We show a worst-case lower bound and a smoothed upper bound on the number of iterations performed by the Iterative Closest Point (ICP) algorithm. First proposed by Besl and McKay, the algorithm is widely used in computational geometry where it is known for its simplicity and its observed speed. The theoretical study of ICP was initiated by Ezra, Sharir and Efrat, who bounded its worst-case running time between Ω(n log n) and O(n 2 d) d. We substantially tighten this gap by improving the lower bound to Ω(n/d) d+1. To help reconcile this bound with the algorithm’s observed speed, we also show the smoothed complexity of ICP is polynomial, independent of the dimensionality of the data. Using similar methods, we improve the best known smoothed upper bound for the popular k-means method to n O(k) , once again independent of the dimension. 1.

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Registration is the problem of bringing together two or more 3D shapes, either of the same object or of two different but similar objects. This chapter first introduces the classical Iterative Closest Point (ICP) algorithm which represents the gold standard registration method. Current limitations of ICP are addressed and the most popular variants of ICP are described to improve the basic implementation in several ways. Challenging registration scenarios are analyzed and a taxonomy of recent and promising alternative registration techniques is introduced. Three case studies are then described with an increasing level of difficulty. The first case study describes a simple but effective technique to detect outliers. The second case study uses the Levenberg-Marquardt optimization procedure to solve standard pairwise registration. The third case study focuses on the challenging problem of deformable object registration. Finally, open issues and directions for future work are discussed and conclusions are drawn. 1

Given the start positions of a group of dancers, a choreographer specifies their end positions and says: “Run!” Each dancer has the choice of his/her motion. These choices influence the perceived beauty (or grace) of the overall choreography. We report experiments with an automatic approach, SAMBA, that computes a pleasing choreography. Rossignac and Vinacua focused on affine motions, which, in the plane, correspond to choreographies for three independent dancers. They proposed the inverse of the Average Relative Acceleration (ARA) as a measure of grace and their Steady Affine Morph (SAM) as the most graceful motion. Here, we extend their approach to larger groups. We start with a discretized (uniformly time-sampled) choreography, where each dancers moves with constant speed. Each SAMBA iteration steadies the choreography by tweaking the positions of dancers at all intermediate frames towards corresponding predicted positions. The prediction for the position of dancer at a given frame is computed by using a novel combination of a distance weighted, least-squares registration between a previous and a subsequent frame and of a modified SAM interpolation. SAMBA is fully automatic, converges in a fraction of a second, and produces pleasing and interesting motions.

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