We propose the idea of simplification envelopes for generating a hierarchy of level-of-detail approximations for a given polygonal model. Our approach guarantees that all points of an approximation are within a user-specifiable distance # from the original model and that all points of the original model are within a distance # from the approximation. Simplificationenvelopes provide a general framework within which a large collection of existing simplification algorithms can run. We demonstrate this technique in conjunction with two algorithms, one local, the other global. The local algorithm provides a fast method for generating approximations to large input meshes (at least hundreds of thousands of triangles). The global algorithm provides the opportunity to avoid local "minima" and possibly achieve better simplifications as a result. Each approximation attempts to minimize the total number of polygons required to satisfy the above # constraint. The key advantages of our approach are...

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 present a simple,robust, and practical method for object simplification for applications where gradual elimination of high frequency details is desired. This is accomplished by converting an object into multi-resolution volume rastersusing a controlled filtering and sampling technique.Amultiresolution triangle-mesh hierarchycan then be generated by applying the Marching Cubes algorithm. We f urther propose an adaptive surface generation algorithm to reduce the number of triangles generated by the standardMarching Cubes. Our method simplifies the topology of objects in a controlled fashion. In addition, at eachlevel of detail, multi-layered meshes can be used for an efficient antialiased rendering.

This dissertation explores some techniques for automatic approximation of geometric objects. My thesis is that using and extending concepts from computational geometry can help us in devising efficient and parallelizable algorithms for automatically constructing useful detail hierarchies for geometric objects. We have demonstrated this by developing new algorithms for two kinds of geometric approximation problems that have been motivated by a single driving problem --- the efficient computation and display of smooth solvent-accessible molecular surfaces. The applications of these detail hierarchies are in biochemistry and computer graphics. The smooth solvent-accessible surface of a molecule is useful in studying the structure and interactions of proteins, in particular for attacking the protein-substrate docking problem. We have developed a parallel linear-time algorithm for computing molecular surfaces. Molecular surfaces are equivalent to the weighted ff-hulls. Thus our work is pot...

Predicate abstraction has emerged to be a powerful technique for extracting finite-state models from infinite-state systems, and has been recently shown to enhance the effectiveness of the reachability computation techniques for hybrid systems. Given a hybrid system with linear dynamics and a set of linear predicates, the verifier performs an on-the-fly search of the finite discrete quotient whose states correspond to the truth assignments to the input predicates. The success of this approach crucially depends on the choice of the predicates used for abstraction. In this paper, we focus on identifying these predicates automatically by analyzing spurious counter-examples generated by the search in the abstract state-space. We present the basic techniques for discovering new predicates that will rule out closely related spurious counter-examples, optimizations of these techniques, implementation of these in the verification tool, and case studies demonstrating the promise of the approach.

Given a family of disjoint polygons P1, P2, : ::, Pk in the plane, and an integer parameter m, it is NP-complete to decide if the Pi's can be pairwise separated by a polygonal family with at most m edges, that is, if there exist polygons R1; R2; : ::; Rk with pairwise-disjoint boundaries such that Pi Ri andP jRij m. In three dimensions, the problem is NP-complete even for two nested convex polyhedra. Many other extensions and generalizations of the polyhedral separation problem, either to families of polyhedra or to higher dimensions, are also intractable. In this paper, we present e cient approximation algorithms for constructing separating families of near-optimal size. Our main results are as follows. In two dimensions, we give an O(n log n) time algorithm for constructing a separating family whose size is within a constant factor of an optimal separating family; n is the number of edges in the input family of polygons. In three dimensions, we show how to separate a convex polyhedron from a nonconvex polyhedron with a polyhedral surface whose facet-complexity is O(log n) times the optimal, where n = jPj+jQj is the complexity of the input polyhedra. Our algorithm runs in O(n4) time, but improves to O(n3) time if the two polyhedra are nested and convex. Our algorithm for separating a convex polyhedron from a nonconvex polyhedron extends to higher dimensions. In d dimensions, for d 4, the facet-complexity of the approximation polyhedron is O(d log n) times the optimal, and the algorithm runs in O(nd+1) time. Finally, we also obtain results on separating sets of points, a family of convex polyhedra, and separation by non-polyhedral surfaces, such as spherical patches.

We show that several well-known optimization problems involving 3-dimensional convex polyhedra and decision trees are NP-hard or NP-complete. One of the techniques we employ is a linear-time method for realizing a planar 3-connected triangulation as a convex polyhedron, which may be of independent interest. Key words: Convex polyhedra, approximation, Steinitz's theorem, planar graphs, art gallery theorems, decision trees. 1 Introduction Convex polyhedra are fundamental geometric structures (e.g., see [20]). They are the product of convex hull algorithms, and are key components for problems in robot motion planning and computer-aided geometric design. Moreover, due to a beautiful theorem of Steinitz [20, 38], they provide a strong link between computational geometry and graph theory, for Steinitz shows that a graph forms the edge structure of a convex polyhedra if and only if it is planar and 3-connected. Unfortunately, algorithmic problems dealing with 3-dimensional convex polyhedra ...

In computer graphics and solid modeling, one is interested in representing complex geometric objects with combinatorially simpler ones. It turns out that via a “fattening ” transformation, one obtains a formulation of the approximation problem in terms of separation: Find a minimumcomplexity surface that separates two sets. In this paper, we provide approximation algorithms for several geometric separation problems, including: • Given a set of triangles T and a set S of points that lie within the union of the triangles, find a minimum-cardinality set, T ′ , of pairwise-disjoint triangles, each contained within some triangle of T, that cover the point set S. • Given finite sets of “red ” and “blue ” points in the plane, determine a simple polygon of fewest edges that separates the red points from the blue points. More generally, given finite sets of points of many color classes, determine a planar “separating ” subdivision of minimum combinatorial complexity, which has the property that each face of the subdivision contains points of at most one color class; • Given two polyhedral terrains, P and Q, over a common support set (e.g., the unit square), with P lying above Q, compute a nested polyhedral terrain R that lies between P and Q such that R has a minimum number of facets. Exact solution of the above problems in polynomial time is highly unlikely: The decision versions of all three problems are known to be NP-hard. We provide polynomial-time algorithms that are guaranteed to produce an answer within a logarithmic factor (O(log n), where n is the complexity of the input problem instance) of optimal. (The error factor is constant in the orthogonal case — coverage by disjoint aligned rectangles, or separation of orthohedral terrains.) We also discuss extensions to higher dimensions. 1

A circle C separates two planar sets if it encloses one of the sets and its open interior disk does not meet the other set. A separating circle is a largest one if it cannot be locally increased while still separating the two given sets. An \Theta(n log n) optimal algorithm is proposed to find all largest circles separating two given sets of line segments when line segments are allowed to meet only at their endpoints. In the general case, when line segments may intersect\Omega\Gamma n 2 ) times, our algorithm can be adapted to work in O(nff(n) log n) time and O(nff(n)) space, where ff(n) represents the extremely slowly growing inverse of the Ackermann function.