Three-dimensional reconstruction of vascular trees. Theory and methodology

In this paper we examine the few-view reconstruction problem as it applies to imaging vascular trees. A fully automated reconstruction algorithm is described that circumvents the traditional "correspondence problem," using only notions of consistency and connectivity. It is assumed that the vascular tree is a connected structure and that its centerlines have been identified in three or more images. The first of three steps in the procedure involves generating a connected structure that represents the multiplicity of solutions that are consistent with any two (different) projections. The second step assigns to each branch in this structure a measure of agreement based on its relationship with one or more additional views of the vasculature. The problem then becomes one of propagating this information, via connectivity relationships and consistency checks, throughout the above structure to distinguish between the branches comprising the imaged structure and the accompanying artifacts. In this paper we present the theory and methodology of the technique, while in a companion paper we address the issue of validation via simulations and experiments. Together, these papers shed some light on why ambiguities arise and often lead to errors in the few-view reconstruction problem. Strategies to handle these errors are described and results are presented that demonstrate the ability to obtain adequate reconstructions with as few as three distinct views.