Interpolation Search for Point Cloud Intersection

We present a novel algorithm to compute intersections of two point clouds. It can be used to detect collisions between implicit surfaces defined by two point sets, or to construct their intersection curves. Our approach utilizes a proximity graph that allows for quick interpolation search of a common zero of the two implicit functions. First, pairs of points from one point set are constructed, bracketing the intersection with the other surface. Second, an interpolation search along shortest paths in the graph is performed. Third, the solutions are refined. For the first and third step, randomized sampling is utilized. We show that the number of evaluations of the implicit function and the overall runtime is in O(loglogN) in the average case, where N is the point cloud size. The storage is bounded by O(N). Our measurements show that we achieve a speedup by an order of magnitude compared to a recently proposed randomized sampling technique for point cloud collision detection.

[1]  N. Amenta,et al.  Defining point-set surfaces , 2004, SIGGRAPH 2004.

[2]  Philip Dutré,et al.  Interactive boolean operations on surfel-bounded solids , 2003, ACM Trans. Graph..

[3]  Marc Levoy,et al.  QSplat: a multiresolution point rendering system for large meshes , 2000, SIGGRAPH.

[4]  Hiroaki Yamamoto,et al.  Stochastic algorithm for detecting intersection of implicit surfaces , 2000, Comput. Graph..

[5]  Gabriel Zachmann,et al.  Minimal hierarchical collision detection , 2002, VRST '02.

[6]  William H. Press,et al.  Numerical recipes in C , 2002 .

[7]  Marc Alexa,et al.  Approximating Bounded, Non-Orientable Surfaces from Points (Figures 5, 6, and 7) , 2004, SMI.

[8]  D. H. McLain,et al.  Drawing Contours from Arbitrary Data Points , 1974, Comput. J..

[9]  Vic Barnett,et al.  Outliers in Statistical Data , 1980 .

[10]  Matthias Zwicker,et al.  Surfels: surface elements as rendering primitives , 2000, SIGGRAPH.

[11]  A. Adamson,et al.  Approximating bounded, nonorientable surfaces from points , 2004, Proceedings Shape Modeling Applications, 2004..

[12]  Gabriel Zachmann,et al.  Point Cloud Collision Detection , 2004, Comput. Graph. Forum.

[13]  C. A. Rogers.  Covering a sphere with spheres , 1963 .

[14]  Marc Alexa,et al.  Approximating and Intersecting Surfaces from Points , 2003, Symposium on Geometry Processing.

[15]  Markus H. Gross,et al.  Contact Handling for Deformable Point-Based Objects , 2004, VMV.

[16]  Gabriel Zachmann,et al.  Proximity Graphs for Defining Surfaces over Point Clouds , 2004, PBG.

[17]  Marc Alexa,et al.  On Normals and Projection Operators for Surfaces Defined by Point Sets , 2004, PBG.

[18]  Bryan L. Shader,et al.  Sphere-of-influence graphs using the sup-norm , 2000 .

[19]  T. S. Michael,et al.  Sphere of Influence Graphs and the L-metric , 2003, Discret. Appl. Math..

[20]  Tony DeRose,et al.  Surface reconstruction from unorganized points , 1992, SIGGRAPH.

[21]  Rajeev Motwani,et al.  Randomized Algorithms , 1995, SIGA.

[22]  W..  Relative Neighborhood Graphs and Their Relatives , 2004 .