Introduction A natural and well-studied problem in algorithmic graph theory and network optimization is that of computing a "shortest path" between two nodes, s and t, in a graph whose edges have "weights" associated with them, and we consider the "length" of a path to be the sum of the weights of the edges that comprise it. Efficient algorithms are well known for this problem, as briefly summarized below. The shortest path problem takes on a new dimension when considered in a geometric domain. In contrast to graphs, where the encoding of edges is explicit, a geometric instance of a shortest path problem is usually specified by giving geometric objects that implicitly encode the graph and its edge weights. Our goal in devising efficient geometric algorithms is generally to avoid explicit construction of the entire underlying graph, since the full induced graph may be very large (even exponential in the input size, or infinite). Computing an optimal

In this survey we consider geometric techniques which have been used to measure the similarity or distance between shapes, as well as to approximate shapes, or interpolate between shapes. Shape is a modality which plays a key role in many disciplines, ranging from computer vision to molecular biology. We focus on algorithmic techniques based on computational geometry that have been developed for shape matching, simplification, and morphing. 1 Introduction The matching and analysis of geometric patterns and shapes is of importance in various application areas, in particular in computer vision and pattern recognition, but also in other disciplines concerned with the form of objects such as cartography, molecular biology, and computer animation. The general situation is that we are given two objects A, B and want to know how much they resemble each other. Usually one of the objects may undergo certain transformations like translations, rotations or scalings in order to be matched with th...

Object recognition, robotic vision, occluding noise removal or photograph design require the ability to perform disocclusion. We call disocclusion the recovery of hidden parts of objects in a digital image by interpolation from the vicinity of the occluded area. It is shown in this paper how disocclusion can be performed by means of level lines structure, which offers a reliable, complete and contrast-invariant representation of image, in contrast to edges. Level lines based disocclusion yields a solution that may have strong discontinuities, which is not possible with PDE-based interpolation. Moreover, the proposed method is fully compatible with Kanizsa's theory of "amodal completion".

We propose an optimal-time algorithm for a classical problem in plane computational geometry: computing a shortest path between two points in the presence of polygonal obstacles. Our algorithm runs in worst-case time O(n log n) and requires O(n log n) space, where n is the total number of vertices in the obstacle polygons. The algorithm is based on an efficient implementation of wavefront propagation among polygonal obstacles, and it actually computes a planar map encoding shortest paths from a fixed source point to all other points of the plane; the map can be used to answer singlesource shortest path queries in O(logn) time. The time complexity of our algorithm is a significant improvement over all previously published results on the shortest path problem. Finally, we also discuss extensions to more general shortest path problems, involving non-point and multiple sources. 1 Introduction 1.1 The Background and Our Result The Euclidean shortest path problem is one of the o...

We study several variations on one basic approach to the task of simplifying a plane polygon or subdivision: Fatten the given object and construct an approximation inside the fattened region. We investigate fattening by convolving the segments or vertices with disks and attempt to approximate objects with the minimum number of line segments, or with near the minimum, by using efficient greedy algorithms. We give some variants that have linear or O(n log n) algorithms approximating polygonal chains of n segments. We also show that approximating subdivisions and approximating with chains with no self-intersections are NP-hard.

We give an O(n log n)-time method for finding a best k-link piecewise-linear function approximating an n-point planar data set using the well-known uniform metric to measure the error, ffl 0, of the approximation. Our method is based upon new characterizations of such functions, which we exploit to design an efficient algorithm using a plane sweep in "ffl space" followed by several applications of the parametric searching technique. The previous best running time for this problem was O(n 2 ). 1 Introduction Approximating a set S = f(x 1 ; y 1 ); (x 2 ; y 2 ); : : : ; (x n ; y n )g of points in the plane by a function is a classic problem in applied mathematics. The general goals in this area of research are to find a function F belonging to a class of functions F such that each F 2 F is simple to describe, represent, and compute and such that the chosen F approximates S well. For example, one may desire that F be the class of linear or piecewise-linear functions, and, for any parti...

Every compact orientable boundaryless surface M can be cut along simple loops with a common point v0, pairwise disjoint except at v0, so that the resulting surface is a topological disk; such a set of loops is called a system of loops for M. The resulting disk may be viewed as a polygon in which the sides are pairwise identified on the surface; it is called a polygonal schema. Assuming that M is a combinatorial surface, and that each edge has a given length, we are interested in a shortest (or optimal) system of loops homotopic to a given one, drawn on the vertex-edge graph of M. We prove that each loop of such an optimal system is a shortest loop among all simple loops in its homotopy class. We give an algorithm to build such a system, which has polynomial running time if the lengths of the edges are uniform. As a byproduct, we get an algorithm with the same running time to compute a shortest simple loop homotopic to a given simple loop.

We consider the problem of approximating a polygonal chain C by another polygonal chain C ′ whose vertices are constrained to be a subset of the set of vertices of C. The goal is to minimize the number of vertices needed in the approximation C ′. Based on a framework introduced by Imai and Iri [25], we define an error criterion for measuring the quality of an approximation. We consider two problems. (1) Given a polygonal chain C and a parameter ε ≥ 0, compute an approximation of C, among all approximations whose error is at most ε, that has the smallest number of vertices. We present an O(n 4/3+δ)-time algorithm to solve this problem, for any δ>0; the constant of proportionality in the running time depends on δ. (2) Given a polygonal chain C and an integer k, compute an approximation of C with at most k vertices whose error is the smallest among all approximations with at most k vertices. We present a simple randomized algorithm, with expected running time O(n 4/3+δ), to solve this problem.

Given a set of obstacles and two distinguished points in the plane the problem of finding a collision free path subject to a certain optimization function is a fundamental problem that arises in many fields, such as motion planning in robotics, wire routing in VLSI and logistics in operations research. In this survey we emphasize its applications to VLSI design and limit ourselves to the rectilinear domain in which the goal path to be computed and the underlying obstacles are all rectilinearly oriented, i.e., the segments are either horizontal or vertical. We consider different routing environments, and various optimization criteria pertaining to VLSI design, and provide a survey of results that have been developed in the past, present current results and give open problems for future research. 1 Introduction Given a set of obstacles and two distinguished points in the plane, the problem of finding a collision free path subject to a certain optimization function is a fundamental probl...

The line simplification problem is an old and well-studied problem in cartography. Although there are several algorithms to compute a simplification, there seem to be no algorithms that perform line simplification in the context of other geographical objects. This paper presents a nearly quadratic time algorithm for the following line simplification problem: Given a polygonal line, a set of extra points, and a real ffl ? 0, compute a simplification that guarantees (i) a maximum error ffl, (ii) that the extra points remain on the same side of the simplified chain as of the original chain, and (iii) that the simplified chain has no self-intersections. The algorithm is applied as the main subroutine for subdivision simplification. 1 Introduction The line simplification problem is a well-studied problem in various disciplines including geographic information systems [Buttenfield '85, Cromley '88, Douglas & Peucker '73, Hershberger & Snoeyink '92, Li & Openshaw '92, McMaster '87], digital...