3D curve interpolation and object reconstruction

Three dimensional objects viewed as surfaces or volumes embedded in R³, are usually sampled along the z-dimension by planes for rendering or modeling purposes. The resulting intersections are curves or planar shapes which may in turn be modeled for parsimony of representation. Each curve or planar shape may be viewed as a point in a high dimensional manifold, thereby providing the notion of interpolation between two curves or two points on this manifold to reconstruct the subsurface that lies between the two slices. We exploit some recent results in formulating this interpolation problem as an optimization problem in R³ to yield a simple interpolating spline, known as elasticae, which when evaluated at intermediate points yields curves which can in turn be instrumental in 3D reconstruction. The approach is particularly suited for interpolation between MRI slices and for modeling and reconstruction of 3D shapes.