this work, we present an investigation that sheds light on the impact of this constraint.

We primarily focus our investigation on the examination of properties of the converged solution. To avoid any possibly confounding bias, the reconstructions are all performed using a provably converging algorithm started from a zero volume. To keep the computational cost manageable, an axial CT scanning geometry with narrow collimation is employed. The investigation is divided into five experimental studies that challenge the non-negativity constraint in various ways, including noise, beam hardening, parametric choices, truncation, and photon starvation. These studies are complemented by a sixth one that examines the effect of using ordered subsets to obtain a satisfactory approximate result within 50 iterations. All studies are based on real data, which come from three phantom scans and one clinical patient scan. The reconstructions with and without the non-negativity constraint are compared in terms of image similarity and convergence speed. In select cases, the image similarity evaluation is augmented with quantitative image quality metrics such as the noise power spectrum and closeness to a known ground truth.

For cases with moderate inconsistencies in the data, associated with noise and bone-induced beam hardening, our results show that the non-negativity constraint offers little benefit. By varying the regularization parameters in one of the studies, we observed that sufficient edge-preserving regularization tends to dilute the value of the constraint. For cases with strong data inconsistencies, the results are mixed: the constraint can be both beneficial and deleterious; in either case, however, the difference between using the constraint or not is small relative to the overall level of error in the image. The results with ordered subsets are encouraging in that they show similar observations. In terms of convergence speed, we only observed one major effect, in the study with data truncation; this effect favored the use of the constraint, but had no impact on our ability to obtain the converged solution without constraint.

Our results did not highlight the non-negativity constraint as being strongly beneficial for diagnostic CT imaging. Altogether, we thus conclude that in some imaging scenarios, the non-negativity constraint could be disregarded to simplify the optimization problem or to adopt other forward projection models that require complex optimization machinery to be used together with non-negativity.