High Order Reconstruction from Cross-Sections

Given parallel cross-sections data of a smooth object in IRd, one of the ways of generating approximation to the object is by interpolating the signed-distance functions corresponding to the cross-sections. This well-known method is useful in many applications, and yet its approximation properties are not fully established. The known result is that away from cross-sections that are parallel to the boundary of the object, this method gives high approximation order. However, near such tangent cross-sections the approximation order is drastically reduced. This is due to the singular behaviour of the signed-distance function near tangent cross-sections. In this paper we suggest a way to restore the high approximation order everywhere. The new method involves a recent development in the approximation of functions with singularities. We present the application of this approach to our case, analyze its approximation properties, and discuss the numerical issues involved.