Joseph O'Rourke

Model-based image analysis of human motion using constraint propagation

A system capable of analyzing image sequences of human motion is described. The system is structured as a feedback loop between high and low levels: predictions are made at the semantic level and verifications are sought at the image level. The domain of human motion lends itself to a model-driven analysis, and the system includes a detailed model of the human body. All information extracted from the image is interpreted through a constraint network based on the structure of the human model. A constraint propagation operator is defined and its theoretical properties outlined. An implementation of this operator is described, and results of the analysis system for short image sequences are presented.

Some NP-hard polygon decomposition problems

The inherent computational complexity of polygon decomposition problems is of theoretical interest to researchers in the field of computational geometry and of practical interest to those working in syntactic pattern recognition. Three polygon decomposition problems are shown to be NP-hard and thus unlikely to admit efficient algorithms. The problems are to find minimum decompositions of a polygonal region into (perhaps overlapping) convex, star-shaped, or spiral subsets. We permit the polygonal region to contain holes. The proofs are by transformation from Boolean three-satisfiability, a known NP-complete problem. Several open problems are discussed.

Finding minimal enclosing boxes

The problem of finding minimal volume boxes circumscribing a given set of three-dimensional points is investigated. It is shown that it is not necessary for a minimum volume box to have any sides flush with a face of the convex hull of the set of points, which makes a naive search problematic. Nevertheless, it is proven that at least two adjacent box sides are flush with edges of the hull, and this characterization enables anO(n3) algorithm to find all minimal boxes for a set ofn points.

Decomposition of Three-Dimensional Objects into Spheres

Algorithms are presented for converting between different three-dimensional object representations: from a collection of cross section outlines to surface points, and from surface points to a collection of overlapping spheres. The algorithms effect a conversion from surface representations (outlines or surface points) to a volume representation (spheres). The spherical representation can be useful for graphical display, and perhaps as an intermediate representation for conversions to representations with other primitives. The spherical decomposition also permits the computation of points on the symmetric surface of an object, the three-dimensional analog of Blums symmetric axis. The algorithms work in real coordinates rather than in a discrete space, and so avoid error introduced by the quantization of the space.

Worst-case optimal algorithms for constructing visibility polygons with holes

EIGindy and Avis [EA] considered the problem of determining the visibility polygon from a point inside a polygon. Their algorithm runs in optimal O(n ) time and space, where n is the number of the vertices of the given polygon. Later their result was generalized to visibility polygons from an edge by EIGindy [Eli, and Lee and Lin ILL I. Both independently discovered O(nlogn) algorithms for this problem. Very recently Guibas e|. ai. [GH] have proposed an optimal 0 ( n ) time algorithm. None of these algorithms work for polygons with holes. In this paper we consider the problem of computing visibility polygon.q inside a polygon P that may have holes. Our first result is an algorithm for computing the visibility polygon from a given point inside P . The algorithm runs in 0 (nlogn) time, which is proved to be optimal by reduction from the problem of sorting n positive integers (Aaano st. al, [AA] have obtained this result independently). Next we consider the problem of determining the visibility polygon from a line segment. As our main result, we establish a worst-ease lower bound of N(n 4) for explicitly computing the boundary of the visibility polygon from a line segment in the presence of other line segments, and design an optimal algorithm to construct the boundary. We also present an 0 (n 2) time and space algorithm if the visibility polygon can be represented as a union of several polygons. The latter algorithm is also proved to be optimal in the worst case.

On Polygonal Chain Approximation

Imai and Iri recently described a clever algorithm for approximating a polygonal chain within a given tolerance. Their algorithm requires O ( n 3 ) time in the worst case. In this note it is shown that their algorithm can be improved to O ( n 2 log n ) by exploiting the geometrical constraints of the problem.

An optimal algorithm for finding minimal enclosing triangles

Abstract Klee and Laskowskis O ( n log 2 n ) algorithm for finding all minimal area triangles enclosing a given convex polygon of n vertices is improved to Θ ( n ), which is shown to be optimal both for finding all minima and for finding just one.

A spherical representation of a human body for visualizing movement

A three-dimensional human body model for displaying body movements on a computer graphics display is described. The surface of the body model is formed from overlapping spheres, yielding a realistically formed and shaded body image on a raster graphics display. An experimental model consisting of 310 spheres is articulated with 19 joints and 20 body segments. The properties of this model include joints which do not deform during movement, simple hidden surface removal, and efficient collision and contact detection. The model may also be placed in a planar polygon environment, displayed in shaded form, and tested for collisions with the environment. Applications in crash simulation and human movement simulation are indicated.

Folding and Unfolding in Computational Geometry

Three open problems on folding/unfolding are discussed: (1) Can every convex polyhedron be cut along edges and unfolded at to a single nonoverlapping piece? (2) Given gluing instructions for a polygon, construct the unique 3D convex polyhedron to which it folds. (3) Can every planar polygonal chain be straightened?

Connect-the-dots: a new heuristic

The problem considered in this paper is that of finding a simple polygon through a given set of points in the plane that is “natural” in some perceptual sense. We propose that a particular geometric object called the minimal spanning Voronoi tree captures the essence of the problem. Despite the fact that we can neither prove the existence of this geometric object nor design an exact algorithm for finding it, a search heuristic results in remarkably pleasing solutions to the problem.