This paper surveys methods for simplifying and approximating polygonal surfaces. A polygonal surface is a piecewiselinear surface in 3-D defined by a set of polygons

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...

Route maps, which depict a path from one location to another, have emerged as one of the most popular applications on the Web. Current computer-generated route maps, however, are often very difficult to use. In this paper we present a set of cartographic generalization techniques specifically designed to improve the usability of route maps. Our generalization techniques are based both on cognitive psychology research studying how route maps are used and on an analysis of the generalizations commonly found in handdrawn route maps. We describe algorithmic implementations of these generalization techniques within BeeLine, a real-time system for automatically designing and rendering route maps. We show that Bee-Line produces route maps that are much more usable than those produced by current computer-based route map rendering systems. Feedback from over 1100 users indicates that over 99 % believe BeeLine maps are preferable to using standard computer-generated route maps alone.

A new hierarchical triangle-based model for representing surfaces over sampled data is proposed, which is based on the subdivision of the surface domain into nested triangulations, called a Hierarchical Triangulation (HT). The model allows compression of spatial data and representation of a surface at successively finer degrees of resolution. An HT is a collection of triangulations organized in a tree, where each node, except for the root, is a triangulation refining a face belonging to its parent in the hierarchy. We present a topological model for representing an HT, and algorithms for its construction and for the extraction of a triangulation at a given degree of resolution. The surface model, called a Hierarchical Triangulated Surface (HTS), is obtained by associating data values with the vertices of triangles, and defining suitable functions that describe the surface over each triangular patch. We consider an application of a piecewise-linear version of the HTS to interpolate topo...

A video sequence can be represented as a trajectory curve in a high dimensional feature space. This video curve can be analyzed by tools similar to those developed for planar curves. In particular, the classic binary curve splitting algorithm has been found to be a useful tool for video analysis. With a splitting condition that checks the dimensionality of the curve segment being split, the video curve can be recursively simplified and represented as a tree structure, and the frames that are found to be junctions between curve segments at different levels of the tree can be used as keyframes to summarize the video sequences at different levels of detail. These keyframes can be combined in various spatial and temporal configurations for browsing purposes. We describe a simple video player that displays the keyframes sequentially and lets the user change the summarization level on the fly with a slider. We also describe an approach to automatically selecting a summarization level that pr...

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 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.

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...

Decomposition is a technique commonly used to partition complex models into simpler components. While decomposition into convex components results in pieces that are easy to process, such decompositions can be costly to construct and can result in representations with an unmanageable number of components. In this paper, we explore an alternative partitioning strategy that decomposes a given model into “approximately convex ” pieces that may provide similar benefits as convex components, while the resulting decomposition is both significantly smaller (typically by orders of magnitude) and can be computed more efficiently. Indeed, for many applications, an approximate convex decomposition (ACD) can more accurately represent the important structural features of the model by providing a mechanism for ignoring less significant features, such as surface texture. We describe a technique for computing ACDs of three-dimensional polyhedral solids and surfaces of arbitrary genus. We provide results illustrating that our approach results in high quality decompositions with very few components and applications showing that comparable or better results can be obtained using ACD decompositions in place of exact convex decompositions (ECD) that are several orders of magnitude larger. 1 ECD Figure 1: The approximate convex decompositions (ACD) of the Armadillo and the David models consist of a small number of nearly convex components that characterize the important features of the models better than the exact convex decompositions (ECD) that have orders of magnitude more components. The Armadillo (500K edges, 12.1MB) has a solid ACD with 98 components (14.2MB) that was computed in 232 seconds while the solid “ECD ” has more than 726,240 components (20+ GB) and could not be completed because disk space was exhausted after nearly 4 hours of computation. The David (750K edges, 18MB) has a surface ACD with 66 components (18.1MB) while the surface ECD has 85,132 components (20.1MB). 1

A constructive solid geometry (CSG) conversion for a polygon takes a list of vertices and produces a formula representing the polygon as an intersection and union of primitive halfspaces. The cartographers' favorite line simplification algorithm recursively selects from a list of data points those to be used to represent a linear feature, such as a coastline, on a map. By using a data structure that maintains convex hulls of polygonal lines under splits, both were known to have O(n log n) time solutions in the worst-case. This paper shows that both are easier than sorting by presenting an O(n log* n) algorithm for maintaining convex hulls under splits at extreme points. It opens the question of whether there are practical, linear-time solutions to these problems.