Three-dimensional reconstruction of curves from pairs of projection views in the presence of error. I. Algorithms.

We have previously described an approach to 3D intracerebral vascular reconstruction that uses an MRA as a reconstruction base. Additional vessels seen only by angiography are added by segmenting 2D curves from projection angiograms and reconstructing these curves into 3D, building upon the MRA. Intracerebral vascular reconstruction is difficult for at least two reasons. First, 2D curves must be associated on projection images even when the human eye cannot do so. Second, 3D curves must be reconstructed in the presence of errors such as misregistration, image distortion, and misdefinition of 2D curves. This paper is the first of two that address the specific issue of reconstruction of a 3D curve from a given pair of 2D curves in the presence of error. The method explicitly separates what can and cannot be determined from a pair of projection views. It is also capable of recognizing interruptions produced by viewplane errors, of continuing reconstruction beyond such interruptions, and of localizing and estimating the magnitude of the interruptions. These measurements can also be used to estimate the lengths of regional disparities between a pair of 2D curves, leading to a quantitative estimate of the capacity of a pair of 2D curves to combine to create a 3D object (match value). Match values can be used, in turn, as part of the strategy for automatically associating pairs of 2D curves. This paper provides methods for reconstructing a given pair of 2D curves into 3D in the presence of error and for calculating match values. Error analysis is given in the companion report.