Automatically Approximating 3D Points with Co-Axisal Objects

In this paper, we investigate the problem of approximating a set S of 3D points with co-axisal objects typically from CAD/CAM (namely, cylindrical segments, cones and conical frustums). The objective is to minimize the sum of volumes of these objects (as well as the number of objects used). The general problem when the objects can have arbitrary axes is strongly NP-hard as a cylindrical segment, a cone and a conical frustum can all degenerate into a line segment. We present a general algorithm which combines a neat doubling search method to decompose S into desired subsets (or components). For each subset S, we present a unified practical approximation algorithm for minimizing the volume of the cone (conical frustum, or cylindrical segment) which encloses points in S. Preliminary empirical results indicate that the algorithm is in fact very accurate.

[1]  Sariel Har-Peled,et al.  Efficiently approximating the minimum-volume bounding box of a point set in three dimensions , 1999, SODA '99.

[2]  Binhai Zhu.  Approximating 3D Points with Cylindrical Segments , 2002, COCOON.

[3]  Franco P. Preparata,et al.  Sequencing-by-hybridization revisited: the analog-spectrum proposal , 2004, IEEE/ACM Transactions on Computational Biology and Bioinformatics.

[4]  Binhai Zhu,et al.  Approximating Convex Polyhedra with Axis-Parallel Boxes , 1997, Int. J. Comput. Geom. Appl..

[5]  G. Jacobs,et al.  Neural Mapping of Direction and Frequency in the Cricket Cercal Sensory System , 1999, The Journal of Neuroscience.

[6]  Timothy M. Chan.  Optimal output-sensitive convex hull algorithms in two and three dimensions , 1996, Discret. Comput. Geom..

[7]  R. K. Shyamasundar,et al.  Introduction to algorithms , 1996 .

[8]  F. Theunissen,et al.  Extraction of Sensory Parameters from a Neural Map by Primary Sensory Interneurons , 2000, The Journal of Neuroscience.

[9]  Timothy M. Chan.  Approximating the Diameter, Width, Smallest Enclosing Cylinder, and Minimum-Width Annulus , 2002, Int. J. Comput. Geom. Appl..

[10]  Claire Mathieu,et al.  Verifying partial orders , 1989, STOC '89.

[11]  Gershon Elber,et al.  Efficient continuous collision detection for bounding boxes under rational motion , 2006, Proceedings 2006 IEEE International Conference on Robotics and Automation, 2006. ICRA 2006..

[12]  Marco Attene,et al.  Hierarchical mesh segmentation based on fitting primitives , 2006, The Visual Computer.

[13]  Michael Ian Shamos,et al.  Computational geometry: an introduction , 1985 .

[14]  Sergey Bereg,et al.  Cylindrical hierarchy for deforming necklaces , 2004, Int. J. Comput. Geom. Appl..

[15]  Subhash Suri,et al.  Analysis of a bounding box heuristic for object intersection , 1999, SODA '99.

[16]  Micha Sharir,et al.  Line Transversals of Balls and Smallest Enclosing Cylinders in Three Dimensions , 1997, SODA '97.

[17]  A. Björklund,et al.  Transplantation of embryonic ventral forebrain neurons to the neocortex of rats with lesions of nucleus basalis magnocellularis—I. Biochemical and anatomical observations , 1985, Neuroscience.

[18]  Marek Teichmann,et al.  Smallest enclosing cylinders , 1996, SCG '96.

[19]  Leif Kobbelt,et al.  Structure Recovery via Hybrid Variational Surface Approximation , 2005, Comput. Graph. Forum.

[20]  Binhai Zhu,et al.  Cylindrical Approximation of a Neuron from Reconstructed Polyhedron , 2004, ICCSA.

[21]  F. Theunissen,et al.  Functional organization of a neural map in the cricket cercal sensory system , 1996, The Journal of neuroscience : the official journal of the Society for Neuroscience.

[22]  Ronald L. Rivest,et al.  Introduction to Algorithms, Second Edition , 2001 .

[23]  Timothy M. Chan.  Approximating the diameter, width, smallest enclosing cylinder, and minimum-width annulus , 2000, SCG '00.

[24]  Ralph R. Martin,et al.  Reverse engineering of geometric models - an introduction , 1997, Comput. Aided Des..