An approximation algorithm for computing minimum-length polygons in 3D images

Abstract

Length measurements in 3D images have raised interest in image geometry for a long time. This paper discusses the Euclidean shortest path (ESP) to be calculated in a loop of face-connected grid cubes in the 3D orthogonal grid, which are defined by minimum-length polygonal (MLP) curves. We propose a new approximation algorithm for computing such an MLP. It is much simpler and easier to understand and to implement than previously published algorithms by Li and Klette. It also has a straightforward application for finding an approximate minimum-length polygonal arc (MLA), a generalization of the MLP problem. We also propose two heuristic algorithms for computing a simple cube-arc within a 3D image component, with a minimum number of cubes between two cubes in this component. This may be interpreted as being an approximate solution to the general ESP problem in 3D (which is known as being NP-hard) assuming a regular subdivision of the 3D space into cubes of uniform size. © 2011 Springer-Verlag Berlin Heidelberg.