Fajie Li

Euclidean Shortest Paths

The introductory chapter explains the difference between shortest paths in finite graphs and shortest paths in Euclidean geometry, which is also called ‘the common geometry of our world’. The chapter demonstrates the diversity of such problems, defined between points in a plane, on a surface, or in the 3-dimensional space.

Approximate shortest paths in simple polyhedra

Since the pioneering work by (Cohen and Kimmel, 1997) on finding a contour as a minimal path between two end points, shortest paths in volume images have raised interest in computer vision and image analysis. This paper considers the calculation of a Euclidean shortest path (ESP) in a three-dimensional (3D) polyhedral space Π. We propose an approximate κ(e) ċ O(M|V|) 3D ESP algorithm, not counting time for preprocessing. The preprocessing time complexity equals O(M|E|+|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 F is the set of faces (triangles) of Π. M is the maximal number of vertices of a so-called critical polygon, and κ(e) = (L0 - L)/e where L0 is the length of an initial path and L is the true (i.e., optimum) path length. The given algorithm solves approximately three (previously known to be) NP-complete or NP-hard 3D ESP problems in time κ(e)ċO(k), where k is the number of layers in a stack, which is introduced in this paper as being the problem environment. The proposed approximation method has straightforward applications for ESP problems when analyzing polyhedral objects (e.g., in 3D imaging), of for flying over a polyhedral terrain.

Convex Hulls in the Plane

Convex hulls in the plane are examples for shortest paths around sets of points, or around simple polygons. Possibly these paths may be constrained by available polygonal regions. This chapter explains a few exact algorithms in this area which run typically in linear or (nlogn)-time with respect to a given input parameter n. However, the problems could also be solved approximately by rubberband algorithms.

An approximate algorithm for solving shortest path problems for mobile robots or driver assistance

Finding a shortest path between two given locations is of importance for mobile robots, but also (e.g.) for identifying unique paths in a given surrounding region Π when (e.g.) evaluating vision software in test vehicles, or for calculating the free-space boundary in vision-based driver assistance. We assume that Π is given as a triangulated surface which is not necessary simply connected.

A fast algorithm for liver surgery planning

Assume that a simplified liver model consists of some vein cells and liver cells. Such a liver model contains two kinds of components, the vein component and the liver components, each of them consists of cells which are 26-connected. The vein component has a tree-shape topology. Suppose that the vein component has already been cut into two parts, and one of them is diseased. Liver surgery planning systems need to design an algorithm to decompose the liver components into two kinds of subsets, one (usually just one component) that has been affected by the diseased vein component while the other one is still healthy. So far, existing algorithms depend heavily on surgeons personal expertise to detect the diseased liver component which needs to be removed. We propose an efficient algorithm for computing the diseased liver component which is based on the diseased vein component, and not on surgeons personal manipulations.

Paths on Surfaces

This chapter presents two RBAs for the calculation of an ESP on the surface of a convex or a general polyhedron Π. Solutions are restricted by specified constraints. First, we consider a convex polyhedron and provide a (kappa(varepsilon)cdot {mathcal{O}}(kn log n) ) RBA for computing a restricted solution. In this formula, k is the number of polygonal cuts between source and target point, and n is the number of edges of Π. Second, we consider the surface of a general polyhedron Π and provide a (kappa_{1}(varepsilon) cdot kappa_{2}(varepsilon) cdot {mathcal{O}}(n^{2}) ) RBA for computing a restricted solution for the surface ESP problem. In this formula, n is again the number of vertices of Π and κ i (e)=(L 0 i −L i )/e, for i=1 or i=2, where L 1 is the length of a shortest path, L 0 1 the length of the initial path, L 2 the length of a restricted shortest path, and L 0 2 the length of an initial path for the restricted path calculation. Both proposed RBAs are easy to implement. Applications are, for example, in 3D object analysis in biomedical or industrial imaging.

Paths in Cube-Curves

This chapter discusses a problem defined in a 3-dimensional regular grid. Such a grid is commonly used in 3D image analysis. We may also assume that a general 3D space (e.g., for a robot) is regularly subdivided into cubes of uniform size. The chapter considers shortest paths in such a cuboidal world.

Deltas and Epsilons

The introduction ended with recalling concepts in discrete mathematics as used in this book. This second chapter adds further basic concepts in continuous mathematics that are also relevant for this book, especially in the context of approximate algorithms.

Partitioning a Polygon or the Plane

The chapter describes algorithms for partitioning a simple polygon into trapezoids or triangles (Seidel’s triangulation and an algorithm using up- and down-stable vertices). Chazelle’s algorithm, published in 1991 and claimed to be of linear time, is often cited as a reference, but this algorithm was never implemented; the chapter provides a brief presentation and discussion of this algorithm. This is followed by a novel procedural presentation of Mitchell’s continuous Dijkstra algorithm for subdividing the plane into a shortest-path map for supporting queries about distances to a fixed start point in the presence of polygonal obstacles.

Safari and Zookeeper Problems

So far, the best result in running time for solving the fixed safari route problem (SRP) is ({mathcal{O}}(n^{2} log n)) published in 2003 by M. Dror, A. Efrat, A. Lubiw, and J. Mitchell. The best result in running time for solving the floating zookeeper route problem (ZRP) is ({mathcal{O}}(n^{2})) published in 2001 by X. Tan. This chapter provides an algorithm for the “floating” SRP with (kappa(varepsilon) cdot{mathcal{O}}(kn + m_{k})) runtime, where n is the number of vertices of the given search space or domain D (a simple polygon), k the number of convex polygons P i in D, and m k is the total number of vertices of all polygons P i . This chapter also provides an algorithm for the floating ZRP with (kappa(varepsilon) cdot{mathcal{O}}(kn)) runtime, where n is the number of vertices of all polygons involved, and k the number of the “cages”. Extensions of the algorithms presented can solve more general SRPs and ZRPs if each convex polygon is replaced by a convex region such as convex polybeziers (beziergons) or ellipses.