Nonrobustness refers to qualitative or catastrophic failures in geometric algorithms arising from numerical errors. Section...

We propose a strategy to decompose a polygon, containing zero or more holes, into “approximately convex” pieces. For many applications, the approximately convex components of this decomposition provide similar benefits as convex components, while the resulting decomposition is significantly smaller and can be computed more efficiently. Moreover, our approximate convex decomposition (ACD) provides a mechanism to focus on key structural features and ignore less significant artifacts such as wrinkles and surface texture. We propose a simple algorithm that computes an ACD of a polygon by iteratively removing (resolving) the most significant non-convex feature (notch). As a by product, it produces an elegant hierarchical representation that provides a series of ‘increasingly convex ’ decompositions. A user specified tolerance determines the degree of concavity that will be allowed in the lowest level of the hierarchy. Our algorithm computes an ACD of a simple polygon with n vertices and r notches in O(nr) time. In contrast, exact convex decomposition is NP-hard or, if the polygon has no holes, takes O(nr 2) time. Models and movies can be found on our web-pages at:

Abstract. We introduce a novel approach to video compression which is suitable for traditional outline-based cartoon animations. In this case the dynamic foreground consists of several homogeneous regions and the background is static textural image. For this drawing style we show how to recover hybrid representation where the background is stored as a single bitmap and the foreground as a sequence of vector images. This allows us to preserve compelling visual quality as well as spatial scalability even for low encoding bit-rates. We also introduce an efficient approach to play back compressed animations in real-time on commodity graphics hardware. Practical results confirm that for the same storage requirements our framework provides better visual quality as compared to standard video compression techniques. 1

We study an engineering approach to computing Voronoi diagrams of points and line segments in the two-dimensional Euclidean space. Our Voronoi code, named vroni, uses standard oating-point arithmetic. It is based on Sugihara and Iri's topology-oriented approach, a very careful implementation of the numerical computations required, an automatic relaxation of epsilon thresholds, and on a multi-level recovery process combined with \desperate mode". Vroni was tested extensively on real-world data and turned out to be reliable. CPU-time statistics document that it is always faster than other popular Voronoi codes. 1 Introduction In a recent editorial, Fortune [2] wrote that \it is notoriously dicult to obtain a practical implementation of an abstractly described geometric algorithm". According to the author's personal experience this remark is particularly true for the implementation of Voronoi diagrams of line segments. This paper discusses the design and implementation of a reliable an...

We present a novel shape completion technique for creating temporally coherent watertight surfaces from real-time captured dynamic performances. Because of occlusions and low surface albedo, scanned mesh sequences typically exhibit large holes that persist over extended periods of time. Most conventional dynamic shape reconstruction techniques rely on template models or assume slow deformations in the input data. Our framework sidesteps these requirements and directly initializes shape completion with topology derived from the visual hull. To seal the holes with patches that are consistent with the subject’s motion, we first minimize surface bending energies in each frame to ensure smooth transitions across hole boundaries. Temporally coherent dynamics of surface patches are obtained by unwarping all frames within a time window using accurate inter-frame correspondences. Aggregated surface samples are then filtered with a temporal visibility kernel that maximizes the use of non-occluded surfaces. A key benefit of our shape completion strategy is that it does not rely on long-range correspondences or a template model. Consequently, our method does not suffer from error accumulation typically introduced by noise, large deformations, and drastic topological changes. We illustrate the effectiveness of our method on several high-resolution scans of human performances captured with a state-of-the-art multi-view 3D acquisition system.

Geometric computations are essential in many real-world problems. One impor-tant issue in geometric computations is that the geometric models in these problems can be so large that computations on them have infeasible storage or computation time requirements. Decomposition is a technique commonly used to partition complex models into simpler components. Whereas decomposition into convex components re-sults 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 work, we have developed an approximate technique, called Approximate Convex Decomposition (ACD), which decomposes a given polygon or polyhedron into “ap-proximately 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 ACD can represent the important structural features of the model more accurately by providing a mechanism for ignoring less significant features, such as wrinkles and surface texture. Our study of a wide range of applications shows that in addition to providing computational efficiency, ACD also provides natural multi-resolution or hi-erarchical representations. In this dissertation, we provide some examples of ACD’s many potential applications, such as particle simulation, mesh generation, motion planning, and skeleton extraction.

Real 3D data acquired from scanning technology provide interesting 3D models for research and industrial applications. However before these models can be used, a surface needs to be fitted to a point cloud of an unknown object, this process might create some undesirable properties, such as triangle normals pointing in incorrect directions. We present a robust algorithm that reliably fixes these triangle normal problems on nonmanifold, and self-interesting surfaces of scanned objects.