Paths in Simple Polyhedrons

Abstract

Since the pioneering work by L. Cohen and R. Kimmel in 1997 on finding a contour as a minimal path between two endpoints, shortest paths in volume images have raised interest in computer vision and image analysis. This chapter considers the calculation of a Euclidean shortest path (ESP) in a 3D polyhedral space Π. We propose an approximate $\kappa(\varepsilon) \cdot{\mathcal{O}}(M|V|)$ 3D ESP algorithm, not counting time for preprocessing. The preprocessing time complexity equals ${\mathcal{O}}(M|E| + |{\mathcal{F}}| +|V|\log|V|)$ for solving a special, but ‘fairly general’ case of the 3D ESP problem, where Π does not need to be convex. V and E are the sets of vertices and edges of Π, respectively, and ${\mathcal{F}}$ is the set of faces (triangles) of Π. M is the maximal number of vertices of a so-called critical polygon, and κ(ε)=(L ₀−L)/ε where L ₀ is the length of an initial path and L is the true (i.e., optimum) path length. The given algorithm approximately solves three (previously known to be) NP-complete or NP-hard 3D ESP problems in time $\kappa(\varepsilon) \cdot{\mathcal{O}}(k)$, where k is the number of layers in a stack, which is introduced in this chapter as being the problem environment. The proposed approximation method has straightforward applications for ESP problems when analysing polyhedral objects (e.g., in 3D imaging), of for ‘flying’ over a polyhedral terrain.

An approximate answer to the right problem is worth a good deal more than an exact answer to an approximate problem.

John Tukey (1915–2000)