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Dynamic Programming Methods for“Connect the Dots” in Scattered Point Clouds

By Xiaoming Huo, Craig Tovey, David L. Donoho and Ery Arias-castro

Abstract

Connect-The-Dots (CTD) is our name for a class of geometric optimization problems in which, given a point cloud, one finds the curve f from a given class F which passes through the maximum possible number of points in the cloud. Examples in 2D include monotone increasing graphs, curves of bounded curvature or length, and so on. In higher dimen-sions we may generalize graphs/curves to surfaces of fixed co-dimension, or bounded surface area. Other generalizations include replacing the “passing through points ” condition by “passing through points in specified directions. ” Potential applications arise in, for exam-ple, image processing (tracking amid clutter) and understanding human vision (perceptual psychophysics), and many more. This report describes a family of dynamic programming algorithms for various CTD problems. Some are their first appearance: e.g., the algorithms for the Hölder-2 functions and the nondecreasing function in high dimension. A software library—CTDLab—that implements many of these algorithms is available on the Internet. Simulations that are based on these tools give insights and new conjectures regarding the asymptotic behavior of solutions when the point (or vector) clouds are i.i.d. uniform

Topics: creasing Subsequence. Limit Distribution. Asymptotic Rate. Hölder-2 function
Year: 2009
OAI identifier: oai:CiteSeerX.psu:10.1.1.135.7409
Provided by: CiteSeerX
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