The convex hull of a set of points is the smallest convex set that contains the points. This article presents a practical convex hull algorithm that combines the two-dimensional Quickhull Algorithm with the general-dimension Beneath-Beyond Algorithm. It is similar to the randomized, incremental algorithms for convex hull and Delaunay triangulation. We provide empirical evidence that the algorithm runs faster when the input contains nonextreme points and that it uses less memory. Computational geometry algorithms have traditionally assumed that input sets are well behaved. When an algorithm is implemented with floating-point arithmetic, this assumption can lead to serious errors. We briefly describe a solution to this problem when computing the convex hull in two, three, or four dimensions. The output is a set of "thick" facets that contain all possible exact convex hulls of the input. A variation is effective in five or more dimensions.

Measuring the similarity between 3D shapes is a fundamental problem, with applications in computer graphics, computer vision, molecular biology, and a variety of other fields. A challenging aspect of this problem is to find a suitable shape signature that can be constructed and compared quickly, while still discriminating between similar and dissimilar shapes.In this paper, we propose and analyze a method for computing shape signatures for arbitrary (possibly degenerate) 3D polygonal models. The key idea is to represent the signature of an object as a shape distribution sampled from a shape function measuring global geometric properties of an object. The primary motivation for this approach is to reduce the shape matching problem to the comparison of probability distributions, which is simpler than traditional shape matching methods that require pose registration, feature correspondence, or model fitting.We find that the dissimilarities between sampled distributions of simple shape functions (e.g., the distance between two random points on a surface) provide a robust method for discriminating between classes of objects (e.g., cars versus airplanes) in a moderately sized database, despite the presence of arbitrary translations, rotations, scales, mirrors, tessellations, simplifications, and model degeneracies. They can be evaluated quickly, and thus the proposed method could be applied as a pre-classifier in a complete shape-based retrieval or analysis system concerned with finding similar whole objects. The paper describes our early experiences using shape distributions for object classification and for interactive web-based retrieval of 3D models.

As the number of 3D models available on the Web grows, there is an increasing need for a search engine to help people find them. Unfortunately, traditional text-based search techniques are not always effective for 3D data. In this article, we investigate new shape-based search methods. The key challenges are to develop query methods simple enough for novice users and matching algorithms robust enough to work for arbitrary polygonal models. We present a Web-based search engine system that supports queries based on 3D sketches, 2D sketches, 3D models, and/or text keywords. For the shape-based queries, we have developed a new matching algorithm that uses spherical harmonics to compute discriminating similarity measures without requiring repair of model degeneracies or alignment of orientations. It provides 46 to 245% better performance than related shape-matching methods during precision--recall experiments, and it is fast enough to return query results from a repository of 20,000 models in under a second. The net result is a growing interactive index of 3D models available on the Web (i.e., a Google for 3D models).

In this paper we give a simple algorithm for constructing sparse spanners for arbitrary weighted graphs. We then apply this algorithm to obtain specific results for planar graphs and Euclidean graphs. We discuss the optimality of our results and present several nearly matching lower bounds.

Measuring the similarity between 3D shapes is a fundamental problem, with applications in computer vision

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We present a new method for user controlled morphing of two homeomorphic triangle meshes of arbitrary topology. In particular we focus on the problem of establishing a correspondence map between source and target meshes. Our method employs the MAPS algorithm to parameterize both meshes over simple base domains and an additional harmonic map bringing the latter into correspondence. To control the mapping the user specifies any number of feature pairs, which control the parameterizations produced by the MAPS algorithm. Additional controls are provided through a direct manipulation interface allowing the user to tune the mapping between the base domains. We give several examples of aesthetically pleasing morphs which can be created in this manner with little user input. Additionally we demonstrate examples of temporal and spatial control over the morph.