Homotopic Object Reconstruction Using Natural Neighbor Barycentric Coordinates

One of the challenging problems in computer vision is object reconstruction from cross sections. In this paper, we address the problem of 2D object reconstruction from arbitrary linear cross sections. This problem has not been much discussed in the literature, but holds great importance since it lifts the requirement of order within the cross sections in a reconstruction problem, consequently making the reconstruction problem harder. Our approach to the reconstruction is via continuous deformations of line intersections in the plane. We define Voronoi diagram based barycentric coordinates on the edges of n-sided convex polygons as the area stolen by any point inside a polygon from the Voronoi regions of each open oriented line segment bounding the polygon. These allow us to formulate homotopies on edges of the polygons from which the underlying object can be reconstructed. We provide results of the reconstruction including the necessary derivation of the gradient at polygon edges.