The specific problem (recovering the 3D geometry of vessel axes from images pairs) seems little researched. Work of direct relevance comes from Wahle et. al. [2], who cite Parker et. al [3] as an influence. Wahle's algorithm generates a matrix of minimum distances between rays back-projected from each pixel in each image, it then picks out a minimum-cost path through the matrix. The method cannot cope with vessels that appear to self-intersect, and is easily disturbed by errors in the input data. Nor does it cope well with sections of the vessel which run along an epipolar plane. Finally, it produces one answer, and there is no guarantee it will be correct in cases of ambiguity. The main problem is that the connectivity of the vessel is implicitly coded into the matrix. A positive feature is that it utilises a weakened epipolar constraint (pixels whose back-projections most nearly cross are assumed to correspond). Other researchers reconstruct blood vessel axes by simultaneously tracking centre lines in each image [4]. We track by finding a path though a graph that represents both images. Others combine the reconstructing a vessel's centre line and cross-section [5], but is too wide a focus for us, we are interested only in centre-lines. None of these algorithms copes with (apparent) self-intersection. Barba et. al. [6] provide an algorithm whose examples do include a self-intersection. However, exactly how they resolve ambiguities is unclear.
More generally, computer vision researchers have looked at recovering the shape of rigid space curves. Papadopoulo and Faugeras [7] do so from a sequence of images. Our work is not premised on particular views, and might feasibly be used over image sequences. However, the work just mentioned would be the algorithm of choice for such problems.
Our problem shares features with work in CADCAM. Lequette [8] describes an algorithm that finds all solids which fit a set of two of three orthogonal images. The images show the object as a wireframe, with no hidden lines. Our algorithm in no way depends on its motivating domain, and may find use in other areas, such as CADCAM. We concentrate on a recovering the 3D shape of a single space curve.