Approximation of 3D objects by piecewise linear geometric interpolants of their 1D cross-sections

In this paper we introduce a method for reconstruction of 3D objects from their 1D parallel cross-sections by set-valued interpolation. We regard a 3D object as the graph of a set-valued function defined on a planar domain, with the given 1D cross-sections as its samples. The method is based on a triangulation T of the sampling points in the planar domain and it is modular: for each triangle in T the corresponding 1D cross-sections are geometrically interpolated by a union of polyhedrons. The union of these interpolants over all triangles of T constitutes the approximating 3D object. Properties of this approximation are studied, in particular we derive the approximation order of the error, measured by the symmetric difference metric.