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...

We review the recent progress in the design of efficient algorithms for various problems in geometric optimization. We present several techniques used to attack these problems, such as parametric searching, geometric alternatives to parametric searching, prune-and-search techniques for linear programming and related problems, and LPtype problems and their efficient solution. We then describe a variety of applications of these and other techniques to numerous problems in geometric optimization, including facility location, proximity problems, statistical estimators and metrology, placement and intersection of polygons and polyhedra, and ray shooting and other query-type problems.

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.

We propose a new approach to the automatic generation of triangular irregular networks from dense terrain models. We have developed and implemented an algorithm based on the greedy principle used to compute minimum-link paths in polygons. Our algorithm works by taking greedy cuts ("bites") out of a simple closed polygon that bounds the yet-to-be triangulated region. The algorithm starts with a large polygon, bounding the whole extent of the terrain to be triangulated, and works its way inward, performing at each step one of three basic operations: ear cutting, greedy biting, and edge splitting. We give experimental evidence that our method is competitive with current algorithms and has the potential to be faster and to generate many fewer triangles. Also, it is able to keep the structural terrain fidelity at almost no extra cost in running time and it requires very little memory beyond that for the input height array. 1 Introduction A terrain is the graph of a function of two variabl...

Given an n-vertex polygonal curve P = [p 1, p 2, : ::, pn] in the 2-dimensional space R 2, we consider the problem of approximating P by finding another polygonal curve P 0

A new framework for analyzing sequential or temporal data such as time series is proposed. It differs from other approaches by the special emphasis on the interpretability of the results, since interpretability is of vital importance for knowledge discovery, that is, the development of new knowledge (in the head of a human) from a list of discovered patterns. While traditional approaches try to model and predict all time series observations, the focus in this work is on modelling local dependencies in multivariate time series. This

Let P be an x-monotone polygonal path in the plane. For a path Q that approximates P let WA(Q) be the area above P and below Q, and let WB(Q) be the area above Q and below P. Given P and an integer k, we show how to compute a path Q with at most k edges that minimizes WA(Q)+WB(Q). Given P and a cost C, we show how to find a path Q with the smallest possible number of edges such that WA(Q) + WB(Q) ≤ C. However, given P, an integer k, and a cost C, it is NP-hard to determine if a path Q with at most k edges exists such that max{WA(Q), WB(Q)} ≤ C. We describe an approximation algorithm for this setting. Finally, it is also NP-hard to decide whether a path Q exists such that |WA(Q) − WB(Q) | = 0. Nevertheless, in this error measure we provide an algorithm for computing an optimal approximation up to an additive error. 1

Fitting a curve of a certain type to a given set of points in the plane is a basic problem in statistics and has numerous applications. We consider fitting a polyline with k joints under the min-sum criteria with respect to L1- and L2-metrics, which are more appropriate measures than uniform and Hausdorff metrics in statistical context. We present efficient algorithms for the 1-joint versions of the problem and fully polynomial-time approximation schemes for the general k-joint versions. 1

Abstract. We investigate the problem of creating simplified representations of polygonal paths. Specifically, we look at a path simplification problem in which line segments of a simplification are required to conform with a restricted set of directions C. An algorithm is given to compute such simplified paths in O(|C | 3 n 2) time, where n is the number of vertices in the original path. This result is extended to produce an algorithm for graphs induced by multiple intersecting paths. The algorithm is applied to construct schematised representations of real world railway networks, in the style of metro maps. 1

Given a polygonal path P with vertices p1, p2,..., pn ∈ Rd and a real number t ≥ 1, a path Q = (pi1, pi2,..., pik) is a t-distance-preserving approximation of P if 1 = i1 < i2 <... < ik = n and each straight-line edge (pij, pij+1) of Q approximates the distance between pij and pij+1 along the path P within a factor of t. We present exact and approximation algorithms that compute such a path Q that minimizes k (when given t) or t (when given k). We also present some experimental results. 1